Df1> IN MEMOR1AM FLOR1AN CAJORI Digitized by the Internet Archive in 2007 with funding from Microsoft Corporation http://www.archive.org/details/elementaryalgebrOOIymarich ELEMENTARY ALGEBRA BY ELMER A. LYMAN Professor of Mathematics in the Michigan State Normal College Ypsilanti, Michigan AND ALBERTUS DARNELL Head of Department of Mathematics, Central High School Detroit, Michigan AMERICAN BOOK COMPANY NEW YORK CINCINNATI CHICAGO COPYRIGHT, 1917, BY ELMEE A. LYMAN and ALBERTUS DARNELL All rights reserved l. and d. el. alg. W. P. I PREFACE The object of this book is to provide a complete course in Elementary Algebra that will satisfy the requirements of courses of study in various states and of the College Entrance Board. Vitality has been given to the subject by imbuing it with the interest that accrues from connection with problems of everyday life and by careful correlation with arithmetic. The utility of algebra is emphasized from the start by showing how much easier it is to solve certain problems by algebra than by arithmetic. Simplicity is the keynote of the book. This effect is gained by omitting exercises of undue difficulty as well as trouble- some phases of the subject that are not essential. A careful development of each new principle anticipates difficulties ; and abundant illustrations and examples give further emphasis to the point at issue. The easy oral exercises assist in developing and fixing in mind the principles and processes. The written exercises are very abundant and range from the simplest type to some of sufficient difficulty to test the pupil's power and to provide adequate drill. Many of the exercises are taken from entrance examination questions set by various colleges and universities, the source being indicated in all such cases. The problems are practical and varied. They include appli- cations to geometry, physics, engineering, agriculture and com- merce, and various interests of everyday life. iii M306039 iv Preface The importance of the equation is recognized throughout by- abundance of practice. Thoroughness and accuracy are secured, first, by many re- views, and second, by the emphasis placed on the checking of results. Exercises designed to encourage an intelligent translation of algebraic language are provided early in the book. The appli- cation of algebraic principles to solutions by formulas has also been emphasized. Graphical representation is introduced in two chapters, but is so arranged that it may be omitted at the option of the teacher without interrupting the sequence of the work. ELMER A. LYMAN. ALBERTUS DARNELL. CONTENTS PAGB I. Introduction ... * 1 II. Positive and Negative Numbers 15 Addition of Signed Numbers 20 Subtraction of Signed Numbers 22 Multiplication of Signed Numbers . . . . .25 Division of Signed Numbers 27 III. Addition 32 Addition of Like Monomials 36 Addition of Unlike Monomials 40 Addition of Polynomials 42 IV. Subtraction 50 Subtraction of Like Monomials 50 Subtraction of Unlike Monomials 52 Subtraction of Polynomials 54 Parentheses 57 V. Multiplication 67 Multiplication of Monomials ...... 69 Multiplication of a Polynomial by a Monomial . . 71 Multiplication of a Polynomial by a Polynomial . . 74 Type Forms in Multiplication 78 VI. Division 92 Division of Monomials . 94 Division of a Polynomial by a Monomial ... 96 Division of a Polynomial by a Polynomial ... 99 VII. Simple Equations 104 The Solution of Problems 116 Rules and Formulas . . . . . . . 123 v vi Contents PA0B VIII. Factoring . . .132 The Solution of Equations by Factoring . . .156 IX. Highest Common Factor and Lowest Common Multiple . 161 Highest Common Factor 161 Lowest Common Multiple 165 X. Fractions . . 167 Reduction of Fractions 168 Addition and Subtraction of Fractions . . . .179 Multiplication of Fractions 186 Division of Fractions 190 Complex Fractions 194 XL Equations Containing Fractions 197 Review of Fractions and Fractional Equations . . 214 XII. Ratio and Proportion 218 Properties of Proportions 222 XIII. Graphs 237 XIV. Linear Simultaneous Equations 250 XV. Square Root . 279 Square Root of Polynomials 280 Square Root of Arithmetical Numbers .... 283 XVI. Quadratic Equations . . 290 Incomplete Quadratic Equations . . . . . 291 Complete Quadratic Equations 294 XVII. Simultaneous Equations Involving Quadratics . . 305 XVIII. Exponents 315 XIX. Radicals .... 331 Reduction of Radicals 333 Addition and Subtraction of Radicals .... 341 Multiplication and Division of Radical Expressions . 344 Rationalizing Factors 849 Division of Polynomial Radicals 350 Rationalizing Denominators 352 Involution and Evolution of Radicals .... 365 Contents vii PAGE XX. Radical Equations * . . .362 XXI. Imaginary Numbers 367 XXII. Quadratic Equations • 373 Complete Quadratics 374 Theory of Quadratic Equations 384 Equations in the Form of Quadratics . . . 388 XXIII. Simultaneous Quadratic Equations 400 Elimination 412 XXIV. Graphical Solution of Equations 418 XXV. The Progressions 427 Arithmetical Progression 427 Geometrical Progression 434 Infinite Geometrical Series — Repeating Decimals . 437 XXVI. The Binomial Formula 442 XXVII. Variation 448 XXVIII. Logarithms 454 XXIX. General Review 469 College Entrance Examinations 490 Appendix 494 Remainder Theorem 494 Factor Theorem 495 Synthetic Division 496 Index 499 ELEMENTARY ALGEBRA L INTRODUCTION 1. Symbols Representing Numbers. Algebra, like arithmetic, treats of numbers ; but in arithmetic, numbers are usually represented by means of Arabic numerals, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, while in algebra they are represented also by letters, as a, b y c, . . . ; . . x, y, z. Numbers represented by letters are called literal numbers. The student has seen how the rules of arithmetic can be abbreviated by the use of letters to represent numbers. When a rule is expressed by means of letters, the result is a formula. The rule, "the number of square units in the area, A, of a rectangle is equal to the product of the number of units in the length, /, and the number of units in the width, w," may be expressed by means of the formula : A = I X w. w 1. Find the area of a rectangle whose Jength A=l xw I is 6 inches and whose width is 4 inches. Rectangle Substitute 6 for I and 4 for w in the formula A = I x w. Thus, J. = Zxto = 6x4=24, the number of square, inches in the area. 2. Find the area of a rectangle whose length is 12 inches and whose width is 9 inches. 3. Explain the use of the following formulas : Area of a triangle, i=|x ax b, where b is the number of units in the base and a is the number of units in the altitude. Triangle Circumference of a circle, C = 2 x irx /? (tt = 3.1416). Interest on money invested, / = p X r x t. Introduction 2. Symbols Representing Operations. The following table shows that the symbols of operation used in algebra are the same, with few additions, as those used in arithmetic : Plus Minus Times Divided by 3+2 a + b 3-2 a — b 3x2 a x b, a • b, ab 4-2, -,4:2 1 2' a -r- 6, - , a : b b The sign for equality, = , is used as in arithmetic. Notice that while with arithmetical numbers multiplication is indicated by the sign x , as 3 x 2, or 3 x 5 X 7, with literal numbers the sign is usually omitted. Thus, 3 a, which is read three a, means 3 x a, and 3 ab, which is read three ab, means 3 x a x b. 3 a + 5 means that 3 times a is to be increased by 5, and is read three a plus Jive. What does 3 a — 5 mean ? If a = 2, what is the value of 3a? of 3a + 5? of3a-5? ORAL EXERCISE 3. 1. What is meant by 3a? by 5 b? 2. What is meant by 3 # + 4 ? by 3 a; + 5y? 3. If x = 2, what is the value of 5 x ? 4. If a = 2, and 6 = 4, what is the value of 4-^a? of 6-s- a? of Sab? 5. What is meant by 5 ab ? 6 xyz ? 2 mnp + rs? 2a-36? 6. What is meant by ab -s- c ? mnp h- r? xy -^ s? 7. When x = 1, and y = 2, find the value of 3 xy ; of x + ?/ ; of 4 a; — 2/ ; of y -7- x. 8. Head 3z + 7 = 12. 9. Read m -i- n = 5 xy. In what other way can the same statement be written ? Introduction 3 10. Read 3 x = x -f- x -f x. 11. Read2a6c + 6; 2ttE; I = prt. 12. Read 3 a; — 1=5; 2x — 5y = ab. 13. Reada + 6 — a6+^- 14. What operation of arithmetic is suggested by each of the following words : sum ? quotient ? product ? difference ? EXERCISE 4. Write, using proper algebraic symbols : 1. The sum of 2 times a increased by 6. 2. The sum of 7 a and 2 b. 3. a times b times c. 4. Two times x diminished by c. 5. The sum of a and two times b. 6. Indicate that two times some number, x, increased by 5 is equal to 25. 7. Indicate the product of the factors 3, a, 6, c. 8. Indicate the sum of 3 times b and a times x. 9. Indicate the product of r and s divided by t. 10. Indicate the quotient of d and I increased by / times g. 11. What does 2 ab + 3 equal when a = 3 and 6 = 4? 12. Find the value of 3 -f 5 a; when x = 2. 13. Find the value of 4 + 5 d when d = 12. 14. Find the value of 5 a -\ h ^ when a = 2. a 2 3 3 15. Find the value of - + 76 when a = - and 6 = 2. a 2 16. If t stands for tens and h for hundreds, what number does 6 ft + 7 £ + 4 represent ? 17. If y stands for yards, / for feet, and i for inches, how many inches does 14 y + 11 / + 5 i represent ? How many inches does 19 y — 16 f + 2 i represent ? Introduction 18. The side of a square is a inches. What will represent the distance around it? 19. If the length of a rectangle is I in- ches and its width w inches, how can you express its area? State this formula for finding the area of a rectangle as a rule. Express the distance around the rectangle. State this result in the form of a rule. 20. The sides of a triangle are x inches, 2 x inches, and y inches. What is the perimeter, that is, the distance around the triangle ? 21. If a, 6, and c represent the units', tens', and hundreds' digits of a number, how may the number itself be represented ? Suggestion. 543 = 100 • 5 + 10 . 4 + 3. 22. Find the value of 2 n when n = 1 ; 2 ; 3 ; 4 ; 5. Does 2 n always represent an even number when n is any integer ? 23. When n is an integer, does 2 n — 2 represent an even or an odd number ? 2 n + 2 ? 24. Do 2 n + 1 and 2 n — 1 represent odd or even numbers when n is any integer ? 25. Does 2a4-3a = 5a when a = 2 ? when a = 3 ? when a = any number ? 26. Does 5 a — a = 4 a when a = 5 ? when a = 7 ? when a = any number ? 27. Does 2x36 = 66 when 6 = 4? when 6 = 10 ? when 6 = any number ? 28. Does 10 a -T- 5 = 2 a when a = 1 ? when a = 8 ? when a = any number ? 29. What number multiplied by 5 equals 25? What num- ber multiplied by 7 equals 35 ? If 5 m = 35, what is the value of m ? If 9 h = 72, what is the value of h? Introduction 5 30. In arithmetic, to find the percentage when the base and the rate are given, we multiply the base by the rate. Express this rule by means of a formula when b, r, and p represent the base, the rate, and the percentage respectively. 5. Factor. If two or more numbers are multiplied together, a product is formed and the numbers are factors of the product. Thus, 7 xy is the product of 7,2, and y\ and 7, x, and y are the factors of the product. 6. Exponent. To indicate that the number a has been used as a factor twice in forming the product ax a, we write a 2 instead of a x a ; for a • a-a we write a 3 . These are read a square, and a cube, respectively. 4 a 3 means 4 • a • a • a and is read four a cube. The numbers 2 and 3 in a 2 and a 3 are ex- ponents. 7. Square Root and Cube Root. The sign V indicates the square root of a number, that is, one of the two equal factors of a number. Thus, Vl6 = 4 ; Vo* = a. The sign -fy~ indicates the cube root of a number, that is, one of its three equal factors. Thus, \/27 = 3; Vx? = x. 8. Symbols of Deduction. In a series of steps one of which is derived from another the symbols of deduction, .-. and v, are used. These symbols are read therefore and since respectively. ORAL EXERCISE 9. Read the following : 1. 5a 2 + 7ab + 2. 5. 3x = x + x + x. 2. 7 • a + a 2 — 2. 6. ax 2 +bx+ c. 3. 3z 2 +3a;-±2. 7. Va+VHa 2 + & 2 . 4. 3x+7 = 12. 8. -.-2.4 = 8, .-.8-5-2 = 4. 6 Introduction 9. Read v 12 + 3 = 15, .-. 15 - 3 = 12. 10. What are the factors of ab? of Sx'h?'? of 5mnp? of 9aW? 11. Express the square of a; the cube of p. 12. Express the square root of a ; of 2 m. 13. Express by using exponents 2-2-2; a x a x a; 6- 6 -c-c; p'p*p'q»q. 14. If the side of a square is a, what is its area ? 15. If the edge of a cube is a inches, express the formula for finding its volume. State the formula for finding the area of its surface. State these two formulas as rules. 10. Some Simple Operations. From the method of writing algebraic numbers we are justified in performing the following simple operations : (a) 2 x -+- 3 x = 5 x f or all values of x. For if two times a number is increased by three times that number, the result is five times the number. Compare this with 2x4 + 3x4 = 5x4. Similarly, 5x — Sx = 2x, and 7 x — x = 6 x. (Note that x is the same as 1 x.) (6) If 2 is subtracted from 5 x + 2, the result is 5 x. For if the sum of two numbers is diminished by one of them, the result is the other. (c) 2x36 = 2x3x6 = 6x6 or 6 6. This is similar to 2x3x4 = 6x4. 5-4?/ = 20?/. 6-3a = ? (d) 2 x -5- 2 = x, f or if the product of two numbers is divided by one of them, the quotient is the other. Also 16m-j-2=8m. ORAL EXERCISE 11. Perform the indicated operations: 1. 2x3o; 4 x5a; 3-26. 2. 2a-f-2; lOy-r-5; 8a-*-4; 16a-*-8j 32m-*-4. Introduction 7 3. Sab -T- a ; 12 xy -+- 4 y ; 18 />g -5- 3 j> ; 25 a&c -r- a&. 4. 4 a; -f- 2 diminished by 2 ; 7m + 3 diminished by 3. 5. Add 3 to 5x + 3. (5x+ 3 + 3 = 5x + 6.) Add 6 to 8p + 2. 6. What number subtracted from 3 x 4- 7" will give 3 a; ? 7. What must be done to the number 3 x + 5 to get 3 a;? 8. What must be done to 3 a; to get x ? 9. 2a; + 13a; = ? 3 x + x + 5 x = ? 10. 5a; — 4a; = ? 8a;- 2 a;- a; = ? 11. If 5 a; = 15, what does x equal ? 12. 7a; + 2a;-3a;=? 9a;- 3a;+ 5a;= ? 13. 10a;-5a; + 8-3 = ? 14. 10a; — 5a; + 8a-3a=? 15. 4aj + 7a; + 12-8=? 16. 4a; + 7a; + 126-8&=? 12. Equation. A statement expressing the equality of two numbers is called an equation. Thus, 2 x + 3 = 9 is an equation. The two equal numbers are the members of the equation. The number written at the left of the sign of equality is the first member, while the other number is the second member of the equation. 13. Unknown Number. A number in the equation whose value is to be found is the unknown number. Thus, in the equation, 2a5 + 3 = 9, 2# + 3is the first member and 9 is the second member, x is the unknown number. That value of the unknown number which, if substituted for it in the equation, will make the two members equal satis- fies the equation. Thus, if 3 is substituted for x in 2 x + 3 = 0, we have 2 • 3 + 3 = 9 or 6 + 3 = 9. Therefore 3 satisfies the equation. Does 2 satisfy 5 x + 1 = 11? Does 3 satisfy 5 x + 1 = 11 ? 8 Introduction 14. Solving Equations. Root. The process of finding the value of the unknown number that satisfies the equation is called solving the equation. The value of the unknown number that satisfies the equation is the root of the equation. Thus,2 is the root of 3 x + 1=7 because 3-2 + 1 = 7. 15. Principles used in Solving Equations : (a) If the same number is added to equal numbers, the resulting numbers are equal. (6) If the same number is subtracted from equal numbers, the result- ing numbers are equal. (c) If equal numbers are multiplied by the same number, the resulting numbers are equal. (d) If equal numbers are divided by the same number, the resulting numbers are equal. Note. Division by zero is not included in this last statement. 16. In the solution of 2 x -f- 3 = 9, the steps are as follows : 1. 2x + 3 = 9. 2. .-. 2x = 6. Subtract 3 from both members of the equation. See § 15, (6). 3. .-. x = 3. Divide both members by 2. See § 15, (d). 17. Check. The solution of an equation may be checked by putting the root obtained in the place of the unknown number in the equation. When* this is done, if the two members are equal, the solution is correct. Thus, to check the answer 3 in the solution of 2 x + 3 = 9, put 3 for x, then 2x3 + 3 = 9. Therefore the solution is correct. EXERCISE 18. Solve the following equations, explaining each step by the statement of the principle involved. Check each solution. 1. 3a; -+ 5 = 20. 3. 5x + 2 = 3. 5. 4 n + 2 = 6. ft. 2x 4- 8 = 13. 4. 3a; -I- 7 = 16. 6. 3 a; + 2 x + 8 = 23. Introduction 9 7. 2z + l = 4. 10. 3v + 6 = ll. 13. 6# + 7 = 13. 8. 5w + 2 = 52. - 11. 9a + l = 3. 14. z + l=»6. 9. ra + l=4. 12. 5 + 6 = 12. 15. 5z + 5=a+9. Solution. 1. 5aj + 5 = a; + 9. 2. .•. 4 a; + 6 = 9. (Subtracting x from each member.) 3. .*. 4aj = 4. (Subtracting 5 from each member.) 4. .-. x = 1. (Dividing each member by 4.) Check. When z = 1, 5 a; + 5 = 10, and a; + 9 = 10. Therefore the solution is correct. 16. 3x + 3 = x + 5. 19. 5a + 3a + 10 = 22. 17. 122/ + 3=7?/ + 18. 20. 8a + 2x+ 9 = 2a; + 20. 18. 12 y + 3 = 7i/ +17. 21. 7z + 16 = 2* + 3 z + 40. EXERCISE 19. Writing Algebraic Numbers and Making Equations. 1. If n stands for a number, what will stand for three times this number? 2. If n stands for a number, what will stand for the num- ber increased by 3 ? 3. If a; is an integer, what will stand for the next larger integer ? 4. If a room is / feet long, how many inches long is it ? 5. How would you express / feet and i inches in inches ? 6. Express p pounds and z ounces in ounces. 7. Express the result of multiplying a number a; by 3 and adding 2 to the product. 8. Indicate that two times some unknown number x in- creased by 5 is equal to 17. 9. Find the unknown number in example 8 : (a) by arith- metic ; (b) by algebra. 10. How can two unknown numbers be expressed if one is double the other ? 10 Introduction • 11. The sum of two numbers is 30, and one of them is twice as large as the other. Find the numbers by arithmetical analysis. Also make and solve an algebraic equation to find them. Suggestion, x and 2 x may represent the numbers. 12. The sum of two numbers is 45, and one of them exceeds the other by 5. What are the numbers ? Solve first by arith- metic, then by algebra. Notice how much easier it is to solve examples 9, 11, and 12 by algebra than by arithmetic. 13. Five times a certain number, increased by 2, is equal to the result obtained by multiplying the same number by 3 and adding 14 to the product. Find the number. Solution. Let x = the required number. Hence 5 x + 2 = the result of multiplying the number by 5, and adding 2 to the result, and Sx + 14 = . . . (Let the student complete the statement.) Then 5a; + 2 = 3x + 14. (By the conditions of the problem.) .-. 5x = Sx+ 12. (Why?) .-. 2 a =12. (Why?) .♦. x = 6. Therefore the required number is 6. Check. In checking the solution of this problem, it will not do to substitute 6 for x in the equation, for an error might have occurred in forming the equation. The answer should be substituted in the original problem. EXERCISE 20. Make and solve equations for the following problems. Check each result by seeing if it satisfies the conditions of the problem. 1. If a certain number is multiplied by 7 and the product is increased by 5, the result is equal to the original number increased by 83. Find the number. Solution. Let x = the number. Hence 7 x + 5 = 7 times the number increased by 5, and x + 83 = the number increased by 83. Then 7 x + 5 = x + 83. (By the conditions of the problem.) .-. 7z = x + 78. (Why?) Introduction 11 .-.6 x = 78. (Why?) .-. x = 13. (Why?) Therefore 13 is the required number. Check. 7 x 13 + 5 = 96, and 13 + 83 = 96. Therefore 13 is the num- ber required by the conditions of the problem. 2. If two times a certain number is increased by 6, the result is equal to the sum of the original number and 9. Find the number. 3. Find three numbers of which the second is double the first, and the third exceeds the first by 8, their sum being 44. 4. The sum of three numbers is 24. The second is double the first, and the third equals the sum of the other two. Find the numbers. 5. Two men have together $68. One of them has $2 more than twice as much as the other. How many dollars has each ? Solution. 1. Let x = the number of dollars one man has. Hence 2 x + 2 = the number of dollars the other has, and x 4- 2 x + 2 = the number of dollars both have. Then x -4- 2 x + 2 = 68. (By the conditions of the problem.) or3x + 2 = 68. (Why?) .-.3x=66. (Why?) .-.a = 22. (Why?) Therefore one man has §22 and the other man has 2 x $22 + $2, or $46. Check. Let the student check the problem. The student should notice that x was not used to represent one man's money, but the number of dollars he had. The dollar sign is not to be placed with any of the numbers in the equation. The equation is ex- pressed in abstract numbers. 6. The cost of a horse is two times the cost of a cow ; the cost of a cow is five times the cost of a sheep. Find the cost of each if a horse, a cow, and a sheep together cost $ 208. Suggestion. Let x = the number of dollars one sheep costs. 7. Divide $55 between A and B so that A shall have $5 more than four times as much as B. Suggestion. Let x = the number of dollars B has. 12 Introduction 8. The sum of the angles of a triangle is 180°. How many- degrees are there in each angle if the largest angle is three times as large as the smallest and the other is twice as large as the smallest ? Suggestion. Let x = number of degrees in the smallest 9. The sum of the lengths of the three sides of a triangle is 17 inches. The second side is two inches longer than the shortest, and the third is twice as long as the shortest. Find the lengths of the sides. Suggestion. Let x = the number of inches in the shortest side. 10. A piece of rope 106 inches long is to be cut into two parts so that one part shall be 10 inches more than twice as long as the other. How long will each part be ? 11. Henry is 5 years older than James, and the sum of their ages is 37. Find the age of each. 12. If f of a number is 72, what is the number ? Solution. Let x = the number. ■x = 72. x = 96. (Dividing both numbers of the equation by f .) 13. The sum of the ages of three boys is 38 years. The youngest is | of the age of the oldest and 3 years younger than the second. How old is each boy ? Suggestion. Let x = the number of years in the age of the oldest. .-. x + f x + f x + 3 = 38. Explain. 14. If an automobile after being reduced 25 % in price costs $ 900, what was its original cost ? Suggestion, x - \ x = 900. (Why '.') 15. A salesman earned $ 20 at 2 % commission. Find the amount of his sales. Suggestion. .02 x = 20. 16. A man bought the same number each of 1^, 2^, and 4P stamps for 70 £ How many of each kind did he buy ? Introduction 13 17. If a debt of $ 144 is paid by using the same number each of $ 1, $ 2, $ 5, and $ 10 bills, how many of each kind of bills is used ? 18. At an' election there were two candidates for the office of mayor. They together received 2360 votes. If one candi- date was defeated by 328 votes, how many votes did each receive ? 19. At an election there were three candidates A, B, and C for a certain office. They together received 3447 votes. If A received twice as many as B, and C 195 more than B, how many votes did each receive ? 20. If a field requires 36 pounds of nitrogen for fertilization, how much nitrate of soda containing 18 % of nitrogen will be needed ? 21. In an algebra class there are 24 pupils. If there are 6 more girls than boys in the class, how many boys are there? 22. A man buys twice as much hard coal as soft coal and pays $ 108. If hard coal is $ 7.50 a ton and soft coal is $ 3, how many tons of each does he buy ? 23. Two trains leave Buffalo at the same time going in opposite directions. One travels 50 miles an hour and the other 40 miles an hour. In how many hours will they be 630 miles apart ? 24. Two trains leave Buffalo at the same time going in the same direction. One travels 45 miles an hour and the other 38 miles. In how many hours will they be 35 miles apart ? 25. A merchant's profits for the second year increased 25 % over the first year's profits. If the total profits for the two years are $ 7623, how much are the profits for each year ? Solution. Let x = number of dollars profit the first year. Hence x + \ x = number of dollars profit the second year. Then x + x + |x = 7623, or fa = 7623. (Why?) . •. x = 3388, the number of dollars profit the first year. 14 Introduction 26. A workman's weekly expenses are f of his wages. How much does he earn each week if he has $ 5 left ? 27. Two pupils together solve 28 algebra problems. One of them solves J as many as the other. How many problems does each one solve ? 28. A rectangular field is f as wide as it is long and its perimeter is 40 rods. Find the length and the width. 29. Divide 90 into two such parts that one part equals twice the other. 30. A farmer raised 3000 bushels of corn, wheat, and oats. If he raised 3 times as much corn as wheat and twice as much oats as wheat, how many bushels of each did he raise ? 31. A farmer has 4 times as many hogs as cattle and twice as many sheep as hogs and cattle together. If he has 210 animals in all, how many of each kind has he ? 32. Three newsboys sold 140 papers. If the first sold i as many as the second and the third twice as many as the second, how many did each boy sell ? 33. A mason and his helper together earn $ 6 a day. If the helper earns i as much as the mason, how much does each receive ? 34. A baseball team won 12 games, which was f of the num- ber of games played. How many games were played ? 35. A boy bought a ball, a bat, and a glove for $ 2.50. The ball cost f as much as the glove and the bat -| as much as the ball. How much did each cost ? IL POSITIVE AND NEGATIVE NUMBERS 21. The first numbers with which we became acquainted were the whole numbers used in counting, such as 1, 2, 3. Later it was found necessary to enlarge our idea of numbers and include fractions, as ^, ^, -f, ^-. Still later it became necessary to express the value of the square roots and cube roots of numbers, as V2, V5. A still further extension of our number system will now be made, introducing negative numbers. 22. A thermometer scale is marked as in the figure. To indicate that the temperature is 10° below zero we write — 10°. To indicate that the temperature is 10° above zero we write + 10°, or simply 10°. 1. At noon on a certain day the temperature was 8° above zero. At night it had fallen 6°. What was the temperature at night ? Will the equation 8° — 6° = 2°, indicate the method of finding the answer ? 2. Suppose the temperature is 8° above zero at noon and falls 12° in the next six hours. What is the tem- perature at 6 o'clock ? The equation, 8° — 12° = — 4°, indicates the method of finding the answer. 3. If the temperature is 10° above zero in the morn- ing and rises 15° during the forenoon, what is the tem- perature at noon ? , 10° + 15° = 25°. 4. If the temperature is 10° below zero in the morning and rises 15° in the forenoon, what is the temperature at noon ? - 10° + 15° = 5°. 15 J0Q 90; - . [80 to; JO 60J ii° »] L«> 1£ 12 10J • 16 Positive and Negative Numbers ORAL EXERCISE 23. Explain and give the answers to the following : 1. 5° + 7° = ? 6. -3° + l° = ? 2. - 3° + 5° = ? 7. 8° - 5° = ? 3. -10° + 7° = ? 8. 10° -12° = ? 4. -8° + 8° = ? 9. 18° -30° = ? 5. 7° -9°=? 10. -5° -2°=? 24. An Extension of Subtraction. In arithmetic the subtra- hend must not be larger than the minuend. Such an opera- tion as 8 — 12 has no arithmetical meaning, for we cannot subtract from a number more units than the number contains. In algebra, however, we do subtract a larger number from a smaller number, and such subtractions ' give rise to negative numbers. Thus, 8- 12 = 8-8-4 = 0- 4, which we write - 4. Also, 5-6 = 5-5-1 = 0-1 or -1. 25. Positive and Negative Numbers. There are many pairs of opposite numbers similar to the numbers of the thermome- ter scale. The fact that numbers are so related to each other can be conveniently represented by the use of the signs + and — . When thus used to represent the quality of a number, these signs are read positive and negative respectively. Thus, + 5 is read positive Jive and — 7 is read negative seven. Num- bers preceded by the sign + to indicate the quality of the number are positive numbers ; numbers preceded by the sign — to indicate the quality of the number are negative num- bers. The student will note that each of the signs + and — may have two distinct uses; they may indicate the operations of addition and subtrac- tion, or they may indicate the quality of a number. 26. We usually omit the positive sign before positive num- bers, writing and reading them exactly as in arithmetic. Positive and Negative Numbers 17 Sometimes, however, for emphasis or for contrast, we write the sign + before a positive number, as (+ 5) or + 5. The negative sign before a negative number is never omitted. To show that these signs are quality signs, and not operation signs, we often write such numbers within a parenthesis, thus (— 3) + (+ 5), read negative 3 plus positive 5. ORAL EXERCISE 27. Read the following, using "positive" and "negative" as the names of these signs when they indicate quality. 1. (_3)+2 + (-3); -3 + 2 +(-3). 2. -3 + 5; (-3)+ 5; 5+(-3). 3.-7-4. 8. 23° +(-4°). 4. (_2)(-3) + 4. 9. 15^+(-5^). 5. (—a)+ &+(—«). 10. —$40 + $17. 6. x+(— y)+y. 11. — 2x— 3a+(— 2x). 7. ra — n+(— m)+a. 12. 5 +(— 7)- a(— b). 13. If we consider north positive, what should we consider south ? If rising temperature is positive, what kind of tem- perature is negative ? 14. What, signs would you associate with each of the fol- lowing: (1) Money earned and money spent? (2) A man's property and his debts ? (3) Distance up and distance down ? (4) Distance to the right and distance to the left ? 28. The Algebraic Number Scale. Draw a straight line and divide it into spaces of equal length. Select some point as zero near the center and name the other points of division as indicated. This arrangement of numbers on a line is the algebraic number scale. (See figure, page 18.) Just as the arithmetical number scale (that part of the alge- braic scale that extends from to the right) is conceived as extending indefinitely to the right, so the negative numbers of the algebraic scale extend from indefinitely to the left. 18 Positive and Negative Numbers 29. Algebraic Numbers. The positive and negative numbers together form the system of algebraic numbers, or signed num- bers. -9-8-7-6-5-4-3-2-1 1 2 3 I 5 6 1 8 \ I 1 1 1 1 1 1 1 1 1 1 1 1 I iii. 30. Addition of Signed Numbers on the Number Scale. 1. To add 3 and 5 on the number scale, begin at 3 and count 5 spaces to the right, arriving at the point 8. This gives the result 3 + 5 = 8. 2. To add (—3) and 5, begin at — 3 and count 5 spaces to the right, arriving at the point 2. .-. — 3 + 5 = 2. 3. Arithmetical numbers can be added in any order ; thus, 3 + 2 = 2 + 3. We shall assume that the order of adding algebraic numbers may be changed in the same way ; thus — 3 + 5=5+(— 3). This suggests that we may add a negative number by counting to the left on the number scale. To verify this begin at 5 and count 3 spaces to the left, arriving at the point 2. Similarly, 8 +(- 3)= 5 and 5 +(- 7) = - 2. Why? 31. These considerations justify the following rules for adding on the number scale : 1. To add any positive number, b, to any number, n, begin at n and count b spaces to the right. 2. To add any negative number, — c, to any number, n, begin at n and count c spaces to the left. EXERCISE 32. Verify the answers given in examples 1 to 10, using the above rule, with a number scale : 1. ;; + 5 = 8. 3. 7+(-5)=2. 2. _4 + 8 = 4. 4. -7+5=-2. Positive and Negative Numbers 19 5. 5+(-6)=-l. 8. 9+(-3) = 6. 6. -3+(-4) = -7. 9. 7+(-8) = -L 7.-5 + 6 = 1. 10.-5 + 8 = 3. Find the answers to examples 11 to 16 by the use of the number scale : 11. 2+(-5)+(-l). 14. -7 + 5 + 3+(-l). 12. (_8)+7+(-l). 15. -5 + 5^6^-(-6). 13. 4+(-3)+2. 16. _7 + (-3)+7+(-7). 33. The essential difference between positive and negative numbers is that they are opposite quantities. In adding a posi- tive number we count to the right ; in adding a negative num- ber we count to the left. Any number of negative units added to the same number of positive units gives zero. If, in adding a positive and a negative number, the number of positive units exceeds the number of negative units, the sum is a positive number, but if the number of negative units exceeds the num- ber of positive units, the result is a negative number. EXERCISE 34. 1. $ 10 gained and $ 12 lost results in an actual loss of $2, or $10 +(-$12)=- $2. 2. Indicate by the addition of signed numbers that a boy has $4 and owes $5. 3. Indicate the change in a man's finances, if he spends $10 in the morning and earns $12 in the afternoon. 4. Indicate by adding signed numbers that a boy won 12 points in a game and was penalized 3 points. What is his score ? 5. In three plays a football team gains 7 yards, is penalized 15 yards, and gains 21 "yards. Show, by adding signed num- bers, the net result of the three plays. 20 Positive and Negative Numbers 6. How does the addition of a negative number compare with the subtraction of a positive number containing the same number of units ? Illustrate the answer, using 8—5 and 8 + (— 5). Give another similar illustration. 35. Absolute Value. The value of a number without its sign is its absolute value. The absolute values of — 2, — 3, 3, 5 are respectively 2, 3, 3, and 5. ADDITION OF SIGNED NUMBERS 36. The rules given in § 31 for adding positive and negative numbers by means of a number scale would be neither convenient nor practical in adding large numbers, or in adding fractions. Following are the first six examples of § 32 with their answers : 1. 3+5 = 8. 4. (-7) + 5= -2. 2. (-4) + 8 = 4. 5. 5 + (- 6) = - 1. 3. 7+(-5)=2. 6. (-3) + (-4) = -7. 37. By observing these examples, and others of the same type, we may deduce the following rules : 1. To add two positive numbers, proceed as in arithmetic. (Ex- ample 1 . ) 2. To add a positive and a negative number, subtract the less absolute value from the greater, and prefix the sign of the number having the greater absolute value. (Examples 2, 3, 4, 5.) 3. To add two negative numbers, add their absolute values and prefix the negative sign. (Example 6.) These rules must be learned. EXERCISE 38. Work out the first Jive examples by the number scale and also by the rules. Solve the remaining examples by the rules. 1. 3 +(-5). 3. -7 + 5. 2 . _8+(-2). 4. -8 + 8. Addition of Signed Numbers 21 5. 10 + (-8). 17. _8 + 10 + 7+(-3). 6. 17 + (-20). 18. 22 + (-54)+ 7. 7. _27 + 30. 19. 23.1 + (- 20.5) + (-1). 8. -357 + (-258). 20. .4 + (-3) + (-2). > 9. 536.5 + (- 233.25). 21. -27 + (-5) + 6. 10. i+(-.5). 22. -5 + (-7) + 11. 11. 2.3 +(-3.4) +5.1. 23. 12 + (-2) + (-5). 12. 144+(-23) + (-7). 24. -l + (-l) + 5. 13. 468 + (-298) + (-200). 25. -8 + (-7) + 9. 14. 31.2 + (-2.01) + (-1.11). 26. 357 + (-252). 15. 4.312 +(-25) +24. 27. - 532 + (- 5) + 224. 16. 3+(-5) + (-2)+7. 28. 75 + 2.3 + (- 5.2). 29. -78 + 37 + (-24). Add the following : 30. - 5 31. 22 32. 21 33. -12 3 -52 -15 - 7 -12 -31 -17 - 5 - 7 27 - 3 18 34. Augustus Caesar ruled the Roman Empire 45 years, be- ginning his reign 31 b.c. Indicate, by the addition of signed numbers, the end of his reign. 35. The Eoman historian Livy was born 65 b.c. and lived to be 82 years old. In what year did he die ? Indicate by using signed numbers. 39. When several numbers are to be added, they may be added in the order written ; or the positive numbers may be added by themselves and the negative numbers by themselves ; then the two results may be added. Thus, 4+(-3)+8+(-5)=l + 8+(-5)=9+(-5)=4; or 4+(-3)+8+(-5) = 4 + 8+(-3) + (-5)=12+(-8)=4. 22 Positive and Negative Numbers EXERCISE 40. Find the sum of: 1. 3+(-5)+7+(-2). 4 . 6.4+5.2+(-2.1)+(-.5). 2. 18+37 + (-52) + (-80). 5. -.7 + 3.2 +(-4) + .25. 3. 25+(-6)+14+(-2). 6. f + $ +(- J)+(-/|). 7. 8+(-6)+5+(-ll). 8. If x + 6 +(- 1)= 14, find a;. 9. What number added to 10 will give 8 ? If y + 10 = 8, what is the value y ? 10. What number added to - 10 will give 2 ? If y +(-10) = 2, what is the value of y ? 11. If a +(— 2) +4 = 6, what is the value of a? 12. A man has $650 in the bank, $45 in his pocket, and another man owes him $135. He owes one man $240 and another man $325. Indicate by addition of signed numbers his financial standing. 41. Algebraic Sum. The result obtained by adding signed numbers is the algebraic sum. SUBTRACTION OF SIGNED NUMBERS 42. In arithmetic, subtraction is defined as the operation of taking one number, the subtrahend, from another larger or equal number, the minuend. The result of subtraction is the difference. This definition would mean nothing in such algebraic sub- tractions as, 5 — (— 8), — 2 — 5, 6 — 15. It is therefore neces- sary to have a new definition of subtraction that will apply to signed numbers. The student will recall the relation, difference + subtrahend = minuend. This relation was used to verify answers in subtraction and is the basis of the following definition of subtraction : Subtraction of Signed Numbers 23 Subtraction is the process of finding one of two numbers when their sum, the minuend, and the other number, the subtra- hend, are given. We shall apply this definition to find answers to a few simple examples in subtraction and from these results shall construct rules for algebraic subtraction. 1.5 — 3=? By definition this means : What number added to 3 will give 5 ? We know that 3 + 2 = 5 and there- fore 5-3 = 2. 2. 4 — 6 = ? According to the definition, this asks the question : What number added to 6 will give 4 ? We know- that 6 + ( - 2) = 4, and therefore 4 - 6 = - 2. 3. 4 — (— 6)= ? The minuend, 4, is the sum of two num- bers, and one of the numbers is (—6). Since (—6)+ 10 = 4, therefore 4 — (— 6)= 10. 4. (—4)— 6 =—10. Let the student explain by using the definition. 5. (-4)-(-6)=2. Why? The student must make sure that he understands the answers in the preceding examples ; that is, he must see that they satisfy the requirements of the definition of subtraction. 43. The method of subtracting by using the definition as a rule would not be practical. We proceed to discover rules that will simplify the process. Collecting, for the sake of comparison, the results of § 42, we have, 1. 5 -(+3) = 2. Compare this with 5+ (-3) = 2. 2. 4 - (+ 6) = - 2. Compare this with 4 + (- 6)= - 2. 3. 4 -(-6)= 10. Compare this with 4+ (+6)= 10. 4. -4 -(+6) = -10. Compare this with _4+(-6) = -10. 5. _4_(_6)=2. Comparethis with _4+(+6) = 2. 24 Positive and Negative Numbers 44. These comparisons indicate that we can change any subtraction to an addition by changing the sign of the subtra- hend. Therefore we have the following rule : To subtract one signed number from another, change the sign of the subtrahend and add the resulting number to the minuend. Examples 1. 13 -(-4)= 13 + 4 = 17. 2. 3-(-4)=3+4 = 7. 3. 4-(-10)=4 + 10 = 14. 4. _5-(+3) = -5+(-3) = -8. 5 . 8-(-3)-(-2)=8 + 3+2 = 13. 6 . 5_(_3) + (_2)=5 + 3+(-2)=6. 7. 243 - (- 500) = 243 + 500 = 743. 8 . -350 -(-250)= -350 + 250 = -100. EXERCISE 45. Find the value of : 1. 7 -(-7). 5. 123 -(-21). 9. -22- (-3). 2. 3-10. 6. 2.75 -(-|). 10. 3.5 -(-2.2). 3. _5-3. 7. -37-15. 11. 0-(-2). 4 . 17 -(-3). 8. .02 -(-.1). 12. 0-(-3). 13. 2 -(-.2). 22. -15+(?) = 12. 14 . _4-4-4. 23. 15-(?)=20. 15. (-4)-(-4)-(-4). 24. (?)- 10 = 17. 16- |-f-(-|). 25. (?)-(- 10)= 17. 17. .5+(-i)-.5. 26. (?)-(-13)=8. 18. 17 -(-3)- 3. 27. (?)-(- 5)= 0. 19. 0-(-3)+2 + 16. , 28. 0-(-10). 20. -17 +(-3)- 16. 29. _(-4)-(-4)-(-4). 21. 15+(?)=12. 30. -(-5). Multiplication of Signed Numbers 25 31. Subtract - 7 from 15. Subtract 218.94 from -123.011. 32. Subtract 12 from - 26. Subtract - 5132 from - 2341. 33. What number increased by 15.123 equals 3.102 ? 34. The minuend is 8.231, the subtrahend is 12.0003 ; find the difference. 35. The subtrahend is —54.265 and the difference is - 2.1981 ; find the minuend. MULTIPLICATION OF SIGNED NUMBERS 46. The result of multiplication is the product. The num- bers multiplied are the factors of the product. 47. There are four cases of multiplication of signed numbers. The indicated multiplication 3x4 is to be read " three times four " ; that is, the first factor is taken as the multiplier. 1. In arithmetic, 3x4 means that 4 is to be added 3 times. Thus, 3 x4 =4+4 + 4 = 12, or (+ a) • (+&)=+«&. 2. Similarly, 3x(— 4) means that (—4) is to be added 3 times. Thus,3x(-4) = (-4) + (-4) + (-4) = -12,or( + a) ■ (-b) = -ab. 3. Since to multiply by +-3 we add the multiplicand 3 times, it is reasonable to assume that to multiply by — 3 we subtract the multiplicand 3 times ; that is, (— 3) x 4 means that 4 is to be subtracted 3 times. Thus, (_3)x4=-4-4-4=-12, or(-a) • (+&) = _«&. 4. As in 3, (-3)x (-4) = -(- 4)-(- 4)-(- 4) = 4 + 4 + 4 = 12, or (- a) . (- b)=ab. 48. Collecting the results in these four cases, we have all possible combinations of signs for two factors. (+3)x(+4)=12, or (+a)x(+&) = + a&. (+3)x(-4) = -12, or (a)x(- b) = -ab. (- 3) x (+ 4) = - 12, or (- a) x (+ b)= - ab. (- 3) x (- 4)= 12, or (- a) x (- 6) = +a&. 26 Positive and Negative Numbers 49. The preceding equations give, in algebraic symbols, the law of signs for multiplication, and the method of multiplying two signed numbers. To find the product of two signed numbers : 1. Find the product of the absolute values of the two numbers. 2. Make the sign of the product positive if the two factors have like signs, and negative if they have unlike signs. Examples 1. 3x(— 7)= — 21. What is the absolute value of the product ? Why is the sign of the product negative ? 2. (— 8) x (— 7) = 56. Why is the sign of the product posi- tive ? 3. (-2)x(-5)x(-2) = 10x(-2)=-20. Explain. ORAL EXERCISE 50. Find the value of: 1. -3x6; -3 X 6a. 2. -3 x(-6); _3ax(-6). 3. 7 x(-'3); 7 x(-Sn). 4. -10x2.5; - 10 a x 2.5. 5. 12x(-7); 12x(-7s). 6. (-3)x(-22); (-3)x(-22m). 7. 8 x ( - 6) X 5. 8. 12x(-2)x(-.3). 9 . ( _6)x5x(-J)x(-4). 10. (-2)x(-2)x(-2)x(-2)x(-2). 11. Given the numbers 2, 5, —3, —2, i, -5, —.25; tell at sight the product of each number multiplied by each one that comes after it. 12. What sign has the product when three negative num- bers are multiplied together ? four negative numbers ? five ? Give an answer that will apply to all cases. Division of Signed Numbers 27 DIVISION OF SIGNED NUMBERS 51. Division is the process of finding one of two factors when their product and the other factor are given. The result of division is the quotient. 52. From the definition of division we derive the following : 1. Since (+ 7) • (+ 3)= + 21, therefore (+21)-h(+7) = +3. 2. Since (+ 7) • (- 3) = - 21, therefore (- 21)-(+ 7) = -3. A negative number divided by a positive number gives a negative quotient. 3. Since (-7) . (-3) = (+21), therefore (+21)-s-(-7)= -3. A positive number divided by a negative number gives a negative quotient. 4. Since (- 7) . (+3) = - 21, therefore (-21) + (-7) =+3. A negative number divided by a negative number gives a positive quotient. 53. Generalizing these results, we have the rule for dividing signed numbers. To divide one signed number by another : 1. Find the quotient of the absolute value of the dividend divided by the absolute value of the divisor. 2. Make the sign of the quotient positive if the dividend and divisor have like signs and negative if they have unlike signs. Examples 1. -12-*- 3 = -4. 2. -12a-=-3 = -4a. 4. -10 -K-5)= 2. 5. (-10a)-(-5«)=2. 6. (_8)x(-2)-(-4)=16--(-4) = -4. 7. (-2) 2 -*-2 3 = 4-f-8 = ^. 8. (-2)3--22 = -8h-4 = -2. 28 Positive and Negative Numbers ORAL EXERCISE 54. Perform the operations indicated : 1. _l2-=-(_3). 5.-1-5-1 9. _7a+(-7). 2. (-12) +4. 6. 3 a -=-3. 10. -32 +(-2)4, 3. 16 +(_4). 7. -7o + a. 11. _!+(_$). 4. -28 + 7. 8. -7a-=-7. 12. 12 a? +(-4). 13. aft -s- (-a). 22. - 39 h- 13 X (- 3). 14. 7x(-6)-(-3). * 23. 4 3 -=-(-4) 2 . 15. 21-(-7)-(-3). 24. (-3) 2 -=-3 3 . 16. 7x(-2r)+2. 25. (_3) 3 -=-(-3) 2 . 17. 12-(-4)x(-l) 3 . 26. 54+(-9)x(-6). 18. -2x(-3) + (-l). 27. (-2) 3 X(-3)X(-1). 19. (-j)+2. 28. (_5a6) + (-a&). 20. -5a-j-(-l)+(-5a). 29. (- 2) 2 +(- 3) 2 -(- 4). 21. _32-(-8) + (-4). 30. -2-f 55. Order of Operations. In a chain of operations involving the signs, +-, — , x, +, the numbers connected by the signs X and -7- must be operated upon first from left to right in the order in which they occur. The results thus obtained should be added and subtracted as indicated by the signs + and — . Thus, 3 + 4x2-6-r3=3 + 8-2 = 9. Also 3 + 6-^2x5 + 7 = 3 + 3x5 + 7. = 3+15 + 7. . =25. EXERCISE 56. Find the value of: 1. 2x(-3)+-(-5)x2. 3. _3x(-4) + (-5)-70. 2. -7-(-6)+-(-12). 4. 0-2x(-3)+-7-(-l). 5. -lx(-2)x(-3)+6x(-2). 6. -5 + 3x7-(-5)x(-4). Division of Signed Numbers 29 7. 24 + 8 x2-(-14). 9. 24-3x4-6x5. 8. 60 - 5 x 3 + 6 - 3. 10. 24-3 + 4-6 + 5. 11. 24-6 + 3 x 5x4. 12. 5 + 6 x 7 - 28 x 2 + 3 x (- 6). 13. 4 - 3 x 2 + 8 x (- 2) + 4 - (- 1). 14. -8x(-2)-15-(-3) + 7x0. 15. 0-4x8 + 7x(-2)-(-20). 16. 15 + (-3) + (-7) 2 -(-8) 2 + 12. 17. l20 + (-3) 3 + (-2) x 8-21-12. 18. _15-(-5) + 8x (-2) -7 + (-3). 19. - 12 x (- 2) + (- 3) x 8 - 10 x (- 1). 20. 0-lOx (-2)-(-4) x 8- 7 -(-7). 21. 12 - 15 - (- 13) - 15 - (- 15). 22. 12 x (- l)-(- 10) x (- 1) - 8 x (- l)-(- 6)(- 1), If a = 6, 6 = — 5, c = — 3, d = — i, e = — -|, find the values of the expressions in examples 23 to 43. 23. ab. 30. ab — c 3 . 37. be — be — 2 a. 24. ac\ 31. b — ad. 38. abede. 25. 36c. 32. -36 + 2 c. 39. a 2 - 6 2 . 26. — a6c. 33. ~(ac)— 4^. 40. 6 2 — (— a) 2 . 27. bed. 34. dc + a + 26. 41. 6a -56 -3c. 28. a + be. 35. 6de + a + 6 — 1. 42. 3c — 2d + 5 e. 29. 6 — c 2 . 36. 6e - 6c + 2 a. 43. 5 a + 6 — 4 d. 44. Does 3a;— 5 = 7 a — 9 when x = 3 ? when cc = 1 ? 45. Does ^-5.t + 6 = when a = 3 ? when a = 2 ? 46. Does X = — 1 satisfy the equation x 2 — 2x— 3 = 0? 30 Positive and Negative Numbers REVIEW EXERCISE 57. 1. What quality signs would you associate with each of the following : north latitude, south latitude ? rising tem- perature, falling temperature ? debts, credits, money lost, money spent, money earned, money found ? a.d., b.c. ? points won in a game, points lost, penalties ? 2. Compare the addition of a negative number with the subtraction of a positive number having the same absolute value. Illustrate. 3. Indicate the net result of $ 10 earned, $3 spent, $2 found, and $ 2 spent. 4. The temperature at 8 o'clock was 28° ; at 10 o'clock it had risen 4° ; at noon it was 5° warmer than at 10 o'clock ; at 2 o'clock it had risen 2° more ; at 4 o'clock it was 3° colder than at 2 o'clock ; at 6 it was 4° below the temperature at 4 o'clock ; and at 8 p.m. it was 7° colder than at 6 o'clock. (1) Indicate by arithmetical additions and subtractions the temperature at 8 p.m. (2) Find the same result by addition of signed numbers. 5. If you walk 3 miles south and 7 miles north, how far and in what direction from the starting point are you? Indicate by adding signed numbers. 6. How far upstream are you if you have rowed 7 miles up and drifted 2 miles down ? Indicate the process of finding the answer in two ways. 7. Pikes Peak is 14,108 feet above sea level. A place in Holland is 161 feet below sea level. How much higher is Pikes Peak than the place in Holland ? Indicate two ways of finding the answer. 8. If a gasoline launch can run 14 miles an hour in still water, how fast can it run up a river whose current flows 4 miles an hour ? How fast can it run downstream ? Review Exercise 31 9. If a person can swim 2\ miles an hour in still water, represent his rate when swimming against a current of 3 miles an hour. Represent his rate downstream. 10. The Roman Empire fell 476 a.d., 622 years after the fall of Carthage. What was the date of the fall of Carthage ? 11. Give the rules for addition, subtraction, multiplication, and division of signed numbers. 12. Define subtraction ; define division. 13. What is the basis of the rule for subtraction of signed numbers ? of the rule for division ? 14. What is the absolute value of a number ? 15. What is the sign of (- l) 10 ? of (- l) 11 ? Can you give an answer that will apply to all such examples ? 16. What is the " order of operations " ? When a = 8, b = — 3 and c = — 9, find the value of: 17. a 4-6+ c 21. a + b+c + c. 25. a 2 + b 2 + c. 18. a—b — c. 22. b — b 2 . 26. ab + be + ac. 19. a — be. 23. ab — be. 27. a 2 — ac. 20. a — b-\-c. 24. abc — c. 28. — a — b — c. Ill ADDITION 58. Algebraic Expression. A number represented by alge- braic symbols is an algebraic expression. Thus, 2 ab, 5 — 3 ab, 4 + 2 b are algebraic expressions. 59. Monomial, Term. An algebraic expression the parts of which are not separated by either of the signs + or — , is a monomial or a term. Thus, 2 ab, — xy,3x + 7 are monomials. 60. Polynomial. An algebraic expression consisting of two or more terms is a polynomial. Thus, 3 ax — 4 c + 7 and m — n + 11 xy — 16 are polynomials. The monomials that make up the polynomial are the terms of the polynomial. Thus, 3 ax, — 4 c, and 7 are the terms of the polynomial 3 ax — 4 c + 7. A polynomial of two terms is a binomial, and one of three terms is a trinomial. Thus, 2 a + b is a binomial, and ax — by + c is a trinomial. ORAL EXERCISE 61. In the following expressions, name (a) the monomials, (b) the binomials, (c) the trinomials, (d) the polynomials: 1. 3a 2 a\ 4. -x 2 -2ax. 7. \at\ 2. 4a 2 + z. 5. 4a-r6. 8. 2x 2 -hSx-l. Z.2b — c + 3d. 6. 4 + a — 6. 9. a - 6 — c. 32 Addition 33 Name the terms in each of the following polynomials : 1.0. 3a— b. 12. |a£ 2 +2a + 7. 14. mxn+m-i-n — 1. 11. 2a 2 -3a6+c. 13. ax 2 + bx + c. 15. - 3 a - 2 & - c. 62. Coefficient. Any factor of a term, or the product of two or more of the factors of a term, is the coefficient (co-factor) of the product of the other factors. Thus, in 2 abc 2 , 2 is the coefficient of abc 2 ; 2 a is the coefficient of &c 2 , etc. What is the coefficient of xy 2 in 5 a;?/ 2 ? of x ? 63. Numerical Coefficient. The numerical factor of a term is its numerical coefficient. Thus, the numerical coefficient of 7 am is 7. la and 7 m are literal coefficients of m and a respectively. When we speak of the coefficient of a term we generally mean the numerical coefficient, including the sign preceding the term. Thus, 2 is the numerical coefficient of 2 abc and — 3 is the numerical coefficient of — 3 ax. Also - is the numerical coefficient of - • 2 2 What are the coefficients of x and y in the equation 3 x 4- 4 y = 7 ? What are the coefficients of x 2 and x in ax 2 -\-bx + c = 0? If no numerical coefficient is expressed, the coefficient 1 is understood. Thus, x is the same as 1 x. What is the numerical coefficient of ab 2 ? of — a 2 ? 64. Power, Exponent, and Base. The product arising from using a number one or more times as a factor is a power of the number. The number written to the right and above another number to indicate how many times the number is used as a factor 34 Addition is the exponent (§ 6) of the power. The repeated factor is the base. Thus, a 4 means the fourth power of a, often read a fourth power, or simply a fourth. 4 is the exponent of the power and a is the base. a 1 means the same as a. The exponent 1 is never written, a 2 and a 3 are read " a square " and " a cube," since if a represents the length of the side of a square or the edge of a cube, a 2 and a 8 denote the area of the square and the volume of the cube respectively. 65. The student, must note carefully the difference between coefficient and exponent. Thus, 3 a means 3 x a, while a 3 means a- a- a, that is, it means that a is used as a factor three times. ORAL EXERCISE 66. 1. How would you write 3 • a • a • b • 6 using expo- nents ? 5 - a-a-a-x-x-x? 1000? 2. How would you write as one term a+ a + a -f-a? Name the numerical coefficients, the exponents, and the base for each exponent : 3. 5a 2 . 6. 4 ran. 9. 3a — \ a 3 . 4. 2xf. 7. -3 a 5 . 10. a 3 - 12 b\ 5. x 2 y n . 8. x. 11. —a 3 . Evaluate (that is, find the value of) the following when a = 2 and, b = — 1 : 12. ab; ab 2 ; ab z . 13. -2a 2 b; -a 2 6 2 ; - a 2 b. 14. a + b ; a - 6 ; a 2 + 6 2 ; a 3 + 6 3 . 15. What is the meaning of m 4 ? of 4 m ? 16. Find the value of a 3 when a = 2 ; of 3 a. 17. Find the value of a 2 when a = — 2 ; of — a 2 ; of — a b . Addition of Like Monomials 35 67. Similar and Dissimilar Terms. Terms that do not differ at all or that differ only in their numerical coefficients are like terms or similar terms. Thus, 2 ab, ab, and 6 ab are similar terms. Terms that differ in other respects than in their numerical coefficients are unlike terms or dissimilar terms. Thus, 2 ab and 12 ab 2 are dissimilar terms. Why ? ORAL EXERCISE 68. In the following list, select all terms that are similar to the first; to the second; to the fourth; to the fifth : 1. 2a 2 x. 4. 5x. 7. 5a*x. 10. — 16 a 2 x. 2. 4a6c. 5. 4 ax. 8. — 3 x. 11. 5 ax 2 . 3. — 4a6c. 6. ±abc. 9. ax. 12. fax. ADDITION OF LIKE MONOMIALS ORAL EXERCISE 69. Add the following : 1. 4 and 7 ; — 2 and 4. 2.-4 and 7 ; — 3 and 8 ; 5 and — 4. 3. 4 and - 7 ; 10 and - 12 ; - 12 and 10. 4.-4 and — 7 ; — 5 and — 7 ; — 3 and 3. 5. 13 inches and 5 inches ; 13 i and 5 1. 6. 4 miles and 7 miles ; 4 m and 7 m. 7. 5 rods, 6 rods, and 11 rods ; 5r, 6 r, and 11 r. 8. $ 5, $ 7, $ 15 ; 5 d, 7 d, 15 d. 9. -11,17,13; -11a, 17 a, 13 a. 10. 8, -9, - 5; 8», -9a?, -5z. 11. m, 10 m, — 7 m, — 4 m. 36 Addition Add the following : 12. 7 ab, 5ab, — 6 ab, 4 ab. 13. 3 x 5, 7 x 5, - 8 x 5, 2 x 5. 14. - 9 x 3, - 4 x 3, 13 x 3, - 6 x 3. 70. These examples suggest the following rule : To add like monomials, add the numerical coefficients and make their sum the coefficient of the common literal part. In applying this rule, the numerical coefficients should be added according to the rules for adding positive and negative numbers. The literal part of each term is thought of as the unit of addition. Terms may be added in any order. Examples 1. Add— 5 a and 7 a. — 5 + 7=2. (Adding numerical coefficients.) .•. — 5 a + 7 a = 2 a. 2. 4a+(-7a)=-3a. 3. ax +(—3ax)-\-(— 5ax)= — 7 ax. Hint. 1 +(- 3) + (- 5) = - 7. ORAL EXERCISE 71 . Add the following : 1. 4 a, 3 a. 9. 3 m, 4 m, —5 m. 2. 4 a, —3 a. 10. —2x,5x,—7x. 3. — 4a,3a. 11. 4r, — 5r, 6r. 4. -4a, -3a. 12. -5t,2t,3t. 5. lip, — 7 p. 13. 5ab, 4a6. 6. —5 s, —6 s. 14. —6xy, —2xy. 7. 6n,5n,2n. 15. 7mn, — 2mn. 8. 6n, -5n, 2n. 16. 10x5,8x5. Addition of Like Monomials 37 18. - 6 x 13, 5 x 13. 21. ax* 22. 7 x — asc 3 — 3 x — 4 aic 3 2x — 9 ax 3 — 5x 17. 11x7, -8 X7. 19. 262 20. 2c -26 2 -3c 6 2 -5c -36 2 -4c 46 2 10 c -4aa? 12 x 23. 3a 2 +(-5a 2 )+(-7a 2 )+2a 2 + a 2 = ? 24. — 5 ax +(— 3ax)-\-ax + 5ax = ? 25. ±d+(-±d)+d+(-2d). 26. Solve3a; + 5a;-8 = 16. Solution. 3x + 5x — 8 = 16. 8x-8=16. (Why?) 8x = 24. (Adding 8 to both members (§ 13, a).) x = 3. (§13,d.) /Sofae Me following equations : 27. 5z-3 = 7. 34. 14a;+(-5a;) = 63 + (-9). 28. 4a;+2a; = 12 + (-3). 35. 4a; + (-3a;)= 10. 29. _5y + 8y = 7 + (-3). 36. 8a; + (- 4a) = -4 + 7. 30. 8n+(— 3w)+w = 12. % 37. 3a; -5 = 7. 31. 15r+(-r)+2r = 20. 38. 7a;-2 = 5l 32. p+ (-p)+3p = l. 39. |a; + 4=10. 33. -3x+10a; = 12. 40. 4a; + 3 = 13. EXERCISE 72. Add the following : 1. .2 6 2. -3 a; 3. wma; 4. 24 r -3.1 6 4 x — 2mna; -12r 4 6 -2.5 2 — 5 mwa; -48r - 6 .08 a; — 4 mna; 122 r .086 12.2 x — 7 mnx -57r Suggestion. When adding several similar terms, we usually first add the positive numbers, then the negative numbers, then the results. 38 Addition Add the following : 5. 425 m, — 321m, — m, — 50 m. 6. 4 a, — 9 a, a, 5 a, — 6 a, 2a. 7. - 250 jpg, 75j>g, 50j9g, 125pg. 8. — 10.1a, .2 a, — a, —2 a, — 5.1a. 9. 210/), -352p,71p, -83p. 10. 14.3 g, - 2.03 q, 17.5 g, - .1 q, q. 11. - 5 ab 2 , - 11 a& 2 , 14 ab 2 , - a& 2 . 12. 13 r, -15r, -r, 73 r, r. 13. a, — 2 a, 3 a, — 4 a, 5 a, — 6 a. 14. 2 a, — 4 a, 6 a, — 8 a, 10 a, 12 a. 15. 6 a, .5 a, — .Ola;, - 1.72a. 16. - 27 ab, - 35 ab, 43 a&, - 20 ab. 17. 44 xy, —12xy, — 24 a?/. 18. 75, - 32, - 70, 23, - 16. ' 19. 784a, -369 a, -Ilia, -53a. 20. 23 xyz, — 24 xyz, — 36 xyz. 21. 1, - 2, 3, - 4, 5, - 6. 22. — 3 a, 7 a, 42 x, — 13 a. 23. What are the units of addition in examples 1 to 8 ? 24. (— 5a) + 7a+(— 12a)=— 10a. ' Is this true when a = 1 ? when a = 2 ? For what other values of a is it true ? 25. In what respect may two like terms differ ? 26. What kind of algebraic expression is obtained by add- ing two like monomials ? 27. Solve 2 a +9 a + (-3 a) = 17 -f-(-l). 28. Evaluate a -f- b + c + d, when a = 2, b = — 3, c = — 5, d=-5. 29. Simplify a + 6 4- c 4- a" when a = — 3a, &=— 5a, c = a, d = 4a. Addition of Like Monomials 39 Solve the following equations : 30. 2.5 x + 12 = 312. 31. 18a> + (-12a>)=15+(-12). 32. 6a?+(-5s)=1.7 + 3.3. 33. 2x + (— 3x) + 4:X = — 5.1. 34. 18aj+(-12»)4-20a; = 164-(-3). 35. 1.25z4-(-.75x)=.95+(-.45). 36. 24 y + (- 12 y)+ Sy = 125 +(- 32) + (- 86) + 48. 37. 3aj + 12aj + 48aj+(-50a;) = 48+2(-4) + (-l). 38. _7 i ,-f-10p+(- j p)=2 x (-3) + (-3)«. 39. _l0r + (-8r) + (-7) + T = 10(-2) + (-5) 2 . 73. In arithmetic we can add several numbers in any order and get the same sum, and we can multiply several factors together in any order and get the same product. Likewise in algebra we can rearrange the terms of a polynomial, or change the order of the factors of a term without changing its value. Thus, 5-2a + 3&=3&-2a + 5 = 3o + 5-2a. Also 6 m*np — 6 nm 2 p. 74. Arranging the Terms of a Polynomial. It is generally convenient, and sometimes necessary, to arrange the terms of a polynomial in some particular order. The most frequent arrangement is in descending powers of some letter that occurs in all, or all but one, of the terms. Thus, 2 x :i + 3 x 2 - 2 x + 8, 4 x 4 - 3 x - 7, ax* + bx + c are all ar- ranged in descending powers of x. An arrangement in ascending powers is sometimes used. Thus, — 7 — 3 a; + 4 x 4 represents one of the above expressions re- arranged in ascending powers of x. If, instead of different powers of the same letter,- we have the same power of different letters, we generally arrange the terms alphabetically. Thus, a 2 + b 2 + c 2 is arranged alphabetically. 40 Addition ORAL EXERCISE 75. Rearrange the following (a) in descending powers, (b) in ascending powers : 1. tf-l+2x*-lx. 4. & + 16b-llb* + 5V-l. 2. 3 a + 7 -x*. 5. p-2p 2 + 7. 3. 3 m 2 — 4 m + 6 -f m 3 . 6. bx -f a# 2 + c. Rearrange the following alphabetically : 7. 26 2 -a 2 + 3d 2 -5c 2 . io. ft 2 - 14/ 2 + 16 # 2 + 7 & 2 . 8. c— b — a — d. 11. y — x + z — v + w. 9. n 2 + 2m 2 -l 2 . 12. p 3 - r 3 + q 3 + n\ ADDITION OF UNLIKE MONOMIALS 76. Just as in arithmetic 2 ft. -f- 3 ft. = 5 ft., so in algebra 2a+3a = 5a; but just as 2 ft. and 3 in. cannot be added without changing them to the same denomination (2 ft. + 3 in. • = 24 in. + 3 in. = 27 in.), so the sum of 2 a and 3 b must be expressed in the form 2 a -f 3 b until we have the values of a and b. ORAL EXERCISE 77. 1. What is the length of a fence around a field 40 rods long and 30 rods wide ? 40 rods long and x rods wide ? a rods long and b rods wide ? 2. A ship travels 300 miles one day and 320 miles the next day. How far has it gone ? If the number of miles had been a and 6, how far would it have gone? What is the value of this result if a = 300, b = 320? 3. There were x boys and y girls in school last term. If a new boys and b new girls enter this term and m boys and n girls leave, how many boys and how many girls are there in school ? how many boys and girls together ? Addition of Unlike Monomials 41 78. The following statement may be regarded both as a definition of the sum of unlike terms and as a rule for adding such terms : The sum of several unlike terms is the algebraic expression obtained by uniting them with their respective signs. Thus, the sum of 2 a, (— 3 6), and 11 c is 2 a — 3 b + 11 c. 79. It is often necessary to use the rule for adding like monomials, along with the above rule for adding unlike mono- mials. Thus, Sa + 5b+(-$b) + (-2a)=a-3b. This kind of simplification is usually called collecting terms. ORAL EXERCISE 80. Add and arrange the terms in proper order : 1. - a 2 , - 4a 2 , 2a 2 . 5. |a 2 , - § 6 2 , Ja 2 , \b\ 2. 3 6, -2 a, a, -5 b. 6. 4 m 2 , -5 n 2 ,. 5 m 2 , 4 n 2 , 2 m 2 . 3. 4a 2 , - 3x, 7, 5x, -2x\ 7. 3x\ - x, 2, 4a 2 , - 1, x, 3. 4. -1, 2x, 3x?, -x% 1. 8. 46, -2 a, -26, 4c, - a; Collect the terms and arrange in order: 9. 46+(-7)+36 + 26+(-36). 10. 8x + y + 2z + 3x + y + 7z. 11. 7y + x +(-2x)+2y+(-z). 12. 7*+(- . 4. Ja4-|6+(-a)+fc+(-f5)+ia+(-fc). s- i»+(-*y)+f»-K-"iiO+(-«). 6. 56a 4 58 p + 218 4- 92 p + 36a + 74 4- 20p + 360. 7. 54m + (-62n) + 18» + (-62m) + (-6a;)+42w + 10m + 18 n'+ (—14®). 8. 10 m + 11 + (- 5 a?) + (- 12) + (- 4 m) + (- 3 a?) + 1 + 9x4- (—5 m). 9. 13» + (-5y) + 8aJ4-(-5aj)4-9y + (-lla;)+(-3a;) I + (-6y)+ft. 10. 5.67) + 18.5 g -f (- 7.25 jp) + 11.5 q 4- 15.5 p + (- 9.4 q). 11. 5a+(-3!&) + 5fc4(-6Ja)49ii&43fa4(-2jic). ADDITION OF POLYNOMIALS 82. 1. Add and compare : 2 ft. + 3 in. 2/4- Si 4 ft. + 2 in. 4/4- 2i 7 ft. 4 5 in. 7/+ 5i 13 ft. + 10 in. 13/ 4 10 1 2. Add 2a-36 + 7c,26 + a-2c, c4-2a-36. 2a-3&4-7c a 4- 2 b — 2 c Rearrange so as to have like terms 2 a — 3 b 4- c in the same column ; then add. 5a-464-6c Addition of Polynomials 43 3.- Add 2 a 2 -3a + 7, -4 a 2 -6 + 5 a, 8 a 2 -9a- 7. 2a 2 -3a + 7 -4a 2 + 5a-6 8a 2 -9a— 7 6a 2 -7a- 6 83. .4c7a*: ORAL EXERCISE 1. 3a + 26 4. 9x -9 7. 4a6 + c 10. 8 z + 9 4a + 96 8x 5r -3 + 3 8. 2 - 5ab — c 3 2. 4 a +3 5. a + 3 11. a; 7# + 8 r -2 5 a -3 ■a; + 3 3. 5m + 3 6. r + 2s 9. 4p + 7 12. a; — 5 8m-2 3r -5s 5: P - £ + 5 13. a+ b- c 20. 4r + 2s— t 2a-2b- 3c -3* — 2* 14. 3a -6 + 2c 2r + 7s a — c 21. iC 2 + # — 1 — X 2 + iC — 1 a 2 - X' + 1 15. 4a — b — 2c 6 + 2c 16. a 2 -2a + 5 22. 3a- 2/ + 2z -2a 2 + 3a- 2 — x + Sy—'Sz 17. 2^ + 3.^ + 9 -2x- 2y + 2z x 2 -9 23. a + 6 — c a — b + c — a+6+c 18. a 2 + a 2a + l 19. a - 3 6 24. a 2 - 2 a + 1 2a - c a* + 2a + l 36 + 2c -2a 2 -1 44 Addition 84. Checking Results. 2 a + 3 a = 5 a for all values of a. The sum of 3 a — 5 b and 4 a + 2 6 equals 7 a — 3 6 for all values of a and b. This fact may be used to check the answers. 1. Add 3 a— 5 b and 4 a + 2 6 and check the result. Addition Check. a=b = 1. 3a-5b 3-5=-2 4a+2fr 4+2 = 6 7a-36 7-3= 4 The work on the right is the result of putting a = 1 and 6 = 1 in the two expressions to be added and in the result. Notice that the final number, 4, is the algebraic sum both of the last line and the right-hand column ; that is, 7 — 3 = 4 and -2 + 6 = 4. 2. Add and check the result, 2 a 2 — 3 a +4, 7 a+3, -2 a 2 +5. Addition Check, a = 2. 2a 2 -3a+4 8-6+4= 6 7a+3 14+3= 17 -2a 2 +5 -8 + 5 =- 3 4 a + 12 8 + 12 = 20 85. To add two or more polynomials : 1. Arrange the terms in order of powers of some letter (or alphabeti- cally), writing like terms in the same column. 2. Add the like terms in each column and unite the partial results obtained with their respective signs. Note. Check, or prove the correctness of the result, by substituting a numerical value for the letters used. Examples t 1. Add and check, 3 a - 2, a 2 - 1, 2 a + 7 a 2 + 3, a - 5. Addition Check, a = 1. 3a-2 3-2= 1 a 2 -1 1-1=0 7a 2 + 2a + 3 7 + 2 + 3= 12 a-5 1-5 =- 4 8a 2 + 6 a- 5 8 + 6-5= 9 Addition of Polynomials 45 2. Add and check, 3 a 2 - 2 ab + b 2 , - 4 aft +2 6 2 - a 2 , a 2 +6 2 . Addition Check, a = 6 = 1. 3 a 2_2a&+ ft 2 2 - a 2 -4a& + 26 2 -3 a 2 + & 2 2 3a 2 -6a6 + 46 2 1 EXERCISE 86. Add the following, rearranging when necessary, and check the results in examples 1 to 7 : 1. 3a + 26 4. lab — 3ac + 46c 4a — 76 4 a6 + 4 ac — 5 6c — 5 a -f 4 6 — 5 a6 — ac -+- 26c 5. 6. 7. 3a 2 + 7a6-26 2 , 96 2 -3a6 + a 2 , 5a 2 + 76 2 . 8. 14a-66 + 3c-5d, 9a + 7 6 - 4c - 9d\ 9. .8 a 2 - 3.47 a6 - 17.25 ac + 3.75 6c, - 7.5 a 2 + .47 a6 + 12 ac - 7 6c. 10. 1.5 x 2 - 3.2 x - .07, 8.04 x - 2.1 + 4 z 2 , .3 a - .75 , - 81.7p - 9.4 m - 8.7 n, 9.76 ra + 4.33p + 9.3 n. 30. 41.6 q - 43.1 a + 37.8 y, .09 ?/ - 5.37 x - 4.05 g, 1.97 x - 4.1 ?/ - .8 7. 31. .3 x 2 + .1 # 2 - .3 yz - .1 z 2 , .2 a# - .3 # 2 -f- .3 yz, - .4 x 1 - .2xy + .1 y 2 + .1 z\ 32. Solve 3 x + 2 x + 5 + 9 = 27 +(-3). 33. Solve 1 x +(_ 3 #) + (-»)= 5+ (-3)+ 12. 34. Find the sum of five numbers, the first number being 2 x and each succeeding number being 3 a greater than the preceding. 35. If a passenger ticket costs x cents a mile and it costs 4 cents to carry a bicycle each 25 miles, how much is the cost of both for 250 miles ? 36. Through how many degrees of longitude does a ship sail in going from - 18° to + 37° ? 37. The oldest known mathematical manuscript was written about — 1700 (1700 b.c). How long ago was it written ? 38. A merchant's capital was diminished by $ 1400 and then amounted to $ 4500. What was his capital at first ? Suggestion. Let x = number of dollars at first. 39. At a certain election A received 113 more votes than B. The number of votes cast for both was 847. How many votes did each receive ? Suggestion. Let x — number of votes B received. Hence x -f 113 = number of votes A received. Then x + x + 113 = 847. (By the conditions.) Review Exercise 49 40. A rectangular field is twice as long as it is wide and its perimeter is 360 rods. Find the leng'th and the width of the field. 41. A ball team played 20 games and won three times as many as it lost. How many games were won and how many were lost ? Suggestion. Let x = number of games lost. 42. A boy paid x cents for a bat, twice as much for a ball, and 20 cents less for a mask than for both ball and bat. How much did each cost him if he spent $ 2.20 all together ? Suggestion. Change $ 2.20 to 220 cents. 43. The larger of two numbers is three times the smaller and their sum is 84. Find the numbers. 44. The larger of two numbers exceeds the smaller by 10, and the sum of the two numbers is 94. Find the numbers. 45. The girls in a certain high school outnumbered the boys by 122. The entire enrollment was 2742. How many boys were there in the school? 46. A woodworking class spent $ 32.50 more for jack planes than for try-squares. If both tools together cost $ 50, find the cost of each kind. 47. One farmer by spraying his potatoes raises 30 bushels more on an acre than his neighbor. If both together raise 400 bushels, how many bushels does each raise ? 48. In 1910 Jerry Moore of South Carolina won a prize in a boys' corn raising contest. In 1913 Walker Dunson of Alabama raised 4 bushels more corn on an acre than Jerry Moore's record yield. The total yield on the two acres was 460 bushels. How many bushels did each raise ? 49. In 1914 the Allred boys, Luther, Clarence, Elmer, and Arthur, of Georgia, raised on four one-acre plots of land 824 bushels of corn. Clarence raised 10 bushels more than Elmer and 7 bushels less than Luther, while Arthur raised 43 bushels less than Elmer. How many bushels did each raise ? IV. SUBTRACTION SUBTRACTION OF LIKE MONOMIALS ORAL EXERCISE 88. 1. Define subtraction. (§ 42.) 2. State the rule for subtracting signed numbers. (§ 44.) Subtract the following : 3. 3-4; 7-8; 10-15. 4. 3-(-4); 9 -(-15); 4 -(-4). 5. -3-5; -7-8; -15-2. 6 . _1_ ( _3); -3-(-4); -5-(-8). 7. 7 ft. -5 ft.; 7f-5f. 8. 20°-30°; 20d-30d. 9. 601b. -351b.; 60 p- 35 p. 10. 85 acres — 42 acres ; 85 a — 42 a, 11. $73- $21; 73 d -21c?. 12. -17°-5°; -17 d-5d. 13. — 10 # — 3 Y 10 x - 2ab — 4 xyz — 27 ary 17. —5 be from — 3 6c. 18. 18 6 5 from -4 6 5 . 19. - 13 a 2 6 from 24 a 2 6. 20. 3 a6 2 from - 7 a6 2 . 21. —5cd? from 14 cd?. 22. — 2.25 m^n from — 3.5 m 2 n. 23. From the sum of 7 a6 2 c and — 11 a6 2 c take — 4 a6 2 c. 24 Take the sum of whi and — 6 m 2 n from the sum of — 4 mfri and 3 m 2 ?i. 52 Subtraction 25. The minuend is and the subtrahend is —3 a;. What is the difference ? 26. The minuend is — 27 xy and the difference is 5 xy. What is the subtrahend ? 27. The subtrahend is — 5.2 x and the difference is .05 x. What is the minuend ? SUBTRACTION OF UNLIKE MONOMIALS ORAL EXERCISE 91. 1. What are unlike monomials ? 2. What does a — b mean ? 3. What length remains if 10 feet are cut from a rope 32 feet long ? if x feet are cut from a rope 32 feet long ? if b feet are cut from a rope a feet long ? 4. How much have you left if you have 16 cents and spend 7 cents ? if you have 16 cents and spend x cents ? if you have a cents and spend x cents ? 5. If you throw a stone vertically upward ft feet, how high is it after it has fallen d feet? How high is the stone if h = 62 and d = 21 ? 6. If the enrollment in a class is m girls and n boys and there are x girls and y boys absent, what is the attendance ? 7. How are unlike monomials added ? 92. To subtract a monomial from an unlike monomial, change the sign of the subtrahend and add the resulting number to the minuend. Examples 1. Subtract 2 a from 3 x. Sx 9 The subtrahend when its sign is changed becomes —2a; adding this to 3 x gives 3 x — 2 a. 3x — 2a The result may be checked as usual, 3« — 2a + 2a = 3a;. Subtraction of Unlike Monomials • 53 2. Subtract — 6 from a. a fi The subtrahend with its we have a -+- 6. a + 6 i sign changed becomes 6. Adding, 3. Subtract — 3 from 4 x. 4:X _3 4z-(-3)=4a; + 3. 4x + 3 EXERCISE 93. Subtract: 1. 116 from 17 a. 3. —5b from 2 a. 2. — 7 a from 5 ?/. 4. b from — 6 a. 5. 7x 6. 5a 2 6 7. ? ' 8. d -7 3a& 2 — p — c i 9. 7 c 2 from 5ab. 11. — 5 a?/ 2 z from — kxhjz. 10. — 4 ?/ from 3 x 2 . 12. — a from a + 2b + a— 2b. 13. - 10 from 3 x + 2 # + 2/- 3 a?. 14. 10 from 3a + 2*/ + (— 2?/) — 7 x. Collect terms: 15. 3a + 5?/-5z-2a + (-3z)-3z-2a. 16. -4 + a-(-a)-f8. 17. 4a-5a+(-2a)+76-(-36). 18. _5+(-7)-3-8-(-17)-(-6). 19 12 a + 3a — 4z— 4a — (— 5z)-z + x — 6 a. 20. 3a + 4?/-(-5z)-2a-3?/ — 4z -(- a)+ # -(- z). 21. 22a -23?/ -(-24a). 22. 44 + 21 z -(-22). 23. 17a-(-lla) + (-13 6)-16&. 24. 21c-15c + 28d-(-6cT). 25. 18a-21a-10d + 8d. 54 Subtraction SUBTRACTION OF POLYNOMIALS ORAL EXERCISE 94. Subtract the following : 1. 111b. 7 oz. 11^ + 73 4. 12 mi. 20 rd. 12m+20r 8 lb. 5 oz. 8p + 5z 8 mi. 8m 2. 7 ft. 9 in. 7/+ 9 i 5. 3 ft. 7 in. 3/+ 7 i 4 ft. 3 in. 4/ + 3 i 5 in. 5i 3. 13 mi. 40 rd. 13m+40r 6. 5a + 2b 8x-5y 11 mi. 28 rd. llm+28?- 2g + 46 7a; + 4y 3a-2b x-9y Let the student check the results in the last example by adding the difference to the subtrahend. 7. 7a; 2 -3a; 11. 3a + 26 + 7c 4 a; 2 — 5x a + b + c 2a6 + 7c 8 ab + 9 c 17 6 + 2p 146 + 3p — 5x + 7y -Sx-Sy 12. 5a-86 + 7c 2a + 36-4c 13. 4a; + 3?/ — 7z 5x-2y-Sz 14. 6x —y — z 2x-\-y — z 10. 95. From these examples we derive the following rule : To subtract one polynomial from another, write the subtrahend under the minuend, with like terms in the same vertical column. Change the sign of each term in the subtrahend, and proceed as in addition. Examples 1. Subtract 6a; 2 - 3a; - 12 from 15 x 2 + 8 x + 1. Subtraction Check, x = 1. 15 a? + 8a;+ 1= 24 This result might be checked by adding the 6 a; 2 — 3 a;— 12=— 9 difference to the subtrahend. The sum should 9 a; 2 + lla;-f-13= 33 equal the minuend. Subtraction of Polynomials 55 2. From 6 mn+3 m 2 + n 2 take 5 n 2 + ra 2 — 3 ran. 3m 2 + 6mn+ n 2 , o , cr o Arrange the terms in descending powers of m. ra 2 — 3ra?i + 5n 2 _ & ... ,. ,.„ "•* _. , , — - Check. Add the difference to the subtrahend. 2 m 2 + 9 ra?i — 4 n 2 3. Subtract 4 a — 3 b -f c from 2 b — 3 c. 2b-3c 4 a — 3 b -f- c Let the student check the result. — 4a + 5 6 — 4c EXERCISE 96. Subtract, and check results as directed : 2. 8.1 r- 1.5 s . 3. 5 a + b 4 r+ .2s 8a 1. 13 ra + 40 r : llra + 89r 4. 5. 3a 2a-6 3a — b— c 2a +5c 6. a — 2b + c 5a-56 7. 2.5r-4s + 3 1.3r-2s 8. 8a 6a-6 + 3c 9. 21d + 14ra + 27s 16d + 18ra + 27s 10. 11 h + 41m + 56s 7 h + 59 m + 34 s 11. 36ra + 37r + 42?/ 25ra + 71r + 84?/ 12. 2a-36 5a -7 13. a ;2 + 3 a ;_7 5 x 2 — a; — 1 14. 2a+ 6 a — 36 + 7c 15. 56 -2 — 4a — c 16. Subtract 2 a — 3 b + c from a + b + c. 17. 4 m 2 — 5 mn — 7 n 2 minus 3 n 2 + ra 2 — 5 mn. 18. The subtrahend is — 2, the minuend is a -f b — 2. Find the difference. 19. The difference is a + b — c and the subtrahend is 2 a — 3 6 + c. Find the minuend. 56 Subtraction 20. From the sum of 4 a 2 + 3 ab + 7 b 2 and b 2 — 7 a 2 take a 2 + 6 2 - a&. Collect terms in examples 21 fo 25. 21. (_7) + 3-(-5)+12+(-12)-8. 22. 5a + 2&-36+(-26)-(-7a). 23. — 8ra+(— 3n)+4ra + 6n — 7ra+n. 24. 10+(-14)-6 + ll+(-4)-(-13). < 25. 7i> + 8g-(-3j9) + (-7g)+4p-llg. 26. From 4 a? 2 -f 2 +(10a-a;). 2. 7d+(4<2 + 5d). 7. 18&+(7&-9&). 3. 19/ + (12/ +7?*). 8. 2r+(10-3r). 4. 17x+(10x-y). 9. (2x-3y) + (-x-y). 5. 7* + (14* -5r). 10. (5p-7g)+(-3p+5g). 102. The expression 11 - (8 + 2) means that 8 + 2 is to be subtracted from 11, or 11 -(8 + 2)= 11 - 10 = 1. But 11 - 8 - 2 = 1. ... ii _(8 + 2)= 11 -8 -2. 60 Subtraction The expression 3 a -{-2 b — (a — 3 6) means that a — 3 b is to be subtracted from 3 a -+- 2 b. The rule for subtraction is : " Change the signs of the subtrahend and add the result to the minuend." To apply this rule in the present case we may remove the parenthesis, changing the signs of the terms within the parenthesis, and collect terms. 3 a + 2 b — (a - 3 6) = 3 a + 2 b — a + 3 b = 2 a + 5 b. Also, 3x 2 -x-(2x 2 -2x + 7) = 3x 2 -x-2x 2 + 2x-7 = x* + x-7. The student may verify the answer by ordinary subtraction. 3 x 2 — x 2 x 2 - 2 x + 7 x 2 + x - 7 A parenthesis inclosing any number of terms and preceded by a minus sign may be removed provided the sign of each term inclosed by the parenthesis is changed. ORAL EXERCISE 103. Remove the parentheses and simplify as much as possible : 1. 32 -(17 + 6). 11. 17 r - (21 r-14r). 2. 32 -(17 -.6). 12. 17r-(-21r + 14r). 3. 15 -(13 + 11). 13. -(7 s -13 s) -15 s. 4. 15 - (13 - 11). 14. - (x + y) — p -J- g. 5. a -(b -f-c). 15. (a — &) + (— a-f 6). 6. a—(b — c). 16. (a-6)-(-a + 6). 7. 12p-(3i> + g). 17. (2 + 3m)-(3-2ra). 8. 12p-(-3p-q). 18. (2-3m) + (3 + 2m). 9. 15m -(6m + 2 m). 19. 23a -(16 + 5a). 10. 15m -(-6m -2 m). 20. 23a +(- 16 - 5a). • 104. Sometimes one or more parentheses are inclosed within a parenthesis. In this case either the outer or the inner paren- Parentheses 61 thesis may be removed first. The beginner will find it ad- visable to remove the inner parenthesis first. = 18 x — {4y — [9aj — 2 y — 3*c + y] j (Removing ( ).) = l$x-\±y-§x + 2 y + 3x-y\ (Removing [ ].) = 18 x — 4 y + 9 # — 2y — 3x + y (Removing \ \ .) = 24 x — 5 y (Collecting terms.) EXERCISE 105. Simplify by removing parentheses and combining like terms. 1. ( x -y-z)-(2x + y-3z). 2. 25 -(3 + 4x2) + 6. 3. 1 + ra — w— (21 — ra+2n). 4. (^-»)-(a; 2 -2a; + 3)-(ar + 2a;-6). 5. ( a r J + x)-(^-l). 6. x 2 -\-2ax + a 2 -(x 2 -2ax + a 2 ). 7. 8m— (4m + 2 7i) + (5m — 6n). 8. a 2 -(a 2 -2a6) + (-2a5 + a 2 ). 9. (4:p-q)-[2p-(q-p)+2p}. 10. y + [(m-») + (m +/>)]. 11. y+[( m + n)-(w+j>)]. 12. y — [(m — n) — (j9 — n)]. 13. y — [(ra + w) — (ra— p)]. 14. 7a-26-[(3a-c)-(26-3c)]. 15. 2a-(3 6 + 2c) -{5&-3a-(a + &)+5c-[2a-(c-2&)]j. 16. 16-a>-{-7»-[8-9a>-(3-6a)]|. 17. « 4 -;4^ 3 -[6^-(4^-l)]j-(^ + 4^ + 6x 2 + 4 i r + l). 18. 4.04 a - [.275 y- (.5 b- 3.875 a) +3.6 y] -(.165 a-. 375 y). 19. ab - [(3 6ce - 2 a&) - (5 6ce - bef) + (3 06 - 3 6e/)]. 62 Subtraction Simplify: 20. l-[_(2-^)] + [4^-(3-n^)] + 4-(6x-5). 21. 3m-38n-(o7p + 15?)-(12|>-38? + 48n-50m). 1 06. Inserting parenthesis : 3 - 2 -f 3 = 3 + (- 2 +3) and 3 - 2 + 3 = 3 -(2 - 3). Also a+b—c+d=a+b +( — c + d) and a + 6 — c-|-d = a-f-6— (c — d). Let the student verify these results by removing the paren- thesis according to the rules of §§ 100 and 102. From these results we draw two conclusions : 1. The value of a polynomial is not changed if any of its terms are inclosed within a parenthesis preceded by a plus sign. 2. The value of a polynomial is not changed if any of its terms are inclosed in a parenthesis preceded by a minus sign, provided the sign of each term inclosed is changed. EXERCISE 107. Inclose in a parenthesis preceded by the plus sign the 3d and 4:th terms in examples 1 to 5. 1. a -f b — c + m. 3. am -f bx — ac — mx + 2 b. 2. 4 m — 3 x -f- y — 2 a + c. 4. a 2 +• 6 2 + m 2 — 2 mxc + 2 a#. 5. ?/ 3 + 3 ax 2 + 3 a 2 * -fa 3 - m 2 + & 2 . Inclose in a parenthesis preceded by the minus sign, the 2d, 3d, and 4th terms of examples 6 to 10. 6. p 2 + q — pq- 4- m 2 — n 2 — 3 mny. 7. m — n—p -\-pq — l y. 8. m 2 + 2 raw -f w 2 -f-p 2 — 2pq. 9. /> 3 — 3 p 2 g -f r/ 3 — n 3 . 10. y 2 + 2 po 2 + p 2 - n 2 + 3 pg. 11. Inclose in a parenthesis preceded by the minus sign, the last three terms of examples 6, 7, 8. 12. Inclose the last four terms of examples 8 and 10 in a parenthesis preceded by a minus sign. Equations 63 EQUATIONS INVOLVING ADDITION, SUBTRACTION, AND PARENTHESES ORAL EXERCISE 108. 1. If 4 x 4 3 x = 7, what does x equal ? 2. If 4 a; — 3 x = 7, what does x equal ? 3. If 2 x = a; 4 2, what does a? equal ? 4. If 3 x 4- 2 = 2 x 4- 3, what does x equal ? /SoZtfe Me following equations and check the results : 5. a?4-2a; = 3. 13. 8q = 15+(—7). 6. 2 a; -a; = 5. 14. 16^4- 9 a; = 50. 7. 7j9-5p = 6. 15. #4-3 = 12. 8. 7p + 5p = 12. 16. 2a; + 6 = 12. 9. 11 a; -7 a; = 8 -4. 17. 2 v + 5 v = 21. 10. 2 + 3a; = 8. 18. 17»-7n = 30. 11. 2r + *5 = 6. 19. 15 a; 4 6 a; = 42. 12. 4m— 4 = 4. 20. 15 a — 6a; = 18. 109. Solve 5 a; -(12 -a?)= 3 4(^4 3). Solution. 5x -(12 — x) = 3 4(^4 3). 5aj — 12 4se = 34a;4 3. ( Removing parentheses. ) 6 x — 12 = 6 + x. (Collecting terms. ) 5 x— 12 = 6. (Subtracting x from both members.) 5 a; = 18. (Adding 12 to both members.) x = 3f. (Dividing both members by 6.) The result may be checked by substituting 3f for x and simplifying. 6x8|-(12-3|) = 8+(8f + 8). 18 _ 8f = 3 + 6$ . 9| = 9f . EXERCISE 110. Solve the following equations and check the results: 1. 5a;- 1 = 14. 3. 7p-f = 13J. 2. 8a;4i = 16i 4. 9^-5^ = 36. 64 Subtraction Solve and check : 5. m-(-2m + 3)=6. 10. 11 x + 14 = x- 16. 6. 2s-5 = 4-(5-s). 11. 16c-(5 + llc)=5. 7. 5y-(3y + 3)=3. 12. 5 a; + 5 = 2x + 6. 8. 15a-10 a = 45. 13. 11 n — 23 = 7» — 4 9. 27&=31-(-20 6+4). 14. J -(-J + 3) = 3. 15. 4y — 11 = 3 — y. 16. 2a+(3 + 4)=a-(5 + 6). 17. m+(3+4m)=3m + 3. 18. a; — 14 = 14 — a:. 19. 4cc + (2a + 2)=2a;+(a} + l). PROBLEMS SOLVED BY MEANS OF ALGEBRAIC EQUATIONS 111. If there are two unknown numbers to be found in a problem, two distinct relations of the numbers must either be given or implied. Generally the method of making the equa- tion is as follows : 1. Introduce some letter as x, to represent one of the unknown num- bers, preferably the smaller one. 2. Express the other unknown in terms of x by using one of the two given relations. 3. Make an equation by using the other relation. EXERCISE 112. 1. The sum of two numbers is 24, and one of them is twice as large as the other. Find the two numbers. Solution. Let x = the smaller number. Hence 2 x = the larger number. Then x + 2 x = 24, (By the first condition stated.) or 3 x = 24. . *. x — 8, the smaller number, and 2 x = 16, the larger number. Problems 65 2. The sum of two numbers is 18 and one is five times as large as the other. Find the numbers. 3. The sum of two numbers is 40, and the larger exceeds the smaller by 10. Find the two numbers. 4. The difference of two numbers is 10, and the larger number is 3 times the smaller. Find the numbers. 5. Find two parts of 53, one of which exceeds the other by 11. 6. Find two parts of 28, one of which exceeds twice the other by 4. Suggestion. If x = the smaller part, 2 x + 4 = the larger part. 7. Find two numbers whose sum is 23 and whose difference is 8. 8. Find two consecutive numbers whose sum is 73. Suggestion. Since the numbers are consecutive the larger exceeds the smaller by 1. 9. Find two consecutive numbers whose sum is 33. 10. Find three consecutive numbers whose sum is 33. 11. Separate 153 into two parts of which the larger exceeds two times the smaller by 30. .12. If the sum of two consecutive numbers is 45, find the numbers. 13. A rectangle is 20 feet longer than it is wide, and its perimeter is 160 feet. Find its length and width. Suggestion. Let x = the number of feet in the width. Hence x + 20 = the number of feet in the length. Then x + x + (x + 20) + (x -j- 20) = 160. Let the student solve the equation. 14. A rectangle is twice as long as it is wide and its perime- ter is 150 feet. Find its length and width. 15. The length of a rectangular lot exceeds twice the width by 50 feet and the perimeter is 364 feet. Find its dimensions. 66 Subtraction 16. If n is the middle one of live consecutive numbers, how would you represent the other four numbers ? Find five con- secutive numbers whose sum is 45. 17. Three men divide $300 so that the second has $25 less than the first, and the third $50 more than the second. How many dollars does each man get ? Suggestion. Let x = the number of dollars the first receives. Hence x — 25 = the number of dollars the second receives, and x + 25 = the number of dollars the third receives. Then x + (x - 25) + (x + 25) = 300. Let the student solve the equation. 18. The combined weight of the largest steam locomotive and the largest electric locomotive in the United States is 381 tons. The steam locomotive weighs 57 tons more than 3 times the weight of the electric locomotive. What is the weight of each? 19. The combined cost of the Panama and Suez Canals was approximately 394 million dollars. The Panama Canal cost 5 million dollars less than 20 times as much as the Suez Canal. What was the approximate cost of each ? 20. It is 1274 miles further from London to New Orleans than it is from London to New York, and the sum of the two distances is 7740 miles. Find the distance from London to each place. 21. Two day-rate telegrams were sent from New York, one to Detroit and one to Winnipeg, Manitoba. The two messages cost $ 1.10. The message to Detroit cost 35 cents less than the one to Winnipeg. Find the cost of each. Hint. Change $ 1.10 to 110 cents. 22. Two six-word Marconigrams (wireless telegrams) were sent from London, one to New York and one to St. Louis. The message to St. Louis cost (in United States money) 36 cents more than the one to New York, and the total cost was $ 2.10. Find the cost of each. V. MULTIPLICATION ORAL EXERCISE 113. 1. What is the law of signs in multiplication ? (§ 49, 2.) Find the products : 2. (-8)(-2).2; (-2)2; (-2)'. 3. (-1)'; (-1)3; (-1)^; (-iy. 4. (-2)(-3) 2 ; (-2)*(-3); (-2)2.3. 5. 2 x 3 yd. ; 2 x 3 y ; 3 X 5 mi. ; 3 X 5 m. 6. 4 x 10 a ; 5x8?/; 7 x 2 a ; 4 x 3 ab. 7. a • ft ; a(- 6) ; (-a) b ; (- a)(- 6). 8. 7- (-3); 7- (-3 a); 5- (-4 a*/); 8-(-3a&). 9. _2.3; -2-3a; -4-76; -5-2a6. 10. -4- (-5); -4- (-5a); -4-(-7a6); -J.(-5xy). 11. 2. (-3 6); -4- 3 aft; - 5 • (- 9 x) ; 4 . (- 2 y). 12. 3 x 4x(-2); 3x4x(-2a); -4x36x2. 13. 2x(-3)x4; 2x(-3)x(-4); - 2 X (- 3) X (- 4). 114. The Law of Exponents in Multiplication. Define expo- nent and base. (§ 64.) Since 2 2 = 2 . 2 and 2 3 = 2 . 2 . 2, therefore 2 2 x 2 s = (2 • 2) x (2 . 2 • 2) = 2 5 or 2 2+3 . Similarly a 2 • a 3 = (a • a) x (a • a • a)= a 5 or a 2+3 . Similarly a 2 • a 2 • a^ = (a - a) x (a - a) x (a - a - a) = a 7 or a 2+2+3 . Also since a m = a • a ••• to m factors and a n = a • a — to ?i factors therefore a w • a" = a m+n . 67 68 Multiplication The equation, a m • a" = a w +", is the law of exponents for mul- tiplication stated in algebraic symbols. In words we have : In multiplying powers of the same base the exponent of any base in the product is equal to the sum of its exponents in the factors. Also since (a 3 ) 2 = a 3 > a? = a 6 , we have (a 3 ) 2 = a 3X2 , and in general (a m ) n = a mn . 115. The student must note that the law a m • a n = a m+n ap- plies only when the bases are the same. It must also be remembered that when no exponent is written, the exponent 1 is understood. Thus, x • x 2 = x 1+2 = x % . The base for any given exponent is the number symbol immediately preceding it. Thus, 3 ab % means 3 a • b • b. If it is desired that the exponent shall affect other pre- ceding numbers, a parenthesis is used. Thus, (3a&) 2 means 3 ab x 3 aft, and is read "the square of 3a6." (a + by is read " the square of the binomial, a + 6." 1. x 2 -x z = x h . 4. 2-2" = 2 n+1 . 2. (xy)*-(xyy=(xy) 7 . 5. (x + 2) 2 (a+2) 5 =(a;+ 2) 7 . 3. (_3)5.(_3)3 = (_ 3 )8 # 6> ( a 4)5 = a 20. ORAL EXERCISE 116. Find the indicated products : 1. a 2 - X 3 . 2. a 3 • a 4 . 3. ra 2 • m 5 4. y 7 ■f- 5. p* ■f. 6. Q n •?<• 7. r*. r 13 . 8. b- b\ 9. x p -X. 10. n r • n 2 . 11. c n+1 • c 3 . 12. X y • X 2y . 13. a x • a y . 14. d m • d n . 15. a 2m • a 2 16. a 2m • a m a. 17. m 1x . m hx % 18. X 2 • X • X 3 . 19. m 2 • m 3 • m 3 . 20. (a+6) 2 -(a+6) 3 . 21. j*2 . o*3 . f^ 22. (2 a) 2 • (2 a) 3 . 23. (2 + a) 2 .(2+z) 6 . 24. 4 3 . 4 2 • 4. Multiplication of Monomials 69 MULTIPLICATION OF MONOMIALS 117. Remember that in multiplying two or more monomials together : The factors of a product may be arranged in any order without chang- ing the value of the product. Thus, 2x3 = 3x2; 2x4x3 = 2x3x4; etc. The product of an even number of negative factors is positive, and the product of an odd number of negative factors is negative. Thus, (— a) x (— b) = ab ; (— m) (— m)(— i») (— m) or (— m) 4 = wt 4 ; (-<*)(-&)(- c) =-abc; (-a) 3 =-a 3 . 1. 2a 2 b x (-3a 3 6 4 c)=2 -(-3) . a 2 . a 3 • b • b 4 • c = -6o 5 5 5 c. 2.-5 ax 2 y X (- 3 x*y n ) = (- 5)(- 3)a • a; 2 • ar» • y • y n = 15 aa^y n+1 . 118. The preceding laws together with the law of exponents, § 114, give the rule. To multiply monomials : 1. Find the product of the numerical coefficients, keeping in mind the law of signs for multiplication. 2. Write after this product the product of the literal factors, giving to each letter an exponent equal to the sum of its exponents in the factors. Examples 1. (-3 a 2 b) • 5 ab 2 c = - 15 a?b z c. 2. 4 a n b • a n b 2 = 4 a 2n 6 3 . 3. a n b . ab = a n+1 b 2 . ORAL EXERCISE 119. Find the products of the following : 1. a 2 a 2 . 5. 4 # 2 . 3^. 9. b 2 c 2 • b 3 c. 2. b 2 b 4 . 6. 5 a 4 - 3 a 2 . 10. -6 2 c 2 .(-6 3 ). 3. x*x\ 7. a 2 b>ab. 11. 3« & .a c . 4. c 5 c 5 . 8. afy-ic 3 ?/ 5 . 12. .v 2a -x a . 70 Multiplication Find the products : 13. a 2 -a?>a\ 22. (- p)(- q)2r. 14. x • x 2 • x*. 23. (— xf • x. 15. 2yy(-y 2 ). 24. (-M) 4 . 16. afr-afc 2 . 25. (- 5) 3 . 17. 3rs-3rs-rs. 26. (— 4) 4 . 18. 5pqx(-p 2 q 2 ). 27. -3a6(-c). 19. c 2 d(- cd 2 )(-c). 28. -7j>(- jd)(- r). 20. x 2 f>2x*y\ 29. (2a)(-2 6)(-c). 21. (-p)(-q)(-r). 30. a 2 (- a) 3 . EXERCISE 120. Find the products of the following : 1. x 2 tf-2x*y\ 9. (-2 ma) (1.26V). 2. ia 4 (-|a; 7 ). 10. .32#2/(-llary). 3. 3a- 2a 2 - 6a 3 . 11. (- 80maf) (.05m?x). 4. a 2 6(-3a 3 c)(-6 3 c 2 ). 12. (- .2x*y 2 ) . hxy\ 5. 3(a+&) 2 -(a + &) 3 -(-4> 13. 2&a(- 5.5&Y). 6. '5oP#-2n&. 14. 12 a 2 z • J aa 4 . 7. ( _ 31 aty) . e a2/ 5 # 15 . .33^ m 2 n . 15 mw s . m7l . 8. (-3a 2 )x(-7aa 5 ). 16. £px* • 5byx 2 . 17. (-15a 2 rc).3&.(-22a6)2a 2 &. 18. (-3ac)(.33|ay)(acv). 19. .4 6..2 6 2 c-26c 2 . 25. (- 40a wl )(- .05 a*). 20. 4ia 2 c 3 .2|ac 4 . 1 Va 4 c. 26. m"" 2 • m 2 . 21. a m 'a n -a?. 27. .82r" 4 (- .4y). 22. a x+1 a z+2 . 28. c* • d'" 1 • c • d 2 " v . 23. yhf. 29. c T d y+1 • cd 2+I/ . 24. 3a n -|a. 30. x n ~ 2 y m -\- x n ^y m ' 1 ). Multiplication of a Polynomial by a Monomial 71 31. ( - \ « m41 ) • 7 a • 2 a 1 ""*. 39. (f^f* 4 • 3 c 2 -^ 4 -*. 32. i5d'"- n • 2d 2 "- m . 40. — m x_2 w 3 • lOm 3 "^^. 33. (- b p c<) • 9 6«c p . 41. 3(x 2 y(-2xy 2 ). 34. a m+ " • a m -». 42. 3 (2 a; 2 ) 2 . (- 3 a;). 35. 5 a 1 " 2 *. 3 a 4 *. 43. 4(2af) 3 (- 4 a 2 ). 36. y 2 z p -3y 2n - 2 -z n - p . 44. 6 (3 # 2 ) 2 ( - 4 a 2 ) 2 . 37. (-2a w+2n ).f a 4 " w . 45. 6(2a&) 2 (- a 2 b) 2 . 38. a 4n a 2w • a?x 2 • a n a m . 46. 5(3 xy)\ — xy 2 ) 2 . MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL 121. 4(3 + 5)= 4- 8 = 32. This result might have been found by multiplying the num- bers within the parenthesis separately by 4. Thus, 4(3 + 6)= 4 • 3 + 4 . 5 = 12 + 20 = 32. 53 j Ordinary arithmetical multiplication, if done without the m abbreviating process of "carrying," shows the same principle. The multiplication at the left, if written in a line, is §52 7(50 + 3) = 350 + 21 = 371. 371 The algebraic law that covers this case may be expressed in algebraic symbols thus, a (b -f- c) = ab + ac. 122. In words this law gives us the following rule : To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial and unite the results with their respective signs. Examples 1. Multiply 2 a 2 b - b 2 c + c 2 by - 3 a 2 b 2 . Multiplication Check, a = b = c = 2. 2 a 2 b - b 2 c 4- c 2 =12 - 3 a*b 2 ==-48 - 6 a 4 6 3 + 3 a 2 6 4 c - 3 a*6 2 c 2 = - 576 72 Multiplication 2. Multiply a m ~ l -f b n ~ l — c p ~ l by — abc. a m ~i + b n ~ l — c*- 1 — abc — a m bc — ab n c + abc p . ORAL EXERCISE 123. Multiply the following : 1. a(6 + c). 11. — 5x(x + y + z). 2. — x(a + 6). 12. — 5 x(— x + y — z). 3. — a(x — ?/). 13. —p(pq — r). 4. 2 6(3 a -f c). 14. r(st-rs). 5. 4 a 2 (3 x 2 - y). 15. pq(p 2 q - pq 2 ). 6.-3 a 2 (2 a + 6). 16. a\bc — ac + a). 7. £c( a + & _|_ c ). 17. -3h{k-hk + h 2 k 2 ). 8. — #(a — 6 — c). 18. 2 x(xy — xz — yz). 9. — 3x(a + b— c). 19. xy(x n ~i — y n ~ l ). 10. -4c(-2c + 3d-6e). 20. - 3x 2 y(x n ~ 2 y - xy n ~ l ). EXERCISE 124. Multiply the following : 1. r*(r + s). 4. -3a6(a 2 + 6 2 ). 2. a 2 (a& — ac). 5. — 4 x(2 x — . 5 y — 3 z). 3. -2aj(a 2 -2a-l). 6. -3«262( a + 6 + c). 7. a(« 2 + & 2 + <*&)• 8. .2 m(.2m 2 + .02 mn + .002 n 2 )n. 9. -3a(-4a 2 -f-2rc -i). 10. - 3 x 2 y 2 (2 xhf - 3 x 2 y 2 + 4 a^ 3 ). 11. 3 ra 2 (2 ra 3 - 7 ??i 2 — m). 12. -x-2/(21aj 2 2/ 2 -14^/ + 7)(-l). 13. 3 par( — pq — 5}or — 7 or). 14. 5 2 (5 + 5 2 + 5 3 ). Multiplication of a Polynomial by a Monomial 73 15 . _9a6c(-|a^|6-|c} 16. 5-137 = 5(100 + 30 + 7)=? "• S-f»-«X-W Multiply the folloioing : 18. 3a 2 -6 2 + 7c 2 - 2 a%W 19. 6^-3^-9^ + 18 .3x 20. .6x 3 -.$x 2 y + 2xy*-2y 3 .bxY 21. 2a 2 6-3cd 3 + £ac 3 -6acW Simplify the following : 22. 4:(2x-7y)+2(x + 14:y). 23. 4a(a& + 6c + ca)-2 6(a 2 + 2ac). 24. «(« 2 -a^ + 2/ 2 )+2/(^ 2 -^ + 2/ 2 )- 25. a(a 2 + «2/ + 2/ 2 )- y(x* + xy + ?/ 2 ). 26. (or* - 3 afy)*/ 3 ~(y z -Sxy 2 ) x*. 27. 12(i^-i2/ + i 2 )-16(^ + i2/-i 2 ). 28. a-2[3a-6-2(6-a) + 3(a-26)]. 29. 2x-8z-3[2y-(2x-z)]-3(x-y-z). 30. 4a(6-3)-56(a-2)+a6 + 7(a-6). 31. a(b — a + d) — 6(a + c - d)+c(a+ b — d). 32. 3 -6a(6-c)-26(9a-c)-2c(6-9c). 33. p[q(s + *) - tf] - «(pg - 1) + *(4 - pq) + j9S*. 34. (8c 2 + 24cd 3 -12c 2 .r-3)|c n d». 35. (3& 2 -6c 2 + 9&c)(-p*c*). 74 Multiplication Simplify : 36. (7 a n - 3 a*" 1 - 2 a*" 2 ) ( - .4 a"" 2 ). 37. (9 x p ?/« — 4 x^y*' 1 +■ 3 x^yi-^y 2 . 38. (8 a 1 " 2 - + & 3_n )(- 5 a 3m 6"). 39. (# m + y p +■ z«)ajy». MULTIPLICATION OF A POLYNOMIAL BY A POLYNOMIAL 125. 1. To multiply 32 by 4, we may first multiply 2 by 4, then multiply 30 by 4, and add the partial products. Thus, 32 = 30 + 2 4 = 4 128 = 120 + 8. 2. To multiply 32 by 24, we may first multiply 30 + 2 by 4 and then by 20, and add the partial products. Thus, 32 = 30 + 2 24 = 20 + 4 128 = 120 + 8 64 = 600 + 40 768 = 600 + 160 + 8. 3. To multiply 2a + 36 by 3 a 4- b, we first multiply 2 a + 3 b by 3 a and then by 6 and add the partial products. Thus, 2 a +3 b 3a + b 6 a 2 + 9 ab 2 aft + 3 6 2 6 a 2 + 1 1 a& + 3 6 2 . 4. Multiply or 5 - 2z 2 - 3x + 2 by 2 a? - 3. An orderly arrangement of the X 3 — 2 X- — 3x-\- 2 terms of the polynomial and of the 9 o partial products is desirable. It is — — a 2 i — A~ usual to arrange in descending powers x ~ ~~ """ of some letter (x in the present case), — 3ar +- off +- \)x — b an( j t0 arran g e the terms of the partial 2a? 4 — 7a? +13 a; — 6. products with like terms in the same vertical column. Multiplication of a Polynomial by a Polynomial 75 126. These examples lead to the rule. To multiply one polynomial by another: 1. Arrange the terms in descending powers of some letter (or alpha- betically). 2. Multiply the multiplicand by each term of the multiplier, writing like terms in the same column. 3. Add the columns of like terms and join the results obtained with their respective signs. 127. To check the answers we may proceed as in addition (§ 84), by using arbitrary values of the letters. Examples 1. Multiply x 2 -3x-2hj x + 5. Multiplication Check, x = 2. a?-3x -2 =- 4 x + 5 = 7 x^^Sx 2 - 2x -28. 5a 2 -15a; -10 ic 3 + 2a; 2 -17a;-10 = -28. Substituting 1 for x in the above example will not, check the exponents. (Why ?) Hence it is better to use some other small number, as 2. 2. Multiply 3 a 3 6 - 2 a 2 b 2 + ab 3 by 2 a 2 - 5 b 2 - ab. Multiplication Check, a = b = 2. 3 a 3 6 - 2 a 2 b 2 + ab 3 = 32 2a 2 -ab -5b 2 =-16 6ft 5 6-4ft 4 6 2 + 2ft 3 6 3 -512. -3ci 4 6 2 + 2a 3 6 3 - a?b 4 -15ft 3 6 3 +10a 2 //-5ftfr 5 6 a b b - 7 a A b 2 - 11 aW + 9 a 2 b 4 -5 a& 5 = - 512. EXERCISE 128 Arrange conveniently for multiplication : 1. (3x* — ±y 2 x — x 2 y + 4 f) (2 xy - 3 y 2 + x 2 ). 2. (4ft6 2 -h6 6 3 -3a 2 6-h7ft 3 )(6 2 + a6 + ft 2 ). 3. (3-ar» + 2a; 2 -x)(a;-3x 2 + 2). 76 Multiplication Multiply the followiny : 4. a + 3 6. c - 2 8. r + 2 10. 2 a; + 1 a — 5 c — 6 r + s 2x — 1 ic + 2/ 9. m + n 11. a; + 1 x — y m +p x + 1 15. 2 6 2 - 2.4 6 + 1.6 106 +20 16. a + 6 — 2 q- 6 + 2 17. m 2 — mn + n 2 m + n 5. 12. a; + 3 7. x + 2 a? -a + 2 a +2 13. x 2 + a; + 1 a> -1 14. 2 a 2 -3a -5 5a -7 18. (2x 2 + l)(2a; 2 -l). 22. (2 a; + 3)(4a; 2 - 6a; + 9). 19. (x + 1) 2 . 23. (3aj-7)(3a? + 7). 20. (a;-4) 2 . 24. (2 R + 3) (72 - 1). 21. (» + l)(a; + 3). 25. (2 m + £>)(- 2 m +p), 26. (l-3x 2 + x)(x 2 + l-x). 27. (5a 2 -3a6-2 6 2 )(a 2 + 2a6). 28. (x 2 + xy + y 2 )(x 2 -xy + y 2 ). 29. (3a 1 - 5 a6 - 2 6 2 )(a 2 - 7 ad). 30. (a; 2 + 7a;-5)(aj 2 + 5-7a;). 31. (af-5aa;-2a 2 )(a; 2 + 3aa; + 2a 2 ). 32. (c 4 - c 2 )(c* + c). 33. (7p 2 -3g 2 )(4p 2 + ? 2 ). 34. (x 4 — afy + x 2 y 2 — xy 3 + 2/ 4 )(a? + y). 35. (m 2 + n 2 + 1) 2 — mw — m/> — np)(m + w + 1>). 36. (3 a; 2 — 2 a^ + ?/ 2 )(3 a; 2 + 2 an/ - ?/ 2 ). 37. (a 3 -aV + a^-x 9 )(o + 4 38. (8a 3 + 4a 2 6 + 2a6 2 + 6 3 )(2a-6). 39. (a + 6 + a; + y)(a + 6 — x — ?/). 40. (a + 6 + c + fi)(a-6 + c-d). Multiplication of a Polynomial by a Polynomial 77 41. (a? + l)(.r 2 + 2)(:c 2 + 3). 42. (2a?-3)(3« + 7)(6«- 5), 43. (3z + 5)(7a- + 5)(2a;-l). 44. (3 a + 2 6)(a -6)+(4a + 5 6)(2 a + 3 6). 45. (w-fv)(2i;-w)-f(M-t;)(H2M). 46. ( x + ±)(x-2)-(x + 2)(x-l). 47. (3 a + 5)(2 a - 3)(a - l)-(x - l)(a + 2)(ic - 3). 48. (a + b) (c - d) - (a - 6)(c + d). 49. (2 aa 2 + 3 a# 2 )(2 az 2 - 3 ay 2 ). 50. (4 b 2 x 2 + 5 ch/ 2 ) 2 . 55. (aj° + #*) 2 . 51. (« + 6) 3 . 56. (af + a;) 2 . 52. (Sx + Ayf. 57. (a; 2a + af + l)(af - 1). 53. (3 aa - 4 6?/) 3 . 58. (a B+1 + a B + a"- 1 )(a 2 + a). 54. (a* - 6 w )(a* + &"). 59. (a 2n + 2 a n + 4) (a n - 2). In examples 60 to 71, A = x*-2x + 4,B = x> + 2x + 4,C=x-2,D = x + 2. Perform the indicated operations: 60. A-B. 64. £ • C 2 . 68. Z) 2 - A 61. A • Z>. 65. B 2 . 69. C 2 - £. 62. A 2 . 66. C 2 • Z) 2 . 70. fe - C 3 . 63. A -IP. 67. A'D-BC. 71. Zte + Ca; -(-4 + B). 72. From the product of x — 3 times a; 2 -f a; — 2, subtract the sum of x 3 -f- 5, and 3 x 2 — 7 a?. 73. Multiply 2 a; - a,- 3 + 7 by the sum of 7 — a; and 2 a; — 10. 74. Solve 7(x + 2) - 3(x - 1) = 2(a> - 1) + 25. Solution. 7(se + 2)- 3(x - 1) = 2(x - 1) + 25. 7z + 14-3z + 3 = 2z-2 + 25. (Why?) 4 x + 17 = 2 x + 23. (Collecting terms. ) 2 z + 17 = 23. (Subtracting 2 z from both members of equation.) 2 a; = 6. (Subtracting 17 from both members of equation.) x = 3. (Why?) 78 Multiplication Solve the following equations : 75. 2(a + 5) = 20. 76. 2(z + 2)+ 3(3 a -3) = 6. 77. S(x- l)+7 = 11. 78. 2(r - 1) + 2(r - 2) = 3(r + 3). 79. 3(^-5)4-8 = 18. 80. 15(x - 3) - 17 = 103. 81. 8(5 x- 37)- 4(3 x -17)= 20. 82. 6(a?-5)H-2a? = 6aj-2(aj-f 10). 83. 17(m-17)-17 = -51. 84. 18a-2(3 + 5a)=10. 85. 7(3 p - 2) + 5(p - 3) - 4(p - 17)= 110. 86. 3(aj-5)- 4(a>-2) + 6 x =15. TYPE FORMS IN MULTIPLICATION 129. Certain multiplications occur so frequently that it is helpful to be able to write the products at sight. Seven such special products are given. They form a sort of algebraic multiplication table, and should be thoroughly learned and understood. The Square of the Sum or of the Differ- II. a - b a — b 130. Types I and II. ence of Two Numbers. I. a +b a +& a 2 H- ab 4- ab 4- b 2 a 2 — ab — ab-\-b 2 a 2 4- 2 ab 4- b 2 a 2 -2ab + b 2 (a 4- b) 2 = a 2 + 2 ab 4- ft 2 . (a - 6) 2 = a 2 -2 a& 4- V. Type I may be stated in words : The square of the sum of two numbers equals the square of the first plus twice the product of the first by the second plus the square of the second. ab 6 2 a* ab Type Forms in Multiplication 79 Let the student state Type II in words. The product (a + b) 2 = a 2 + 2 ab + b 2 may be represented by a figure. a Examples 1. (2 x 2 + 3) 2 =(2 x 2 ) 2 + 2(2 z 2 ) • 3 + 3 2 = 4 x* + 12 x 2 + 9. a 6 In applying the type form to find the square of the binomial 2 x 2 + 3 we note that the a of the type is to be replaced by 2 x 2 , and b by 3. Therefore in the second member of the equa- tion, (a + b) 2 = a 2 + 2 ab + 6 2 , we shall put 4 a 4 for a 2 , 12 x 2 for 2 a6, and 9 for b 2 . 2. (2 a - 3 ?/) 2 =(2 x) 2 - 2(2 s)(3 y) + (3 y) 2 = ±x 2 -12 xy + 9 y\ Explain. 3. (x 2 — ±y) 2 = x A — 8x 2 y + 16y 2 . Explain. 4. 132 = (10 + 3) 2 = 100 + 60 + 9 = 169. Explain 5. (-a + 6) 2 = [(-a)+ 6] 2 = a 2 - 2 a& +6 2 . Explain. 6. (-a-6) 2 = [(-a)+(-6)] 2 = a 2 + 2a& + & 2 . Explain. EXERCISE 131. Square the following binomials by inspection, using Types I and II : 1. (x + y) 2 ; (x-y) 2 ; (y - x) 2 . 2. (c - a) 2 ; (c + a) 2 ; (a - c) 2 . 3. (r + s) 2 ; (r-s) 2 ; (-r + s) 2 . 4. (m + n) 2 ; (m — rc) 2 ; (— m — n)\ 5. O + 9) 2 ; 0>-?) 2 ; fe-p) 2 - 6. 52 2 =(50 + 2) 2 ; 48 2 = (50 - 2) 2 . 7. 25 2 = (20 + 5) 2 ; 25 2 = (30 - 5) 2 . 8. (a + 2) 2 ; (a-2) 2 ; (2 - a) 2 . 9. (4+6)2; (4-6)2; (6-4) 2 . 80 Multiplication 10. (z + 5) 2 ; (a -5)2; (_x-5)2. 11. (6 + n)2 ; (6 - n)2 ; (- 6 - n)\ 12. (a+7)2; (a; -7) 2 ; (7 - a;) 2 . 13. (a: 2 + 3)2; (aj 2 -3) 2 ; (3-af)* 14. (2 a+36)2; (3a-26)2; (26-3a) 2 . 15. (az + y) 2 ; (ax + y 2 ) 2 ; (a» + 2Z 3 ) 2 . 16. (3 a + 2)2. 22. 492 = (50 - 1) 2 . 28. (5 a 3 - 2 6 4 ) 2 . 17. (4 + 2/)2. 23. (-a -by. 29. (_9p-3g) 2 . 18. (8 - m) 2 . 24. (4a-56) 2 . 30. (2^ + 3) 2 . 19. (a*/ + z) 2 . 25. (a -lO) 2 . 31. (m 2 -^) 2 ' 20. (_2 + x2) 2 . 26. (7c-4d 2 )2. 32. (7 — 4a£)». 21. 972 = (100-, 3) 2 . 27. (2 w + u) 2 . 33. (7 ahj - 3 aa)2. 34. (20i) 2 = (20+^)2 37. (2z + $yy. 35. (3 6c - 2 cd) 2 38. 322 = (30 + 2) 2 . 36. (1«-^) 2 . 39. 65 2 =(60 +5) 2 . Of what binomial is each of the following trinomials the square ? 40. x 2 + 2xy + 2/ 2 . 45. 9 m 2 - 24 mp + 16p 2 . 41. m 2 + 12rap + 36p 2 . 46. 9 - 6 a + a 2 . 42. a; 2 + 4 a +4. 47. 4a 2 — 4a + 1. 43. r 2 - 14 r + 49. 48. a; 4 + 2a; 2 + 1. 44. x 2 y 2 — 6a^ + 9. 49. x 2 y 2 — 16 xy + 64. 132. Type III. The Product of the Sum and the Difference of Two Numbers. a + 6 a — b a? + ab -ab-b* a 2 -62 (a + 6)(a-6)=a 2 - G 2 Type Forms in Multiplication 81 Type III may be stated in words : The product of the sum and the difference of two numbers equals the square of the first minus the square of the second. Examples 2. (3* - 5y) (Sx + 5y) = (3x) 2 -(By)* = 9x 2 -25 y\ In this example a of the type form is replaced by 3 x, and b, by 5 y. Therefore, in the product we shall have to replace a? by (3 x) 2 , and b 2 by (5 y) 2 . This will give the result obtained above. 3. (x + 2 y)(2 y - x) = (2 y + x)(2y - x)=±y 2 - x 2 . 4. (- 2 + *)(- 2 - *)=[(- 2)+ «][(■- 2)- a]= 4 - aft EXERCISE 133. Multiply by Type III: 1. (x + y)(x-y). 11. + 2?)(p_2?). 2. (c + d)(c — d). 12. (7i-f 5fc)(ofc-ft). 3. (r + s)(r-s). 13. (l + 4m)(4m-l). 4. (a + 5)(a-5). 14. (2a+36)(2a-36). 5. (4 + «)(4-a). 15. (3*-fr2y)(2y-3*). 6. (2a + 6)(2o-6). 16. 22 xl8 = (20+2)(20-2). 7. (a* + 2)(tf-2). 17. 27x33=(30-3)(30+3). 8. (3«2 + 2)(3a2-2). 18. 49x51. 9. (2x + y)(2x-y). 19. 68x72. 10. (3a + c)(3a-c). 20. 103x97. In ivhich of the examples 21 to 28, may we apjily Type III ? Give reason in each case. 21. (2a + 3 6)(2a-36). 25. (-a + &)(& + a). 22. (2 a + 6)(2 a 2 - 6). 26. 29 x 31. 23. (x + y)(x 2 - y 2 ). 27. 25 x 35. 24. (1 + x)(\ — »). 28. (m - ?i)(a - 6). 82 Multiplication Multiply : 29. (a 2 — 6)(a 2 + b). 33. (ab + cd)(cd - ab). 30. (2*-z 2 )(2;c+z 2 ). 34. (3 a 2 - 6) (3 a 2 + 6). 31. (im + ir)(im-ir). 35. (- 2 + 3a)(2 + 3 a). 32. (2a 3 -6 2 )(2a 3 + 6 2 ). 36. 87-93. 37. (a + 6 + c)(a + 6 — c). Solution, (a -j- 6 + c) (a + 6 — c) = [(o + 6)+c][(a + 6) -c] = (a + b) 2 -c 2 = a 2 + 2 a& + & 2 - c 2 . 38. (a- b + c)(a + b + c). Hint, (a - b + c)(a + b + c) = [(a + c)— 6][(a + c) + 6], etc. 39. (a — & + c)(a + 6 — c) = [a — (& — c)][a + (6 — c)], etc. 40. (a — 6 + c)(a — b — c). 41. (m — n +p)(m ■+- w — p). 42. (/)-29 + 3r)(j9-f-2g-|-3r). 43. (2a;-y-3«)(2a;-y + 3z). 44. (2x — y — 3z)(2x + y + 3z). 45. (x 2 + y 2 -\-xy)(x 2 + ?/ — xy). 46. (a-^)(a + ^)(a 2 + ^ 2 ). 47. (x - y)(x + y)(x 2 + y 2 )(x A + y 4 ). 48. (^+2/ n )(« n -2/ n )- 49. (m* — rn^Xm 2 * + m*>). 50. (a?' + p 2 )(a; + p)(a; — #). 51. [(a-l)(a+l)]> 52. (2 6-c)(2 6 + c)+(c-2 6)(c + 2 6). 53. [(x + y)(x-y)Y + [(y-x)(y + x)J. 54. (2r-3o 2 )(3a 2 + 2r). 55. (-r-Ss)(r-?>s). 56. (2r 2 -4s)(-4s-2^). Type Forms in Multiplication 83 Write each of the following expressions as the product of two binomials : 57. a 2 — 4. 61. x 2 y 2 -a 2 b 2 . 58. 9a 2 -16 6 2 . 62. xhfz 2 - 25. 59. 36 -Six 4 . 63. 25-16. 60. 4a 2 -16 6 2 . 64. x 2n 134. Type IV. The Product of Two Binomials having a Com- mon Term. The two binomials are of the form x + a and x + b. By multiplying, and arranging in order of powers of x, we have x + 2 x + a x + 3 x + b x 2 + 2x 3a + 6 x 2 + ax bx + ab x 2 + 5x + 6. x 2 -f-(a + b)x + ab. This gives (x + a)(x + b) = x 2 -f (a + &)* + a&. In words : x The product of two binomials having a com- mon term equals the square of the common term, plus the product of the common term by the sum of the other terms, plus the product of the other terms. The product (x + a){x + b) = x 2 + (a + b)x + ab, which is the same as x 2 + ax + bx + ab, may be represented by the figure. Examples 1. (x + 3)(x + 7) = x 2 +10x + 21. Here a = 3 and b = 7, a + 6 = 10 and a& = 21. 2. (a? - 3)(ar+ 2) = a; 2 + [(- 3) + 2> +(-3) X 2=x 2 -z-6. 3. (x + &)(aj - 2) = .T 2 + (6 - 2)x - 2 6. 4. (.T — p)(o; — r) = x 2 — (/) + r) 49. (ay — 55)(o# + l). 30. (6 2 - 3)(& 2 -2). 50. (xy — 5 z)(#y —3 2). Type Forms in Multiplication 85 51. (2 x + 7 y) (2 x + 3 y). 56. (10 + 2)(10 + 8). 52. (2x + 7y)(2x + 5). 57. (60 + 2)(60 + 1). 53. (ar + 5 x)(ar - 7 x). 58. (30+ 5)(30 + 6). 54. (ar + 5x)(ar-7y). 59. (10 +6)(10 + 2) ; 14x16. 55. (a2x + 5x)(a 2 x + 3x). 60. (20 + 3) (20 + 4) ; 26 X 28. 61. Explain how Types I, II, III may be regarded as spe- cial cases of Type IV. 136. Type V. The Square of a Polynomial. a + 5-f-c a + b +c a 2 + ab -h ac a& + 6 2 + be ac + bc + (2^+3 w+4r) 2 . 3. (z 2 + y 2 - z 2 ) 2 . 7# (tf + tf^iy. n. ( a 2 - 2 ao + 6 2 ; 2 . 4. (a& + bc + ac) 2 . 8. (aj — xy + 2) 2 . 12. (a + b + c+a") 2 . 13. (m + n-p-q)\ 17. (a - 6 + 2 c + d) 2 . 14. (a-;?/-z-w) 2 . 18. (3a-2 6+5c) 2 . 15. (10 m + 5 »i + 6) 2 . 19. (xy — yz — zx)*. 16. (2m + 3n-4g+p) 2 . 20. (a& - 6c 4- cd - da) 2 . 138. Types VI and VII. The Cube of a Binomial. By actual multiplication we can find the value of (a -f 6) 3 to be a 3 H- 3 a 2 6 + 3 a6 2 + 6 3 . Let the student perform the mul- tiplication. This gives us Type VI : (a + by = a 3 + 3 a 2 b + 3 aft 2 + ft 3 . Similarly, Type VII : (a - b) 3 = a 3 - 3 a 2 b + 3 aft 2 - ft 3 . In words : The cube of the sum of two numbers equals the cube of the first plus three times the square of the first multiplied by the second, plus three times the first multiplied by the square of the second plus the cube of the second. Let the student make the corresponding statement for Type VII. Examples 1. (3 x + 2 yf = (3 xf + 3(3 x)\2 y) + 3(3 x) (2 y y + (2 y y = 27 x 3 + 54 afy + 36 xf- + 8 ?/ 3 . 2. (2 a 2 - 6) 3 = (2 a 2 ) 3 - 3(2 a 2 ) 2 6 + 3(2 a 2 )6 2 - 6 3 = 8a 6 -12a 4 6 + 6a 2 6 2 -6 3 . 3. (-a + 2) 3 =(2-z) 3 = etc. Summary of Type Forms 87 4. (_a_6)3 = [C-a) + (-&)] 3 = (_a)3+3(_a) 2 (-&)f3(-a)(-&) 2 + (-&) 3 = -a 3 -3a 2 o-3a& 2 -& 3 . EXERCISE 139. Multiply by Types Viand VII: 1. (x + y)\ 11. (a + 1) 3 . 21. (be - a& 2 ) 3 . 2 (a - y)» 12. (a + 2) 3 . 22. (6 + 5 a 3 ) 3 . 3. (c + d) 3 . 13. (2 a- l) 3 . 23. (9 m 3 - 5p 3 ) 3 . 4. (c-d) 3 . 14. (a 2 + a) 3 . 24. (10 - a 2 ) 3 . 5. (m + 2) 3 . 15. (-a + &) 3 . 25. (2p 2 - 3) 3 . 6. (m-3) 3 . 16. (-a-26) 3 . 26. (6 - 5p 4 ) 3 . 7. (2 a + 1) 3 . 17. (2a + 3) 3 . 27. (m 2 - p*qf. 8. (a 2 + l) 3 . 18. (ab + bc)\ 28. (7a 2 -2) 3 . 9. (a 2 - 2) 3 . 19. (abc - l) 3 . 29. (ab - 6 3 ) 3 . 10. (4 + a) 3 . 20. (6 2 c-ao 2 ) 3 . 30. (a^z - 3a? 2 z) 3 - SUMMARY OF TYPE FORMS 140. The student should carefully memorize the following type forms that have been developed in this chapter : I. (a + 6) 2 = a 2 + 2a& + & 2 . II. (a-6) 2 = a 2 -2a&+6 2 . III. (a-\-b)(a- b) = a 2 - ft 2 . IV. (x + a)(x+b)=x*+(a + b)x+ab. V. (a + b + c) 2 = a 2 + & 2 + c 2 + 2 a& + 2 ac + 2 &c. (a -h &- c) 2 = a 2 + ft 2 + c 2 + 2 a&- 2 ac- 2 6c. VI. (a + a) 3 = a 3 + 3a 2 & + 3a6 2 +& 3 . VII. (a - 6) 3 = a 3 - 3 a 2 6 + 3 aft 2 - 6 3 . Let the student state each type form in words. 88 Multiplication REVIEW EXERCISE 141. Give the answers to examples 1 to Jfi by referring each to the proper type form : 1. ( [ax — by) (ax + by). 21. (x + y-z)(x + y + z). 2. { [ax — 3)(ax — 4). 22. (x-y-z)(x + y + z). 3. 1 [ax - 3)(ax - 3). 23. (x — pr + q) 2 . 4. [(a + b)(a - &)]*. 24. (v + 5 wu)(/v — 5 ww). 5. ( [5dy -3f. 25. (v -f- 5 wy) 2 . 6. [1-2 m) 2 . 26. (2 mn — Ipq) 2 . 7. [ab - 3 c) 2 . 27. (a — b + x)(a + b — x). 8. [a 2 -3x)\_-(3x-a 2 )']. 28. (x 2 + 7)(x 2 -8). 9. (2p + q)(-2p + q). 29. (7m-6#) 2 . 10. (2a-7)(2a-9). 30. (2/ + 2)(y - m). 11. (2 a + 6 - 3 c) 2 . 31. (a + m)(o+p). 12. (3 xy- 2 z)\ 32. (2c-l + d)(2c-l-d) 13. (3xy-2zf. 33. (2 c - 1 + c?) 2 . 14. (3xy + 2z)(3xy-2z). 34. (2 a 2 - 3 6) 3 . 15. (4 + 3 a 2 ) 2 . 35. ( r + 2i-3) 2 . 16. (m — w)(ra + n)(m 2 -f w 2 ). 36. (d» - 6) 2 . 17. Jra — n) (m -h n) (m 2 — w 2 ). 37. (a 2 a 2 + 5) (a 2 x 2 - 3). 18. (2 a - 6 2 ) 3 . 38. (a> + 2y-c-2d) 2 . 19. (a + 6 + 3) 2 . 39. Qf + a;) 2 - 20. (3 3)-(a+-l) 2 = 9. Solution. (a + 2)(a + 3) -(a + l) 2 = 9. (a 2 + 5 a + 6)-(a 2 + 2 a + 1)=9. a 2 + 5 a + 6 - a 2 - 2 a - 1 = 9. 3a + 5 = 9. 3a = 4. a = f 71. (a? + l) 2 = a 2 + 12. 72. 12m 2 +-2m + l-12(m 2 +-l)=0. 73. 3 + ^ 2 -(9 + 2/ 2 ) + 22/ = 0. 74. 4(z-l)+l = 7-2(2z-3). 75. 17(l+aj)-8(aj + 2) = 26. 76. -3(a?4-2)rf7(a? + l)=3. 77. 3(m + l)=2(m + 3). 78. 7(a?— 1)— 2(a? + 2)=aj — 3. 79. 5(«-2)+2(aj-f 3)=17 4-2(1-*). 80. A certain fertilizer contains 1^ times as much potash as nitrogen and 4 times as much phosphoric acid as nitrogen. Find the amount of each element in 130 pounds of fertilizer. 81. If 10 is added to a certain number, the sum is three times the original number. Find the number. 82. One number is 32 greater than another. When 3 is added to each number the greater is 5 times the smaller. Find the original numbers. Solution. Let x = the smaller number. Hence x + 32 = the larger number. Also x 4* 8 as the smaller number increased by 3, and x + 35 = the larger number increased by 3. Then x 4- 86 = 6(x + 8), (By the conditions of the problem.) or a; + 35 = 5* + 15. (Why?) .-. 35 = 4 z + 15. (Why?) .-. 20 = 4*. (Why?) /. x = 5, the smaller number, and 5 + 32 = 37, the larger number. Review Exercise 91 83. If a certain number is multiplied by 8 and the product is increased by 14, the result exceeds 5 times the original number by 28. What is the number ? 84. A boy had twice as much money as his sister ; but after each had spent 12 cents he found that he had 3 times as much as his sister. How much had each at first? 85. One number is 5 times another. If 15 is added to each number, the greater will be 3 times the less. Find the original numbers. 86. A rectangle is 3 times as long as it is wide. If both dimensions are increased by 4 inches, it will be twice as long as it is wide. Find its dimensions. 87. A rectangle is 3 inches longer than it is wide. If both dimensions are increased by 3 inches the area will be increased by 54 square inches. Find the dimensions. 88. A box of candy contained a certain quantity at 35 cents a pound, twice as much at 50 cents a pound, and 3 times as much at 55 cents a pound. If the mixture cost $ 3, how many pounds of each quality did it contain ? Solution. Let x = the number of pounds @ 35 ^. Hence 2x — the number of pounds @ 50 ?, and 3 a; = the number of pounds @ 55^. Then 35 x + 50 . 2 x + 55 • 3 x= 300. Let the student complete the solution. 89. A grocer blended a certain quantity of coffee at 35 cents a pound with twice as much at 32 cents a pound and 4 times as much at 25 cents a pound. If the total value was $ 15.92, find the number of pounds of each in the mixture. 90. A certain number of 4^ stamps, 3 times as many 2^ stamps, and 10 times as many 1^ stamps cost $2.00. How many of each were bought ? VL DIVISION 142. Division has been denned as the process of finding one of two factors when their product and the other factor are given. The product is the dividend, the given factor is the divisor, and the factor sought is the quotient. 143. What is the rule for dividing signed numbers ? (See § 53). ORAL EXERCISE 144. Divide the following : 1. 8-K-2); -8 + 2; -8-(-2). 2. _ 18-6; -18 -(-6); 18 -(-6). 3. 36-h(_9); -36 -5- (-6); 36-*-(-4). 4. 8 +(-12); -9 + 12; -5 + (-15). 5. -lO-s-5; 5 +(-10); -5 + 10. 6. 12 ft. --3; 12/+ 3; 12 an- 3. 7. $10 + 2; 10eZ+2; 10z + 2. 8. 12 yd. -3; 12 y + 3; 12 6-3. 9. 20 mi. + 4 ; 20 m + 4 ; 20 x + 4. 10. 3 x 6a; 18a + 3; 126-3. 11. 5 x7r; 35r + 5; 27fc + 3. 12. 21a -7; 28p + 2; 50s -25. 13. $18 + $9; 18d + 9d; 15d + 5d. 14. 26ct.-2ct.; 26c + 2c; 18r + 6r. 15. 24T-4T; 24* + 4*; 28L + 7L. 16. 5 x8a; 40a + 8a; 40a + 5. 92 Division 93 17. 8 x7ft; 56k+7k-, 56k + 8. 18. 4x(-2o); -8a-*-(-2a); -8a-r4. 19. 7x(-3a); - 21a-s-(- 3a) ; -21a-=-7. 20. -8x7<; -56*-*-(-8); -56*-^7*. 21. -2x(-5r); 10r-s-(-2); 10r-=-(-5r). 22. 3ax(-2a); -6a 2 --3a; -6a 2 --(-2a). 23. 21fc-*-(-3fc); -8a 2 -2a; - 18a -*-(- 6«). 145. Integral Algebraic Expression. An algebraic expression is an integral algebraic expression if there are no literal numbers in a denominator. Thus, a 2 + 2 ab, \ x — *-, a + are integral algebraic expressions, and — , -, a are fractional algebraic expressions. 3 6a c 146. The Law of Signs in Division. The student should remember that in dividing one number by another : 1. The quotient of two numbers having like signs is positive. 2. The quotient of two numbers having unlike signs is negative. 147. The Law of Exponents in Division. Since a 3 • a 2 = a 5 , therefore a 5 -s- a 2 = a 3 or a 5-2 , and a 5 -f- a 3 = a 2 or a* -3 . Similarly •.• a 8 • a 3 = a 11 , .*. a 11 -5- a 8 = a 3 or a 11-8 . and a 11 -s- a 3 = a 8 or a u ~ 3 . In general, a m -j- a" = a*"'". The equation' a™ -rfl» = a^-", gives in algebraic symbols, the laiv of exponents in division. In words, this law may be stated : The exponent of any base in the quotient is equal to its exponent in the dividend minus its exponent in the divisor. 94 Division Examples 1. a 7 + a 5 =: a 7 " 5 or a 2 . 4. — 2 7 -=- 2 4 = — 2 3 . (Why ?) 2. a 2 -=- - a = - a. (Why ?) 5. a 3 * -3- a* = a 2 *. (Why ?) 3. - A; 6 -=- - fc 2 = k\ (Why ?) 6. a r+2 -f- a r = a r+2 ~' = a 2 . ORAL EXERCISE 148. Find the quotients : 1. a 4 h- a 2 . 6. - r 10 -*-(- r 6 ). 11. c 7 -s- c 6 . 2. m 6 -i-ra. 7. — /i 5 -=-(- /*). 12. — d8^_(_^ p 3. a**-s-(-afy 8. -W + -W>. 13. (abf + ab. 4. 2/ 7 -f-y 4 . 9. 6 8 --(-6 5 ). 14. 5 7 -^5 4 . 5. — J9 6 -7- p 4 . 10. - t 9 -j- £ 7 . 15. — 2 8 -j- 2 7 . 16. (2a) 8 -(2a) 6 . 20. m*+ 3 -*-m*. 17. (a + 6)6-s-(aH-6) 8 . 21. ra* +3 ■*■ m*. 18. as 2 " -*- as". 22. 3 n+1 -+■ 3. 19. x br -r-a? r . 23. a k + a l . 24. a m+n -h a w-w . DIVISION OF MONOMIALS 149. State the definition of division. Define dividend, divisor, and quotient. (§§51, 142.) 8a 2 & 3 c-=-( — 2a?b)= — 4& 2 c is an immediate consequence of the definition of division since (—2 a 2 b) x (— 4 b 2 c) = 8 a 2 b 3 c. ORAL EXERCISE 150. {Jswmj only the definition of division give answers to the following and, explain : 1. 21-=- 7. 6. aW* -s- aW*. 2. a 5 -=-<*. 7. 6p 7 q 5 + 2pY> 3. 3 aft-*- a. 8. — 18 a 2 ?/ 2 z 5 ■+• 9 xyz. 4. 5 a^%! -^ xyz. 9. 42 a& 2 -s- (— 7 aft). 5. - 7 m 3 w 2 2) -?- mw'p. 10. (- 33 a & 2 c 3 )--(- 11 abc). Division of Monomials 95 151. When the examples are simple, the definition of division along with our previous practice in multiplication will enable us to find the quotients. A rule can be stated, however, that will help us to perform divisions. To divide a monomial by a monomial : 1. Divide the numerical coefficient of the dividend by that of the divisor, keeping in mind the law of signs. 2. Subtract the exponent of any letter in the divisor from the ex- ponent of that letter in the dividend to find its exponent in the quotient. 3. Omit from the quotient any letter whose exponent in the dividend is the same as its exponent in the divisor and write unchanged in the quotient any letter that occurs only in the dividend. Examples 1. - 28 a b bc). 13 ' 15 a 6c' 5. ^a*bc*+(-ia*bc). ^ _2a 2 6c 6. J|»y + 4«^. 4a 2 6c 15. 8. — ax 9 -7- 3 ax 6 . — 2 xyz 9. 7aWc+(-Zabc). 33 ^y 10. 5040 a«6»-4- 720 ab 2 . ' (-3x)(-y) 96 Division Find the quotients : 17 -4a5.(-7q 2 6 (- 2)2 . aW 18 a h W(-aby - a 4 b 4 * 19. 20. 21. 23. 24. 3»+2 a n+8 2i2g 7n4 2 2 10 a 2 (-<*)(- yx-c)« (_ a) 2 ( _ 6) 3 ( _ c) 4- ( _ a2)( _ a) 2- -3(-a6) 2 3 2 a 3 -3 4 5 e 3 3 (- 5)2* 2»+2 . 3n+3 25. 2» . 3^+2 26. 3 aa 2 ?/ • 6 a?/ 3 -h ( — 9 alxhf). Write the work for this example as follows : 22. ~ v v ~ ™) m 3 ax*y ■ 6 ay 9 = 18 a^y 4 18 a% V -r ( - 9 aWy 2 ) = ? 27. 4 aty -r- 2 a*/ + 17 afy 2 -h 2 a% 2 . 28. ^(a-fcy-f-a^a-fr) 4 . 29. 4a^(8- ra) 3 -=-4a?>(8-m). 30. 126(?/-z) 5 -[-4&(?/-z) 4 ]. 31. 15m 2 (x 2 -l) 4 --3m(ic 2 -l) 3 . 32. 15 cd 4 -=- 5 <& + 22 c 5 d 8 X cd -=- ll^d 8 . 33. 10 a 2 6 X a& 2 -- 5 ab - 5 a 4 6 3 -*• ( - 5 a 2 6). 34. a m -r- a n . 37. o 2n h- a n . 35. 2a m -5-a B . 38. 33 a s+2 6 3 -r- 3 a s b. 36. a m+n -5-a n . 39. 28a 8+2 6 3 -s-(— 4a 2 6). DIVISION OF A POLYNOMIAL BY A MONOMIAL 153. 1. Since 2x(a + 6) = 2a + 26, therefore (2a + 26)-*-2 = a + &, by the definition of division. %. Since a (# + y) = asc + ay, therefore (ax + ay) -r- a = x -\- y. Division of a Polynomial by a Monomial 97 3. (2x + ±y)+2 = x + 2y. Check. 2{x + 2y) = 2x + ±y. 4. (4a 2 + 10a)-=-a = 4a + 10. 5. (5a - 106 + 15c)-*-(- 5)=- a + 26 -3c. ORAL EXERCISE 154. Find the quotients : 1. (cy + dy)+y. 2. Q>g + rg)-*-g. 3. (cd-6a")-J-d. 4. (4a-8 6)-K4. 5. (a 2 + 2a)-=-a. 6. (a? + a)+a. 7. (15p + 20g)-5. 8. (18 ft- 27fc)-s- (-9). 9. (21x*-Ux) + 7x. 10. (-18m 2 - 24 m)-?-6m. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. ax + bx + cx)-i-x. dx — dy — dz) -s- (— d). 2a 3 -6a; 2 + 4a;)-^2z. 9p>+12p) + (-3p). a 2 6 + a6 2 )-f-a6. 2a 2 -8a + 10)-(-2). a6c-2 6 2 c)-r-(-6c). 14a-166+18c)-(-2). a n + a 2n ) -j- a n . a n+1 — a n ) -s- a w . 155. From these examples we derive the following rule : To divide a polynomial by a monomial, divide each term of the poly- nomial by the monomial and unite the results with their respective signs. Examples 2. (15 a 2 - 5 a + 5)-*-(- 5). - 5)15a«-5a + 5 -3a 2 + a-l 1. (3a 2 -6a)-?-3a. 3a )3a 2 -6a a — 2 Check. 3a(a - 2)= 3a 2 - 6 a. 3. (3 a 3 6 3 - 12 a 2 6 2 - 3 ab) + (- 3 ab). - 3a6 )3a 3 6 3 -12a 2 6 2 -3a6 — a 2 6 2 + 4 a6 + 1 The simplest verification of such exercises is by using the relation, divisor x quotient = dividend, (d X q = D.) 98 Division EXERCISE 156. Find the quotients : 1. 4 a 3 - 6 a 2 b - 12 ab 2 ) -- (- 2 a). 2. (2^-8afy + 10^ 2 )H-2a;. 3. (21 a&c - 35 bed - 42 acd) --(-7 c). 4. (15 a 2 6c - 27 ab 2 c - 33 a&c 2 ) -=-(- 3 abc). 5. (17a 2 6 -13rt6 2 )-=-2a7>. 6. (21az 2 + 15aV)--7«a;. 7. (3 m 2 + 4 wwi - 9 ?i 2 ) ■+•(— 3). g 14a-76 + 7 5p-5g 7 ' 5 9 a pq-3bpq+pq m(a + b)-2 m(2 a -b) — 2pq m 12. D = S x 2 y + 5xy 2 , d = 2 xy, find q. 13. q = a + & - 3, d = - 2 ab, find D. 14. D = 2 a 2 cc 2 — 6 asc, ti = 2 ax, find g. 15. Z> = 15 a 2 - 9 a 5 + 18 a 9 , d = 3 a 2 , find g. 16. (8 a h b - 24 a 4 b 3 + 16 a 7 6 8 ) -s- ( - 8 a 4 &). 17. (25 a?x 2 + 60 a 2 x* - 25 or?/ 2 ) ■#-(- 5 »> 18. (21 a 3 -14 a 2 -a) -(-a). 19. (36 x*y 2 - 24 ayz - 18 xy 2 z) -*> (- 6 ax/ 2 ). 20. (36 a 10 - 24 a 6 + 21 a 5 ) -5- ( - 6 a 5 ). 21. (100 a 2 bc - 75 ab 2 c + 50 a&c 2 ) *•(- 25 a&c). 22. (35 c 2 xy + 42 ex 2 - 56 cxy) -+- 7 ex. 23. (12 a n+ * - 15 a n+2 - 27 a" +1 ) -3a. 24. (12 + y). Solution. * + *K*±jQ«±I*+jrt& a + 6 31. [( a + &)a>+(a + &)y]-s-(a+&). 32. [?* 2 (m + n) — 2 r(ra + ») -f (m + n)] -5- (ra + n). 33. [12 z 2 (« + 6) 3 - 32 o?y(a + &) 2 ] ■*-[— 4 x(a + &)*]. 34. [2 m 2 (a - ?/ 2 ) 3 - 3 m(x - y 2 ) 2 - (x - y 2 ) ] -5- (x - y 2 ). 35. [ - 8 a 2 b(x - y) 2 + 9 ab 2 (x - y)]^ a6(a? - 2/). 36. [> 5 (a 2 + 6 2 ) - 2 z 2 (a 2 + 6 2 )] +x\a 2 + 6 2 ). 37. [12 &(a 2 - ?/ 2 ) - 15 b 2 (x 2 - ?/ 2 )] h- 3 b(x* - y 2 ). DIVISION OF A POLYNOMIAL BY A POLYNOMIAL 157. This kind of division will be understood best by study- ing an example Divide 2 a 3 - 7 a 2 + 10 a; - 8 by a: - 2. 1. 2a*»-7a; 2 -l-l0a;-8| a;-2 2. 2 x 2 (x - 2)= 2 a? -4 a 2 | 2 a? 2 - 3 x + 4 3. _ 3 a? + 10 a; - 8 4. _3a(a-2)= - 3 x 2 + 6 a; 5. 4 x - 8 6. 4(z-2)= 4a;- 8 1. Both dividend and divisor are arranged in descending powers of x. St. The first term of the dividend is divided by the first term of the divisor to obtain the first term of the quotient, 2 x 2 . The entire divisor is then multiplied by the first term of the quotient. 3. The product obtained is subtracted from the dividend. 4. The first term of the remainder is divided by the first term of the divisor, to obtain the next term of the quotient, — 3 x. The entire divisor is then multiplied by this second term of the quotient. 5. The product is subtracted from the last remainder. 6. The process described in the last two steps is repeated until, in exact division, a remainder zero is obtained. 100 Division 158. The explanation just given may be regarded as a rule for the division of a polynomial by a polynomial. It is of the greatest importance that a proper arrangement of the terms of the polynomials be made at the beginning and that the same arrangement be observed in all the remainders obtained in the course of the work. Let the student explain how the next term of the quotient is obtained. Also explain all the operations involved in steps 5 and 6. To check examples in long division the relation d • q = D may be used, or arbitrary values of the letters may be substi- tuted. If the latter method is employed, values of the letters should be chosen which will not make the divisor 0. Examples 1. Divide (x 2 + Sx - 4) by (x - 1). Division a2 + 3a;-4 x 2 — x x — 1 Check : When x = 2, D = 6, x + 4 d = 1, q = 6, and 6 -=- 1 = 6. 4» — 4 4sc — 4 2. Divide 8 a 3 + 27 6 3 by 4a 2 -6a6 + 96 2 . 8a 3 + 27 6 3 8a 3 -12a 2 & + 18a& 2 4a 2 -6a&+96* 2a + 36 12a 2 6-18a& 2 + 27& 3 12a 2 5-18a6 2 + 27 6 3 Check. Multiply the divisor by the quotient. EXERCISE 159. Find the quotients : 1. (<*-5» + 6)-!-(*_2). 4. (a 2 -6 2 )-Ka-&). 2. (a 2 -8a + 15)-j-(a-3). 5. (a 2 - a6)-5-(6 - a). 3. (4 6 2 -46+l)--(26-l). 6. (7 a - 14) -5- (2 - a). Division of a Polynomial by a Polynomial 101 7. (3a; 2 _ 4 a? + 20) -r- (a; -2). 8. (6a 3 - 23a 2 6 + 25a6 2 - 6 6 3 )-j-(2a- 3 6). 9. (30ap-6 6p + 12cp)-i-(5a-6 + 2c). 10. (20 ac -Wad- 126c + 9&d)-s-(5 a- 36). 11. (3 a6d - 3 cd + a6c - c 2 ) -r- (aft - c). 12. (6 a 3 6 + 9 a6 2 + 3 abc + 2 a 2 c + 3 be + c 2 )-5-(3 a6 + c). 13. (a^ + a?-4ar J + 5a;-3)-i-(l-a? + a 2 ). 14. (27a?-8y*)-*-(3a>-2y). 15. (8 a 3 6 3 - c 3 ^ 3 ) -f- (4 a 2 6 2 + 2 a6ca" + c 2 d 2 ). 16. (a 2 4- 6 2 + c 2 + 2 ab - 2ac - 2 6c)-s-(a + 6 - c). 17. (5a 6 + 15a 5 + 5a + 15)-(a + 3). 18. (2a 4 -6a 3 + 3a 2 -3a + l)-(a 2 -3a + l). 19. (42 a 4 + 41 a 3 - 9a 2 - 9a - 1)-(7 a 2 + 8a + 1). 20. (2m 4 -6m 3 4- 3m 2 - 3m-f-l)-r(m ! -3m+ 1). 21. (6a?x - 17 a?x 2 + 14:0,0? - 3x*) + (2 a - 3x). 22. (2 x 4 + afy - 13 afy 2 - 3 a?/ + y*) + (x* -2xy- if). 23. (15 a 5 + 10 a 4 6 + 4 a 3 6 2 + 6 a 2 6 3 - 3 a6 4 ) -=- (5 a 3 + 3 a6 2 ). 24. (21 a 4 - 16 a 3 6 + 16 a 2 6 2 - 5 a6 3 + 2 6 4 ) + (3 a 2 - a6 4- 6 2 ). 25. (20a 6 - 53a 7 4- 45a 9 - a 8 ) -- (4 a 2 - 5 a 3 ). 26. (a 6 - 5x*y- 10 bY 4-lOafy 2 4- 5a;?/ 4 - ?/ 5 )-r- (a; 2 - 2a;?/ + y*). 27. (a 4 4-2aV 4- z 4 - 6 4 )-=-(a 2 + # 2 + 6 2 ). 28. (6 a 2 4- a6 4- 7 ac - 12 6 2 + 19 6c - 5 c 2 ) -- (2 a 4- 3 6 - c). 29. (15a; 2 -29an/4-122/ 2 -222/z-602! 2 )-^(5a;-32/+103). 30. (48 x*y A - 80 x*tf - 8 a;?/ 5 + 200 afy 2 ) -r- (20 a?y 2 - 4 a;?/ 3 ). 31. (343 aW - 64 6 3 a*) -- (49 oV + 28 a6a? 4- 16 6V). 32. (20a^ + 32a;-51a* J -12a; 2 )-=-(4a; 2 -7a;-8). 33. (32 a 2 + 456 2 4- 60 c 2 4- 76a6 4- 88ac + 1046c) -(8 a + 9 6 + 10c). 34. (1.2 a 2 + 1.17 a6 - 11.34 6 2 )-4- (1.5 a 4- 5.4 6). 35. [x 2 4- (a 4- c)a; 4- ac] -f- (a 4- a;) . 36. [f-(a-b)y-ab']^(a-y). 102 Division 160. Division with a Remainder. If the dividend is not the product of the divisor multiplied by some integral algebraic expression, we shall have a remainder. 1. Divide 6 x 2 - lSx - 3 by 2 x + 1. Division 6 x 2 -13a 6x 2 + 3x 2a; + l 3 x — 8, quotient -16a;-3 - 16 x - 8 5, remainder. 2. Divide a? + 3 a; 2 + 7 by a; 2 -2a; + 2. Check 3a; -8 2a; + l 6 x 2 — 16 x 3x- -8 6 a; 2 -13 a;- -8 5 6a; 2 -13a; -3 ar J + 3a; 2 + 7 aj3_2a; 2 4-2a; a; 2 _2a;4-2 x -f 5, quotient. 5a;2_ 2a;+ 7 5a; 2 -10a; +10 8 x — 3, remainder. Unless otherwise directed, perform all divisions in descending powers of some letter, and continue the division until the exponent of the highest power of the letter of arrangement in a remainder is less than that of the highest power of that letter in the divisor. EXERCISE 161. Divide, and check by the relation a* • q-\- r =D. 1. (a; 2 -3a; + 5)--(a; + l). 2. (4 - 3 x 2 + 2 x) -(2 + x). 3. (ar J -l)--(a; 2 -a; + l). 4. (3a-a 3 + 2)-(l-a 2 ). 5. (7a 2 + 6a 3 + 5a-7)-;-(3a-l). 6. (7a 2 + 6 a* + 5 a -7)-s-(2 a 2 + 3 a + 2). 7. (x 3 -8a 3 -2a 2 a;)-r(2a-x). Division of a Polynomial by a Polynomial 103 8 . (_ 73 #2_ 25 + 56x 4 + 95 a: -59 a 3 ) -=-(- llaj + 7a?-3a*+l). 9. (49a 3 -72.a«/ 2 + 28^)--(7a-3 ? /). 10. (4 m 4 — m 2 n 2 + 6 mn 3 — 9 w 3 ) -r- (2 m 2 - mn + 3 ?i 2 ). 11. »»-f-(a? — 1). 12. For what value of k is x 2 — 3 x + A: exactly divisible by 05+1? Solution, x 2 — 3x + k \x + 1 s 2 + a _ laj — 4 — 4x + & -4s-4 & + 4 = the remainder. The division is exact if the remainder is 0, or if k + 4 = 0, that is, if Jfc = -4. 13. Determine k so that ar } +3ar J + 2.T + fc shall be exactly divisible by x — 2. 14. Determine k so that a + 1 shall be an exact divisor of a^ + A:. 15. For what value of k is x — 1 an exact divisor of x* + A; ? VIL SIMPLE EQUATIONS 162. An equation is a statement expressing the equality of two numbers. (Review §§ 12-16.) There are two essentially different kinds of algebraic equa- tions as illustrated by the following : 1. (z + 2)(a;-2)=z 2 -4. 2. x + 2 = 5. Equation 1 is true for all values of x ; equation 2 is satisfied when x equals 3, and not otherwise. 163. Identity. An equation that is true for all values of the letters involved is an identical equation, or simply an identity. The symbol = is sometimes used to indicate an identity. The most frequent use of the identical equation is to indi- cate the result of some operation performed upon algebraic expressions. The following are examples of identical equations : 2 x + 3 x + 5 x = 10 x. (x + 3y=x 2 + 6x + 9. (a-b)(a + b) = a 2 -6 2 . 6m (m — n) =6m 2 -6w»n. In the identical equation, if the indicated operations are performed and the like terms are collected in each member, the two members will be exactly alike. 164. Conditional Equation. An equation that is true for only certain values of the letters involved is a conditional equa- tion or simply an equation. 104 Simple Equations 105 A conditional equation expresses a relation between an un- known number and certain known numbers. The problem suggested by a conditional equation is that of rinding for what value of the unknown number the relation expressed in the equation is true. The following are examples of conditional equations : 2x — 7 = x + 3. True when x = 10, and not otherwise. 3a + 7 = 4a + 7. True when a = and not otherwise. 2 ax = 4 a 2 . True when x = 2 a and not otherwise. ORAL EXERCISE 165. 1. Is x + 1 = 2 a conditional equation or an identity? 2# + 3 = 7? 2. Is 2x — (x'+ 1)= x — 1 a conditional equation or an identity? (x - l)(x + 1)= x* - 1 ? 2#-l = #? 3. State the four principles used in solving equations. (See § 13.) 4. What is the root of an equation ? (See § 16.) 5. What is the root of 3 + 2 = 7? of #-2 = 7? of 2# = 3? ofi.r=5? 6. What value of x satisfies the equation x — 2 = 3 ? /SftoM? £to £^e following are identities by reducing the two mem- bers to the same expression : 7. a(x — y) = ax — ay. 8. (x + a)(x + b)=x 2 +(a + b)x + ab. 9. 5y + 3- 4?/ = ; y + 3. 10. llz-(5 + 10z)=z-5. ySoZve tae following conditional equations : 11. #-3 = 2. 16. 2^-8=3. 12. 2/4-7 = 9- #. 17. m? -|-4 =-10. 13. 2x-l=5. 18. 2n=-6. 14. Ia>+1=4. 19 - 4x-2a + 3=-3, 3 20. 5n-4 = -14. 15. 3 # — 4 = 5. 106 Simple Equations EXERCISE 166. Show that equations 1 to 4 are identical equations by re- ducing the tivo members to the same expression. 1. a(6 — c) + b(a — c)=2ab — c(a + 6). 2. (a + 6 - c)(a + b + c) = a 2 + 2 a6 + 6 2 - c 2 . 3. (a 2 - 2 _ 2)(a 2 + x -2) = (a 2 - 3 x + 2)(« 2 + 3 a + 2). 4 - 0* + V)(y + «)(« + ») + xyz =(x + y + 2!)(ajy + y« + «»). ify substituting 1, 2, awe? 3 /or a; m equations 5 to 8, s^oiy £ta eacft is a conditional equation. 5. 2a- 5 = a;-3. 7. (a - l)(a + 2)=a 2 . 6. (a-4) 2 +2 = (a-5) 2 -3. 8. 8a + 7-a = 14. Solve the following equations : 9. x -4 x + 3 = 0. 10. 5j> + 18 = 3(p + 10)-2. 11. 31 -7a = 41 -8*. 12. 5x + 13 -2a = 100 -20a -18+ 12 a; -15. 13. 16^ + 10-21^ = 45-10?; -15. 14. 7y-9-3y + 5 = lly-2(3 + 2y). 15. -40 = 5-30a + 35-40a. 16. 6 = 6- 9 + 36+2. 167. It is a common practice in algebra to use x, y, and z to represent the unknown numbers in an equation and to use a, 6, c etc. to represent numbers that are regarded as known. Thus, in the equation ax = 3 a 2 b. x is the unknown number and the value of x is to be found in terms of the other letters involved. The value of x is found by dividing both members of the equation by a, giving x = 3 ab. The equation x = 3 ab can be solved for a or for b. Thus, dividing both members by 3 b gives a = ~- Solve the equation for b. 3 b Simple Equations 107 168. Integral Equation. An equation in which the unknown number does not occur in any denominator is an integral equation. 2 x Thus, Zx — 4 =-x — 5 and ax + b = Sb are integral equations. 3 a In the present chapter all equations are integral. 169. Solving Equations. 1. Solve 3 a; -(5 -a) =11. Solution. 3x— (5 — x)= 11. 3 x — 6 + as = 11. (Removing parentheses.) — 5 + 4 x = 11. (Collecting like terms.) 4 x = 16. (Adding 5 to both members.) x = 4. (Dividing both members by 4.) Let the student check the answer by putting 4 for x in the original equation. 4 x = 16 is the simplified form of the equation and the work done to reduce the equation to this form is called simplifying the equation. 2. Solve (x -3)(x-2) = (x- 4)2. Solution. (x - 3)(x - 2) = (x - 4)2. x 2 - 5 x + 6 = x 2 - 8 x + 16. (Multiplying.) -bx + 6=- 8x + 16. (Why?) 8z-5a; + 6 = 16. (Why?) 3x=10. (Why?) x = V - (Why?) Let the student check the answer as in example 1. What is the simplified form of this equation ? 170. Simple Equation. An equation that can be reduced to an integral form containing the first power of the unknown number and no higher power is a simple equation. Thus, 5x — 2(3x — 1) = 1 is a simple equation. Also x{x — 5) = (x — 3) (x — 7) is a simple equation, for it reduces to 5 x = 21. Simple equations are frequently called first degree equations, also linear equations. 108 Simple Equations 171. The type form of the simplified equation of the first degree is ax = b. By this we mean that x, with any coefficient it may have (represented in the type form by a), constitutes the first member of the equation, and the known term or terms, represented by b, constitute the second member. Thus, 3 x = 7 is in the form ax = b ; here a = 3, and 6 = 7. 172. The steps required to reduce an equation to the form ax—b are illustrated in the solution of examples 1 and 2 of § 169. The principles involved are those stated in § 15. EXERCISE 173. Solve the first 20 equations orally. 1. 2 x = 15. 8. x + 3 = -, 7. 15. — x — 7 =-8. 2. a; -2 = 0. 9. 3 x + 6 = 9. 16. 7 = 2+a. 3. -25 = 2. 10. _ x - 7 = 0. 17. —5 a; =.5. 4. 3a = 0. 11. -2a + 8 = 0. 18. ax = 2a 2 . 5. — x + 3 = 0. 12. 7 * - .56. 19. bx — 2 ab = 0. 6. x + 1 = 5. 13. Ax = 2. 20. — a» -f 7 a = 0. 7. 2a + l = 5. 14. 5 x - 7 = 13. 21. ia-3 = 2. 22. a- 3=2- - X. 27. .5 a: -.05= .2. 23. x — 5 — 5 - ■ X. 28. -100=20+(2x-25). 24. -2x-f = -0. 29. .82 x -f- .1 x = .3 sc. 25. 17 -(16-! r)=l. 30. 2x - 7 = 4 - 2 x. 26. -(18 + a) = = 19. 31. ±x _(7-2z)=4a + 5. 32. 6a + 4 = 3(a + 3)+ 2(3 - a). 33. 5(2 x -3)- 2(3 -2x)+ 2 = 0. 34. 2a-(4a-7)-3(5-7a;)=100. 35. 7a-[3-2(a;-5)+3]=2. 36. 5 -2[2 a -(3 -10 x) -4]= -225. Simple Equations 109 174. It is sometimes desired to solve equations when some of the numbers that are regarded as known numbers are repre- sented by letters. The method is the same as that used in the equations already solved. Example. Solve 3 ax — a(2 c + x) = 2 ab — 4 ac. Solution. 3 ax — a (2 c 4- x) = 2 ab — 4 ac. 3 ax — 2 ac — ax = 2 ab — 4 ac. 2 ax — 2 ac = 2 ab — 4 ac. 2 ax = 2 aft — 2 ac. x = b — c. EXERCISE 175. #o£i;e the following equations, regarding the last letters of the alphabet as the unknown numbers : 1. 3 nx — nx = 2 n 2 — 4 n. 4. bx — 2ba = b 2 . 2. 4 c# — 5 c = ex + c. , 5. ex — cd -\- c 2 = c 3 . 3. — a* + 7 a = a 2 . 6. 3 x — (4 a + 7) = 2 a + a. 7. 5a-3(2c-4d)=2d + 4c. 8. 3a«- 2(x = 2x— 5-3. (Subtracting 3 from both members. ) 3. 5 x — 2x = — 5 — 3. (Subtracting 2 x from both members.) 4. 3 x — — 8. (Collecting terms gives type form ax = b. ) x = — If we compare equation 3 with equation 1, we shall see that the term containing x in the second member of equation 1 appears in the first mem- ber of equation 3 with its sign changed; also that the term not containing K, that was in the first member, is now in the second member with its sign changed. 177. Transposing Terms. Any term of an equation may be changed from one member of the equation to the other by changing its sign. This process is known as transposing terms. In writing the solution of equations the student may omit step 2 as given in § 176, and write equation 3 immediately from equation 1, describing the process, as " transposiiig all terms containing the unknown number to the first member and all terms not containing the unknown number to the second member." The mechanical process of transposing is a simple one, but great care must be taken not to lose sight of the principles which underlie the process. 178. The rule for solving linear equations will now be stated more fully. To solve linear equations : 1. Perform all indicated operations, removing all parentheses in both members. Simple Equations 111 2. Transpose so that all terms containing the unknown shall be in the first member and all terms not containing the unknown shall be in the second member of the equation. 3. Collect the terms. 4. Divide both members by the coefficient of the unknown. Examples 1. Solve Ix -8 = 4 -(2 - 10a>). Solution. 7x - 8 =4 -(2 - 10 x). 7 x — 8 = 4 — 2 + 10 x. (Removing parenthesis. ) 7 x — lOx = 4 - 2 -f 8. (Transposing terms.) — 3 x = 10. (Collecting terms. ) x = - 3£. (Dividing by - 3.) Check. Substitute x = — 3| in both members of the original equa- tion, 7x-8 = 4-(2-10x). Thus, 7(- 8*)- 8 = 4 - [2 - 10(- 8*)]. - ¥ = - ¥• x = — 3£ satisfies the equation. 2. Solve (x - 5)(x -7)=(x- 4)2 - 1. Solution. (x- 5)(»- 7) = (x - 4) 2 - 1. a? _ 12 X + 35 = x 2 - 8 x + 16 - 1. (Why ?) -12x + 8x= 1(5 -35-1. (Why?) -4x=-20. (Why?) x = 5. (Why ?) Let the student check the result. 3. Solve x(x-6) = (x-3)(x-2)-6. Solution. x(x - 6) = (x - 3)(x - 2)- 6. x 2 -6x = x 2 - 5x+6-6. (Why?) -6x+5x = 0. (Why?) - x = 0. ■ (Why ?) x = 0. (Why ?) Let the student check the result. EXERCISE 179. Solve the following equations: 1. 3x-5 = x + 2. 3. (5a-3)-(6a + 8)=0. 2. 4y-7 = 12y + 2. 4. _[»-(2-3ar)]=14. 112 Simple Equations Solve : 5. 7x -25 = 15(21 -3 z)+24. 6. 4-(2a>-7)=3-4(4-5a>). 7. m(m — 5) = (m + 2) 2 + 5. 8. 4(a> + 2) -2(a; + l)- 3(7 -»)=<>. 9. (p-4)(p + 4)-(p + 2)(p + 3) = -23. Solution. p 2 - 16 -(p 2 + 5p + 6) = - 23. (Why?) ^•2_ 16-^2-5^ -6 =-23. (Why?) -5p =-23+ 16 + 6. (Why?) _5j9 = -l. (Why?) P = *. (Why ?) Note . In simplifying the product - ( p + 2 ) ( p + 3) , it is better to per- form the multiplication first — (j? 2 + 5 p + 6) and remove the parenthesis afterward, as the sign — affects the whole result of the multiplication. 10. (v + 2)2 - (y +- l) 2 =(v- 2){v - 1) - v\ 11. (2a> - l)(3a> + l)-(6x- 12) (« + 3)= 0. 12. (4aj-7)(9a>-48)=12(3a + l)(a>-6). 13. (2 6-5)(2 6 + 5)-(4 6-ll)(6 + l)=0. 14. (8z + 5)(2a;+-7)-(4a-3)(4a;+-3)=0. 15. (2x- l)(lUx + 5)- 26a =(8 a? + 1)36 x + 11. 16. (* + 4) 2 - *(* + 6) = 22. 17. (2xy + 5x(x + 7)=(3xy + 70. 18. (x + l) 2 +(» - 2)2 -(x - l)(x + 5)- & = 0. 19. (g - 1)2 + (g - 3) 2 - 2(q - 7)(q + 15) = 0. 20. ^(.t - 1)0 + 7) - + l)(x + 2)(» + 3) = 0. 21. (z+-l) 3 = z 3 + 10 + 3z(z+2). 22. x v _ i _l7 =ii(3v + 1) . Solution. £ * - J - \ | = ^(3 • + 1). *«-i-tf = tt" + tt- (Why?) *«-}*« = ** + *+*!• (Why?) i v = |. (Why?) o = f + \ = 6. Let the student check the answer. 23. \x — \x + \x — \x= 13. Simple Equations 113 24. |(2 + 5aj)=i(9aj + 2). 25. 30(m-2) + im=-L(5m + l)+30. o Id 26. |(5x + l)-K 4 ^ + 5 )=i( 3a; - 1 )-A(6^ + 4). 27. |(5a?-l)-8 = -J(4 »- 2). 28. i(l-aO-K2_a>)=i(3+*). 29. .2(o? - .3) -(a; - .l) 2 = #(.25 - x) + .005. Solution. .2(x - .3) - (* - .l) 2 = x(.25 - x) + .005. (.2x-.06)-(x 2 - .2 a; + .01) = (.25x-x 2 )+-005. (Why?) .2sc-.06-x 2 + .2x- .01 = .25x-x 2 + .005. (Why?) .2z + .2x- .25 x = .06 + .01 + .005. (Why?) .15 a; = .076. (Why?) x = .5. Let the student check the answer. 30. .25(4 c - 6) - .4(5 c - 7) = 0. 31. 5 a- 1.7 = 5.4a; + 0.8. 32. 0.123 d + 0.138 = 0.876 - 0.123 d. 33. 7.5 x- 2.5 -1.5 x = 4.5. 34. .45 2/ - .75 = - .125 - .45 y. 35. .7(.8 x - 3)- .39 = 1.11 a - 3(.2 a? - .5). 36. .02^ = 2(.6 - .046/) - .2(.5 gr - 2). 37. 2fa+ia;=2i+5i. 38. = .75&-2&- .6fc + 5&-9. 39. |(7 x - 10) - |(50 - x) m 20. 40. 5 = 3v + |(v+3)-i(llv-37). 41. |(3aj-5)-l = Kll-2*) + «. 42. l- 3(7^+^+7(1*- |)+t*= a 43. 3 -(.3 - .07d) + .5(.ld + 1) = 4 - .2(7 - .3d). 44. (1 + 6 a;) 2 +(2 + 8a;) 2 -(1 + 10 a?) 2 = 0. 45. 9(2 x - 7) 2 +(4 x - 27) 2 = 13(4 a; + 15)(aj + 6). 46. 9[7(5 + {3*-2}-4)-6]-8 = l. 114 Simple Equations Solve : 47. 3[3(3 + J3 7i-2j-2)-2]-2 = l. 48. 2 + 2(2a;+3)+3(a; + 2)=12-(a; + l). 49. i(27c + 3)-23 = l(6k-5). 50. .25(2 x + 1)+ .2(3 x - 1) = J(7» - 1)- x. 51. (8-9)(* + 9) = (s+ 6) 2 4-s. 52. = 4(10- 2a;)- 3(a;- 5). 53. = 3(9-2$- 5(2 g- 9). 54. 7(4a;-3)+3(7-8a;)=l. 55. 8(3 i- 2)- 7t- 5(12 -3t)= 131. 56. 7(3a; - 6) -f- 5(a; - 3) -4(a;- 17) =11. 57. 6aj-7(ll-aj)+ll=4aj-3(20-aj). 58. 44 a? .- 32 = 84 + 31.5 a; + 4.2 a; + 16.8. EXERCISE 180. The equation x — 15 = 24 states in algebraic language that some number diminished by 15 is equal to 24, or that some number is 15 greater than 24. In the same way translate into verbal language each of the following algebraic equations : 1. x + 10 = 25. 4. 2x-15 = 3x. 2. x + 296= 5a. 5. 25 -5a; = 11. 3. 3?/- 14 =40. 6. 72 - 2a; = 9a;. Write in algebraic language, that is, make algebraic equations for the following : 7. 36 is greater than some unknown number by 10. 8. Three times a number is 14 less than 5 times the number. 9. 25 is divided into two parts the larger of which is n. What is the smaller part ? Make an equation that indicates that 3 times one of the parts is equal to 2 times the other part. Simple Equations 115 10. 786 diminished by two times an unknown number is 270. 11. If 296 is added to an unknown number, the sum is 5 times the number. 12. A boy is three times as old as his sister and the sum of their ages is 16 years. Let x = the number of years in the sister's age. 13. If 7 is taken from 5 times a number, the remainder is 53. 14. If 42 is added to 7 times an unknown number, the sum is 54. 15. A rectangle whose length is a feet, and whose width is 3 feet less than its length, has an area of 54 square feet. 16. 60 is divided into two parts, the smaller of which is § of the larger. 17. 50 is divided into two parts such that 40 % of one part is equal to 20 % of the other part. 18. A house and lot are together worth $ 8500. The value of the house exceeds 3 times the value of the lot by $ 1000. Find the value of each. 19. The frame of a picture is 3 inches wide. The picture is 4 inches longer than it is wide and the area of the frame is 252 square inches. Find the dimen- sions of the picture. Hint. The area of the picture is ic(cc -f 4). The area of the picture and frame is (x + 6)(x + 10). (Why?) The area of the frame is the difference of these areas. 20. A square lot has a walk around it that is 6 feet wide. The surface of the walk contains 2256 square feet. Find the length of a side of the square inside the walk. 116 Simple Equations THE SOLUTION OF PROBLEMS 181. A problem is a question proposed for solution. It in- volves the finding of one or more unknown numbers from relations stated in the problem. 182. In solving problems the following suggestions will be found useful : 1. The problem should be carefully read and the conditions stated in the problem should be carefully analyzed. Before the solution is attempted, the student should see clearly the relations existing between the unknown number and the known numbers. 2. Represent one of the unknown numbers by some letter. If more than one unknown number is involved, represent them in terms of the same letter. 3. Translate the verbal language of the problem into alge- braic language in the form of an equation. 4. Solve the equation. 5. Check the result by testing whether the number or num- bers found by solving the equation satisfy the conditions stated in the problem. It is not sufficient to determine whether these numbers satisfy the equation obtained, as an error might occur in forming the equation. PROBLEMS 183. The following problems will illustrate the above sug- gestions : 1. If to 7 times a given number 12 is added, the sum is 54. What is the number ? Solution. There is but one unknown number involved. Let n = this number. Then by the conditions of the problem, 7 n + 12, or 7 times the number with 12 added = 54, or the sum. Therefore the verbal language of the problem translated into algebraic language gives the following equation : The Solution of Problems 117 7 ft + 12 = 64. .-. 7ft = 54-12, (Why?) or 7 n = 42. (Why ?) .-. » = 6. (Why?) Check. 7 x 6 + 12 = 54. 2. A certain number is 3 times another number. The sum of the two numbers is 28 less than twice the larger number. What are the numbers ? Solution. Two unknown numbers are involved in this problem. Let x = the smaller number. Hence 3 x — the larger number. Then x + 3x = 6x — 28. (By the conditions of the problem.) .-. -2 ac = -28. (Why?) .-. x = 14, the smaller number, and 3 x = 42, the larger number. Check. 14 + 42 = 84 - 28, or 56 = 66. 3. I bought for my library 3 volumes at a certain price, 5 volumes at double the price, and 4 volumes at J the price. For all I paid $24. How much did each volume cost? Solution. Three unknown numbers are involved in this problem. Let x = the number of dollars paid for one of the 3 volumes. Hence 2 x = the number of dollars paid for one of the 5 volumes, and | x = the number of dollars paid for one of the 4 volumes. Then 3 a; + 10 a; + 3 x = 24. (Why?) Let the student finish the solution and check the answer. 4. The sum of the digits of a number of two figures is 9. By interchanging the digits the resulting number will be 27 greater than the original number. What is the number ? Solution. Two unknown numbers are involved in this problem. Let x — the units' digit. Hence 9 — x = the tens 1 digit. Therefore the original number is 10(9 — x) + x, (Why?) and the number with the digits interchanged is 10 x -4- (9 — x). The second number is 27 greater than the original number. Then 10(9 - x)-f x = lOx +(9 - x)- 27. Let the student solve and check. 118 Simple Equations 5. If a certain number is doubled and 7 is added, the re- sult is — 1. Find the number. 6. The number of boys in a certain school after being doubled and further increased by 10 is 60. What was the number at first ? 7. A man walked for a certain number of hours at 4 miles an hour and then for twice as long a time at 3 miles an hour, covering 20 miles in all. How long did he walk at each rate ? Hint. Let x = the number of hours at 4 miles an hour. Hence 2x= the number of hours at 3 miles an hour, and 4 x = the number of miles at 4 miles an hour. Let the student complete the solution and check. 8. Find a number such that 10 times the number is 14 less than 3 times the number. 9. Find the three consecutive numbers whose sum is 15. 10. The sum of three consecutive odd numbers is 33. Find the numbers. 11. The sum of four consecutive even numbers is 44. Find the numbers. 12. Divide $ 880 between A and B so that A shall receive $ 50 less than twice as much as B. 13. Divide 120 into two parts such that 7 times one part equals 8 times the other part. 14. Find a number such that 15 times the number is 10 times as great as the sum of the number and 4. 15. What price does a dealer pay for 6 dozen lead pencils if he sells them for 5 ^ each and makes a profit of 90 ^ ? 16. Find the number such that, if you add 3 and multiply the sum by 5, the result is 1 greater than if you add 5 and multiply by 3. 17. A bag contains an equal number of dollars, half dol- lars, quarters, dimes, and nickels. If the amount contained in the bag is $ 47.50, how many coins of each kind are there ? The Solution of Problems 119 18. If 20 is added to a number the result will be 3 times as great as if 4 is subtracted from it. Find the number. 19. My neighbor's orchard contains 8 more trees than mine and together they contain 34 trees. How many trees does each orchard contain ? 20. The sum of three numbers is 32. The second exceeds the smallest by 2 and the largest is 2 less than twice the smallest. Find the numbers. 21. The tens' digit of a number is 3 times the units' digit and the number exceeds 7 times the sum of its digits by 9. What is the number ? (See problem 4.) 22. Two towns are 60 miles apart. A starts from one and walks 3| miles an hour toward the other town until he meets B who has started from the other town at the same time and is driving an automobile at 16 miles an hour. After how long will they meet ? How far will each have gone ? 23. If in the last problem A had started at 8 o'clock and B at 10 o'clock, at what time would they have met ? 24. What time is it if the number of hours past noon equals ^ of the number of hours to midnight ? 25. A man left half his property to his wife, one fifth to his daughter and the remainder, $6000, to his son. How much property did he leave ? 26. The deposits in a bank during 2 days amounted to $ 21,000. The deposits on the second day were ^ larger than on the first day. Find the deposits for each day. 27. The Washington Monument is 73 feet higher than the Great Pyramid in Egypt and the sum of their heights is 1037 feet. Find the height of each. 28. The sum of the three angles of any triangle is 180°. If in a right-angled tri- angle (having one angle 90°) one acute angle is twice as large as the other, how large is each angle ? 120 Simple Equations 29. How large is each angle in a right-angled triangle if one acute angle is 10° less than twice as large as the other ? 30. How large is each angle in a triangle if the second angle is 10° larger than the smallest and the largest angle is equal to the sum of the other two? (See problem 28.) 31. An Iowa produce dealer ships eggs to the city of New York. The expense of shipping and selling the eggs is § of the original cost of the eggs. If the eggs are sold for 25 ^ a dozen, how much does the produce dealer pay for them ? 32. A retail dealer received ±^ more from the sale of a beef than he paid the packer. How much did he pay the packer if he received $84.20? 33. The cost of shipping wheat from Kansas to Philadelphia was ||- of the price paid to the Kansas farmer. How much a bushel did the farmer receive if the shipper received $1.17| a bushel in Philadelphia ? 34. If the price of wheat in Kansas is J of the price de- livered in Liverpool, and the Kansas farmer receives 90^ a bushel, what is the price of wheat in Liverpool ? 35. The perimeter of a triangle is 60 centimeters. The second side is twice as long as the shortest and the longest side is 6 centimeters less than the sum of the other two sides. Find the length of each side. 36. A rectangular tennis court is 20 feet more than twice as long as it is wide and the distance around the court is 220 feet. Find the length and the width of the court. 37. The height of one of the big trees in California is 43 feet more than twice the distance around it at a point six feet from the ground. The sum of its height and girth is 466 feet. Find the height and the girth. 38. The largest package that can be sent by parcel post must not exceed 72 inches in length and girth combined. The Solution of Problems 121 What is the largest box with square ends that can be sent, if the box is twice as long as it is wide ? Hint. If x = the number of inches in width, 4 x = the number of inches in girth. 39. A star is added to the flag of the United States for each new state. There is one bar on the flag for each of the orig- inal colonies. What is the number of states and of original colonies if the number of stars is 9 more than three times the number of bars, and if the number of stars plus the number of bars is 61 ? 40. The cost of a cable message from New York to London is 25 $ a word. The rate from San Francisco to London is 1 ^ more than 3 times the difference between the rates from New York and San Francisco to London. What is the rate from San Francisco ? Hint. Let x = number of cents per word from San Francisco. ' Thena = 3(x-26)+l. (Why?) 41. The total railway mileage of Ohio, Indiana, and Illinois in 1911 was approximately 28,000 miles. The mileage of Ohio exceeded that of Indiana by 2000 miles, and Illinois had 4000 miles less than the other two states together. Find the mileage of each state. 42. The steel bridge from New York to Long Island is the longest single arch in the world. The length of the arch exceeds twice the height of Washington Monument by 7 feet, and the sum of the length of the arch and the height of the monument is 1672 feet. How high is the monument and how long is the bridge ? 43. The annual precipitation (rainfall and snow) of Michi- gan is 4 times that of Nevada and is f as great as that of Washington. The sum of the numbers of inches in the three states is 93.5 inches. Find the number of inches for each state. 122 Simple Equations 44. A mark (a German coin) is worth 4 J ^ more than a franc (a French coin). How much is each one worth in our money if two francs and three marks are worth $ 1.10? 45. The distance from the earth to the moon is about ^ of the diameter of the sun, and the sum of the distance to the moon and the diameter of the sun is 1,128,000 miles. Find the distance to the moon and the diameter of the sun. 46. A man bought 200 acres of land for $15,200. For some of it he paid $ 70 per acre and for the rest he paid $ 85 per acre. How much did he buy at each price ? Solution. Let x = the number of acres at $ 70. Hence 200 — x= the number of acres at # 85. Also 70 x= the number of dollars for 1st part, and 85(200 — x) = the number of dollars for 2d part. Then 70 x + 85(200 — as) ss 15,200. (By the conditions of the problem.) Let the student complete the solution. 47. A grocer bought 70 pounds of coffee for $ 19.20. Part of it cost 24 ^ a pound and the rest cost 30 ^ a pound. How many pounds of each kind did he buy ? 48. A grocer wishes to mix coffee that he sells at 28 cents a pound with other coffee that he sells at 35 cents, to get a blend that he can sell at 30 cents a pound. How many pounds of each should he take to get 70 pounds of the mixture ? 49. How many pounds each of 50 ^ tea and 75 $ tea should be mixed to get 20 pounds worth 60^ a pound ? 50. How many pounds each of white Dutch clover seed worth 40 i a pound and blue grass seed worth 20^ a pound, should be used to make 100 pounds of a lawn grass mixture worth 25^ a pound ? 51. A man loaned $ 2000, part at 6 % and part at 4 %. The interest on each part was the same. How much was loaned at each rate ? 52. A man loaned % 1000, part at 5 % and part at 6 %. His interest was $ 57. How much was loaned at each rate ? Rules and Formulas 123 RULES AND FORMULAS 184. A rule can often be more easily remembered if ex- pressed in algebraic language by means of a formula. Thus, i = prt (where i = the interest, p = the principal, r = the rate, and t = the time in years) is a formula by means of which the interest on a sum of money can be found when the principal, the rate, and the time are given. A — - bh is a formula by means of which ■ the area. A, of a triangle can be found when the base, b, and the altitude, h, are given. 185. Translating Rules into Formulas. & The area of a trapezoid equals the sum of the parallel sides multiplied by i the altitude. The formula A^lhCb + bz) is a short way of writing the rule, where A represents the area, h the altitude, and b x and b 2 (read b sub one and b sub two) represent the two parallel sides of Tl ' the trapezoid. ORAL EXERCISE 186. Express each of the following rules as a formula : 1. The area, A, of a circle is equal to -n- (3.1416) times the square of the radius, r. 2. The area, A, of a rectangle is equal to the product of its two dimensions, a and b. 3. The diagonal, d, of a square is equal to one of its sides, s, multiplied by V2. 4. The distance, d, that a train goes is equal to the product of the rate per hour, r, multiplied by the number of hours (t). 5. The profit, p, is equal to the selling price, s, minus the cost, c. 6. The rate per cent of profit, r, is equal to the quotient of the selling price, s, minus the cost, c, divided by the cost. /\ c /' //b 124 Simple Equations 187. Translating Formulas into Rules. The formula for find- ing the area of a rectangle is A = ab, where A is the area, and a and 6 are its two dimensions. Hence this formula is an abbreviation for the rule : The area of a rectangle is equal to the product of its two dimensions. EXERCISE 188. Express each of the following formulas as rules : 1. C = 2 irr, where C represents the circumference of a circle and r its radius. 2. V= a>b - c, a formula for the volume of a rectangular parallelepiped whose dimensions are a, 6, and c. 3. rt=d, d-^r—t and d-i-t=r, where d, r, and t represent distance, rate, and time respectively. 4. A = irr 2 , where A represents the area of a circle and r its radius. 5. V=\ hirr 2 where V= volume of a cone, h = altitude and r = radius of base. 189. The Use of Formulas. In using a formula the problem may be merely that of evaluating an expression, or it may involve the solution of an equation. 1. Find the area of a triangle whose base is 6 inches long and whose altitude is 5 inches. Solution. Substituting 6 and 5 for b and h respectively in the formula A = lbh, we have ^1 = | . 6 • 5 = 15. .*. the area of the triangle is 15 sq. in. 2. Find the altitude of a triangle whose area is 20 square inches and whose base is 10 inches long. Solution. 20 = \ . 10 • h or 20 = 5 h. The altitude of the triangle is 4 inches. Rules and Formulas 125 EXERCISE 190. 1. Find the area of a triangle whose base is 9 inches and whose altitude is 7 inches. 2. Find the base of a triangle whose altitude is 5 inches and whose area is 18 square inches. In problems in simple interest, if p represents the principal, r the rate of interest, t the time expressed in years, % the interest, and a the amount (principal plus interest), we have the following formulas : (1) i = prt, (2) a = p+ior a=p + prt 3. What is the interest on $ 900 at 6 % for 2 years ? 4. What principal will produce $288 interest in 4 years at 6 % ? Hint : Substituting in i = prt, 288 = p x .06 x 4, or 288 = .24p. Solve. 5. How long will it take $ 1200 at 5 % to produce $ 210 ? 6. At what rate will $ 1800 produce $ 252 interest in 2 years ? 7. What principal will amount to $ 1220 in 4 years at 5^f ? Hint. Substituting in a = p +prt, 1220 =p -f .22 p. Solve. 8. What principal at 5 % will yield an annual income of $350? 9. Using the formula of § 185 find the area of a trapezoid whose parallel sides are 12 inches and 15 inches and whose altitude is 8 inches. 10. Using the same formula, find the altitude of a trapezoid knowing that the parallel sides are 8 feet and 4 feet long and that the area is 50 square feet. Hint. Substitute the numbers given for the proper letters of the formula and solve the resulting equation for h. 11. In a trapezoid ^1=72 square inches, 6 X «= 17 inches, h = 2| inches ; find b 2 . 126 Simple Equations To find the perimeter of a rectangle, we have the formula : 12. Find p, when a = 12 inches, and b = 7 inches. 13. Find a when p = 12 feet and b = 20 feet. A formula for the approximate length, /, of an open belt pass- ing around two pulleys, as in the figure, is given by the equation / = 2 d + 3^(7? + r), where d is the distance between the centers of the pulleys and R and r are the radii of the pulleys. 14. Find the length of the belt when the pulleys have radii of 2 feet and 1 foot respectively, and the distance between their centers is 7 feet. 15. Find d, when I = 27 § feet, R = 2 feet, r = 11 feet. 16. Find R if / = 80.5 feet, d = 24 feet, r = 4 feet. 17. Z = 12 feet 3 inches, R = 10 inches, r = 6 inches, find d. The formula for a crossed belt is l=2d + 3l(R + r). 18. Find the length of a crossed belt when the centers of the pulleys are 8 feet apart and their radii are 1.5 feet and .9 foot. 19. Find d if the length of the crossed belt is 26 feet and the radii of the pulleys are 1.8 feet and 1.2 feet. REVIEW EXERCISE 191. 1. Define algebraic expression, monomial, polynomial, binomial, trinomial. Illustrate each. 2. Define and contrast factor and term ; degree and power ; exponent and coefficient. Review Exercise 127 3. What are the four principles used in solving equations ? 4. What is the base in (-3) 2 ? in — 3 2 ? Compare the values of 3 2 and (- 3) 2 ; of 3 3 and (- 3) 3 ; (- 3) 2 and - 3 2 . 5. If a series of numbers are connected by the signs 4- , — , x , -r- , in what order must the operations be performed ? 3 + 2-5- 3(- 3)=? 6. Give the rules for adding algebraic expressions. How can results be tested? 7. What kind of expression is obtained by adding two like monomials ? two unlike monomials ? 8. What is the rule for subtraction? How may results be tested ? 9. Give the rule for finding the sign of a product. What sign has the product in (— l) 3 • (— 2) 5 • (— 7)? 10. State the law of exponents in multiplication. Without using the law of exponents explain why a 2 • a 3 = a 5 . 11. Give the rules for multiplying, (a) two monomials ; (b) a polynomial by a monomial ; (c) two polynomials. How can you test the correctness of the product ? 12. Give the rules for dividing, (a) one monomial by an- other ; (6) a polynomial by a monomial ; (c) one polynomial by another. How can you test the correctness of the quotient? 13. In multiplication, two factors are given and the product is required. In division, which two of these three numbers are given and which is required ? 14. How may the subtrahend and the difference be com- bined to get the minuend ? How may the divisor and the quotient be combined to get the dividend ? 15. If the minuend is positive and the subtrahend is nega- tive, what is the sign of the difference ? 16. What is the sign of the sum of two negative numbers ? of the difference ? of the product ? of the quotient ? 128 Simple Equations 17. If the product and the multiplier have the same sign, what is the sign of the multiplicand ? If they have opposite signs, what is the sign of the multiplicand ? 18. If the dividend and the divisor have the same sign, what is the sign of the quotient ? If they have opposite signs, what is the sign of the quotient ? 19. In the expression a — 3 m + 4p — 7 b +n — 15, inclose the third and the fourth terms in a parenthesis preceded by the minus sign, then inclose this parenthesis and the term imme- diately preceding and the one immediately following it in brackets preceded by the minus sign, leaving the final expres- sion of the same value as the original polynomial. 20. How can you test the correctness of the factors of an algebraic expression ? 21. What is the difference in meaning between Sx and sc 3 ? Illustrate when x = 5. 22. What is the difference in meaning between the square of the difference of two numbers and the difference of the squares of the same numbers ? Illustrate when the numbers are a and b. 23. Why do we change signs when removing a parenthesis preceded by the minus sign ? 24. Give the rule for squaring a binomial. 25. When is a binomial the product of the sum and the difference of two numbers ? 26. What must be added to x 2 4- 4 x to make it an exact square ? What must be added to x 2 + 6 x -f 4 ? 27. Is the product changed if an even number of its factors have their signs changed ? Compare the value of (a — b) 2 with (b-ay. 28. State the rule for cubing a binomial. 29. Define equation ; identical equation ; conditional equa- tion. Review Exercise 129 30. What is the root of an equation ? Are any of the num- bers, 1, 2, - 3, 5 roots of x 2 - 2x - 15 = 0? 31. Describe briefly the steps used in solving an equation. What is meant by transposing ? What principles are used in transposing ? 32. What important difference is there between the equa- tions (x — l)(x + l)=o; 2 -l and x 2 — 1 = 0. 33. Subtract 1 — x 4- 2 x 2 from x 3 . Subtract the same ex- pression from 0. 34. Divide x 2 — 7 x -f k by x — 2, giving quotient and re- mainder. How long should such divisions be continued? For what value of k will this division be exact ? In examples 35 to 45, A = a 2 + 3 ab - 4 b 2 , B — a z + 4 a 2 b -a& 2 -46 3 , C=a + 4b, D = a 3 + 64 6 3 . Perform the indicated operations : 35. B-(Ab + D). 37. B - AC. 36. D+C. 38. Aa - D + Cb\ 39. The minuend is B and the difference is D ; find the sub- trahend. 40. The divisor is C, the quotient is A, and the remainder is 16 b 3 ; find the dividend. 41. Find the value of B when a = — 2 and b = — 3. 42. Find the value of B when a = b. 43. Multiply B by C and verify the result by using a = 1, 6=2. 44. J. is quotient, C is divisor ; find dividend. 45. C 2 = ? Z> 2 = ? 46. Expand by type forms : (a) (3 s 3 -4 a) 2 , (d) (4a* + 5# 2 ) 2 . (6) (a> + 2 + 12) (a) - 12) -(a? + 8) 2 = 0. 59. Using x as the unknown number, write equations whose solutions will answer the following questions : (a) What number added to 23.7 gives 14.81 ? (b) What number subtracted from 12.84 gives 14.81 ? (c) What number multiplied by 98 gives 12.25 ? (d) What number multiplied by -l gives 12.25? (e) To what number must | be added if the result is to be equal to that obtained by multiplying the number by f ? 60. Solve the equations of 59. 61. Divide a 4 - b A by a — b. 62. What must be added to a? 4 — 3 a? — x + 5 to produce x 3 — x — 1 ? 63. Solve (4z-l)(a; + 3)-4a3 2 -(-10a; + 3) + 6 = 0. Review Exercise 131 64. (a 5 - 48 - 17 a 8 + 52 a + 12c**)-s-(a - 2 + a 2 )= ? 65. -bind (# -f l) 3 — (x — l) 3 when # = — |. 66. Simplify a - [3 a - b - 2(6 - a) + 3(a - 2 6)]. 67. Divide a,* 3 by x + 1. 68. State in algebraic symbols the type forms of multiplica- tion given as special products. 69. Find the quotients : (a) [3x + 3y + a(x + y)-]+(x + y). (b) (l-9a 2 e 6 )-r-(l- 3 at 3 ). (c) (l-9^ + 8^y)-(l-8^). 70. Electric light bills are paid at the rate of 14 ^ each for the first few units used and 4^ each for the remainder. A bill for 35 units was $2. How many units at each price were paid for ? 71. Think of a number, double it, add 13, subtract 5, divide by 2. Show that the final result will always be 4 greater than the number you first thought of. 72. Think of a number, multiply it by 3, add 6, divide by 3, subtract the original number. Show that the result will always be 2. 73. Divide x 3 — 10 x + 17 by x — a until the remainder does not contain x. Compare the remainder with the dividend. 74. Divide x 3 — 5 by x — a until the remainder does not con- tain x and compare as in example 73. 75. Divide x* — 5 by x — 2. VIIL FACTORING 192. If two or more algebraic expressions are multiplied together, the result is their product, and the expressions multi- plied are factors of the product. Thus, 3x5 = 15. .-.3 and 5 are factors of 15, also m(x + y) = mx + my. Here mx + my is the product of which m and (x + y) are the factors. Note. Unless otherwise stated, expressions containing fractions or indicated roots are not considered as factors. Thus, although 3 = 5 x f , or V3 x V3, we shall not in this chapter consider these expressions as factors of 3. 193. Prime Number. A number which has no integral fac- tors except itself and 1 is a prime number. Thus, 7, 23, a + b, a 2 + 3 b' 2 are prime numbers. Prime numbers used as factors are prime factors. Thus, a and a + b are the prime factors of a 2 + ab. 194. The student will recall that division is the process of finding one of two factors when their product and the other factor are given. In factoring it is required to find both fac- tors when only the product is given. Thus factoring, like division, is an inverse of multiplication. In arithmetic we learned a multiplication table and could factor all products that occur in the table from memory. For example, 42 = 6 x 7. Corresponding to this we have in algebra some type forms of multiplication (Chapter V), and we shall be able to factor the corresponding products from memory. 132 Factoring 133 Thus, & - y 2 = (x + y)(x-y), and a 2 + 2ab + b 2 = (a + b) 2 . Success in this kind of factoring depends upon ability to recognize these type products. ORAL EXERCISES 195. Factor the following : 1. ra 2 - n 2 . 5. a 2 -4. 9. h 2 + 2hk + k 2 . 2. p 2 - q 2 . 6. 4a 2 — 9. 10. a 2 + 2a + l. 3. & - d 2 . 7. a 2 + 2 xy+ y 2 . 11. p 2 -2pg4-9 2 . 4. h 2 - k\ 8. m 2 -2mp+i> 2 . 12. c 2 -2c-f-l. 3. a 2 + 4« + 4. 16. s 2 — 4 s£ 4- 4 £ 2 . 4. x 2 + 6x + 9. 17. /i 2 + 10 /* 4- 25. .5. ra 2 -8? 71-1-16. 18. a 2 &2 4. 2 ab + 1. 196. When we try to factor products not found in the multiplication table in arithmetic, we generally look for an exact divisor, following certain rules regarding divisors. Thus, 195 is clearly divisible by 5. If we divide, we get a quotient 39. Therefore 195 = 5 x 39. Similarly/ in an algebraic expression, if we can find a divisor, we can factor the expression. For example, 3 a 2 + 6 ab is clearly divisible by 3 a, and the quotient is a + 2 6, hence the factors of 3 a 2 + 6 ab are 3 a and a + 2 b. Also a(x + y)- b(x + y) = (x + y)(a -6). We proceed to classify some of the simpler types of factoring. 197. Case I. Factors of Monomials. Square Root. The factors of monomials are generally evident. If the two factors of a product are equal, either of them is the square root of the product. The radical sign (V ) is used to indicate that the square root of a number is to be taken. Thus, V9 x* = VS x 2 • 3 x 2 = 3 x 2 . 134 Factoring ORAL EXERCISE 198. Factor the following : 1. x* = x (?). 5. « 5 = a 2 (?) = a 4 (?). 2. 3 x 2 y=xy (?) = 3 y (?). 6. cr+ 2 = a n (?) = a n+1 (?). 3. a b b 3 = a6 (?) = a 2 b (?). 7. x 2n = a" (?) = #-i (?). 4. 72 a% 2 =9 a? (?) = 8 ay (?). 8. m%* = ra 2 ™ (?). 9. 6 e* +4 = 2 e 3 (?) = 2 e*+ 2 (?). 10. 39 aW = a&c (?) = 13 a 2 6 2 c 2 (?). Find the indicated roots : 11. VI. 19. V49 a 10 bu OQ /625 a 2 W 12. Vtf. 20. V121 mW. A 225 TO ""' 13. V4tf. ai. V169^. 27 - V i^. 4 - 14. V9aW. 22. Vl44a 4 6 8 . 28. -v/^TT" . ^ a 2 y 4 15. V25 ay. 23. V^V p4 — 16. V3tW. 24. VS- 29 ' Wrf&S" it. vsw 25 fiy. 30 /49Z. 18. V3 2 • 5 2 • x*yK yl n 2 ' \ 3 2 199. Case II. Type Form ab + ac, — Polynomials with the Same Monomial Factor in Each Term. 1. Factor 4 a 4- 6 b — 10 c. Solution. 4 a + 6 b - 10 c = 2(2 a + 3 b— 6 c). 2. Factor 2 a 3 ?/ + 6 a; 2 ?/ 2 - 8 xy 3 . Solution. 2x s y + 6 x 2 y 2 — 8 xy 3 = 2 xy{x 2 + 3 xy — 4 y 2 ) . 3. Has x 2 4- #?/ 4- y 2 a monomial factor ? 4. How may results in factoring be verified ? Factoring 135 200. To factor polynomials of the form ab + ac : 1 . Find the greatest monomial factor of every term of the polynomial. 2. Divide the polynomial by this monomial. 3. The factors will be the monomial and the quotient obtained by dividing. EXERCISE 201. Factor: 1. cy + dy. 8. — 5p + log. 15. 3a 2 + 4a&. 2. mp+np. 9. —7a -21 a. 16. 3x 2 + 6x. 3. rs + ps. 10. — pr— qr. 17. 4# 2 — 8 #. 4. /aA; + mk. 11. 2 ax -f- 6 ax. 18. jwj + qx 2 . 5. 2p + 2r. 12. 5 ]>q- 10 pq. 19. 35 - 14 a 2 . 6. 4s + 8.y. 13. Sax-6bx. 20. 42 - 28 p 2 . 7. 6r-12r. 14. -6cfc-18 67c. 21. 48 a + 6 a 2 . 22. a 3 - 3 a 2 + 7 a. 30. 2 x*y - 6 a 2 ?/ 2 - 8 xty 5 . 23. 3 ar»- 21 a 2 - 15 x. 31. 5 m 6 - 2 m 4 n + 10 m 3 ?i. 24. 27 x 2 - 3 a?/ + 15 y 2 . 32. a 2 — afc + ac — a. 25. ab 2 + a 2 6 + a 2 6 2 . 33. a 2 — ay — 6 a + 6 xz. 26. ax-\-bx — a. 34. ac — 6c + ac 2 d — bdc. 27. a n + a n+1 . 35. ax— bx — axy + #. 28. a 3 + a 2 6 — ab 2 . 36. x* — x— xhj — xy 2 . 29. x 2 — xy — xf. 37. a 3 + 4 a 2 + 3 a. ORAL EXERCISE 202. 1. State in algebraic symbols and in words the rule for squaring the sum or the difference of two numbers. (See § 130.) Find the indicated squares : 2. {x + yf. 7. (a-2) 2 . 12. (5p-6) 2 . 3. (x-yf. 8. (m + 5) 2 . 13. (2 a- 3 b)\ 4. (m + nf. 9. (7 - r) 2 . 14. (-3 + 2 m) 2 . 5. (p-q) 2 . 10. (0--9) 2 . 15. (2m-3) 2 . 6. (h + k)\ 11. (2z+3) 2 . 16. (_a;_2 2/) 2 . 136 Factoring Square : 17. O-10w) 2 . 20. -(x + y)\ 23. (_4m+3a 2 ) 2 . 18. (2mn-5p) 2 . 21. -(2-3 a) 2 . 24. (- m 2 w -p 2 ?) 2 . 19. (4aj» + 5y)*. 22. (-5a 3 -2 6 4 ) 2 . 25. (-a 2 -a) 2 . 203. If a trinomial contains tivo terms that are perfect squares, and if the absolute value of the other term is tivice the product of their square roots, the trinomial is the square of a binomial. Note. To make a 2 +b 2 a perfect square we add 2 ab (twice the product of the square roots of a 2 and b 2 ). To make a 2 4- 2 ab a perfect square we add b 2 (the square of the quotient 2ab+2a). To make 16p 2 +2bq 2 a perfect square we add 2 x 4p x bq = 40jp#. IQp 2 + 40 pq + 26 q 2 — (4 P + o +( ) + 16. (/) a 2 b 2 -6abc + ( ). (6) 4a 2 + ( )+96 2 . fo) 16m 2 -( )-f-25n 2 . (c) 25a 2 -f 10a+( ). (ft) 36p 4 + 24p 2 + ( ). (d) ( )+8a + 16. (0 ( )+16a 2 6 + 6 2 . (e) ( )-8z + 4. (/) 49 m 2 7i 2 + ( ) + 25/> 2 . Factoring 137 205. Case III. Type Forms a 2 + 2 ab + ft 2 and a 2 - 2 ab + 6 2 , — the Square of a Binomial. Factor a 2 4- 4 a + 4. Solution. The first term is the square of a, the last term is the square of 2, and the middle term is twice the product of a and 2. ... a 2 + 4 a + 4 = ( + 2) (a + 2) or (a + 2) 2 . 206. To factor a trinomial that is the square of a binomial : 1. Arrange the trinomial in order of the powers of some letter. 2. Extract the square roots of the first and last terms and connect the results with the sign of the middle term. The square of this binomial equals the trinomial. In algebraic symbols this rule may be written : a 2 + 2ab + & = (a + b) 2 or a 2 - 2 ab + ft 2 = (a - b) 2 . Examples 1. Factor 4 x 2 y 2 — 12 xyz + 9 z 2 . Solution. The arrangement is in order of powers of x. Step 2 of the rule gives 4 x 2 y 2 - 12 xyz + 9z 2 =(2xy-3 z) 2 . 2. Factor 9^ + 4^+12^. Solution. 9 x* + 12 x 3 4 4 x 2 = (3 x 2 + 2 x) 2 . EXERCISE 207. Factor the following : 1. x 2 - 20a 4- 100. 9. x 2 + 12 x + 36. 2. m 2 -f-6m + 9. 10. 6 4 -46 2 + 4. 3. 4a 2 +4a + l. 11. a 2 a 2 - 12 aa + 36. 4. 6 2 -106c + 25c 2 . 12. & 4 -46 2 c + 4c 2 . 5. x l + 2x 2 + l. 13. 9a 2 + 6a; + l. 6 . a s _ 40 a 4 + 400. 14. m 2 + 14 mn* + 49 n 6 . 7. x y 4- 4 a# + 4. 15. r%* - 10 rs£ 4- 25 £ 2 . 8. a 6 4- 6a 3 4- 9. 16. a 2 + 4 c 2 — 4ac. 138 Factoring Factor : 17. 2ax + a 2 +x>. 24. 81a + 18a 2 +« 3 . 18. 49 p 2 — 28pq + 4g 2 . (Hint. First apply Case II.) 19. x 2 b 2 + a~y 2 - 2 aba*/. 25. 49 a 2 b 2 c + 28a&c + 4 c. 20. 1 - 20a + 100a 2 . 26. 9a 4 6 2 +4c 2 d 4 -12a 2 6cd 2 . 21. 9- 12 a + 4a 2 . 27. 8 a 3 - 16 a 2 + 8 a. 22. 25 a 2 + 4 a 2 c 2 - 20 a 2 c. 28. aV + 4aa; 2 + 4. 23. 81a 2 & 4 -18a6 2 + l. 29. 9 x 2 - 42 a + 49. 30. 3a 2 x + 6ax 2 + 3^. 31. m 2 + 2mw + ?* 2 + 2(ra + w) + 1. 32. 4:X 2 + £xy + y 2 + 2(2x + y)z + z 2 . 208. State in algebraic symbols and in words the rule for multiplying the sum of two numbers by the difference of the same two numbers. See § 132. ORAL EXERCISE 209. Find the products : 1. (x + 2a)(x-2a). 4. [(a + &)+ c][(a + &)- c]. 2. {xy- z 2 )(xy + z 2 ). 5. [x -(a + &)][> + (a + 6)]. 3. (6aa-9a 2 )(6aa + 9a 2 ). 6. (12a 2 - 6 2 )(12a 2 + 6 2 ). Find the quotients : 7. (x 2 -y 2 ) + (x-y). 10. (25 - 16a 2 )--(5 - 4a). 8 . ( a 2_9)_j_( a +3). n. ( a 4_i 6 ) H _( a 2_4). 9. (9r 2 - l)H-(l+3r). 12. [(a + &) 2 -c 2 ]-=-[(a4-&)-c]. What binomial ivill exactly divide each of the following ? What is the quotient ? 13. a 2 -16. 15. 144 r 2 -121s 2 . 14. l-4a 2 . 16. a 4 -b\ Factoring 139 Factor the following : 17. g 2 - h\ 20. x 2 - y 2 z 2 . 18. a 2 -4. 21. 1 -81 a* 19. 4-9 c 2 . 22. x* — 2/ 6 . 210. Case IV. Type Form a 2 — b 2 , — the Difference of Two Squares. 211. To factor the difference of two squares : 1. Find the square roots of the squares. 2. Use the sum of the square roots for one factor and their difference for the other factor. In algebraic symbols this may be written : a*-&=(a + b)(a-b). Examples 1. 16a 2 -9=(4a + 3)(4a-3). 2. x 4 -y 2 =(x 2 + y)(x 2 -y). 3. (a-&) 2 -9c 2 =(a-& + 3c)(a-6-3c). EXERCISE 212. Factor the following : 1. 9 a 2 -49. 11. 25 a 2 6 4 -c 6 . 2. a 4 -4 a 2 . 12. 121 x 4 - 144 y\ 3. 25a 2 6 2 -c 4 . 13. 16 a 4 - 1. 4. a 4 -49. 14. 25 a 6 - 16 b\ 5. 16c 4 -25#. " 15. x 4 -x\ 6. a?y 2 - h 2 x\ 16. a % — b\ (Four factors.) 7. x 4 -y\ 17. 9 a 2 b 4 - 25 c 4 d 6 . 8. 9-16 a 2 & 4 . 18. a* - 100 afyV. 9. 144 -81, 19. 1 -400 a; 4 . 10. 64 - x*. 20. 9 - a 2 . 140 Factoring Factor : 21. 3 -27 a 2 . 23. 169 - z 4 . 22. 6 m 2 - 24. 24. 16 - a 4 b\ (Three factors.) 213. Case IV, a. Sometimes polynomials of four or six terms can be written as the difference of two squares. 1. Factor ra 2 + 2 mn + n 2 — x 2 . Solution, m 2 + 2 mn -f n 2 — x 2 = (m + n) 2 — x 2 = (m + n + x) (m + n — x) . 2. Factor a 2 -x 2 + 2xy - y 2 . Solution. a 2 — x 2 + 2xy — y 2 = a 2 —(x 2 — 2 xy + y 2 ) '= (a + x-y)(a-x + y). 3. Factor a 2 + 2 a& + b 2 — c 2 + 2 cd - d 2 . Solution, a 2 + 2 a& + b 2 - c 2 + 2 cd — d 2 = (a 2 + 2 a& + 6 2 )-(c 2 - 2 dc + + 2). 28. 3(ra -f nf — 5(m + n) 2 + m + w. 29. ri 2 (2m-l)-2n(2ra-l) + (2m -1). 30. p 2 +p + q+pq. 31. a 3 + a 2 -6a-6. 32. a# 4- bx + c# -f a -f 6 + c. 33. ac — 5 6c + a — 5 & — 6 c — 6. 34. a 2 -(a +&)# + «&• Hint. Remove parenthesis. 35. x 2 + (a — b)x — ab. 36. y 2 -(a-2)y-2a. 37. a 2 +(7 -y)x- ly. 38. p 2 -(a 2 -a)p-a 3 . 39. 4r 2 + 2(d-c)r-cd. 40. a 4 +(&-4)a 2 -4&. 146 Factoring 222. Case VI. Type Form jc 2 -f- bx + c, — the Product of Two Binomials having a Common Term. x + 2 x + p x + 5 x-\- q x 2 -\- 2 x x 2 -f- px 5x+ 10 qx + pq x 2 + 7.x + 10 x2+(i9 + q)x+pq. It is readily seen that x 4- 2 and x + 5 are the factors of se 2 -f 7 a; + 10, and x + p and a; + q are the factors of x 2 + (p + q)x + pq. In factoring a trinomial of the type form x 2 + bx + c (sometimes called a quadratic trinomial) the first term of each factor is x and the sum of the second terms of the factors is b and their product is c. (See § 134.) ORAL EXERCISE 223. Multiply the following : 1. (a + 3)0 + 4). 6. (xy-&)(xy + 7). 2. (a>-l)(& + 3). 7. (o6 + l)(a& + 3). 3. (a + 2)(a + 5). 8. (m 2 + 3)(m 2 - 7). 4. (a - 2){x + 3). 9. (m 2 - 3)(m 2 + 7). 5. (a 2 + 7){a 2 - 9). 10. (x -f a) (a + &)• 11. Find two numbers whose sum is 5 and whose product is 6. Factor a? + 5 a; + 6. 12. Find two numbers whose sum is — 5 and whose product is 6. Factor x 2 — 5 x + 6. 13. Find two numbers whose sum is — 3 and whose product is - 10. Factor a 2 - 3 a - 10. 14. Find two numbers whose product is — 6 and whose sum is — 1. Factor x 2 — x — 6. 15. Find two numbers whose product is 6 a 2 and whose sum is 5 a. Factor x 2 + 5 ax -f- 6 a 2 . 224. To factor a trinomial of the form x 2 + bx + c. 1. Find two numbers whose product is c and whose sum is b, the coefficient of x with its proper sign. Factoring 147 2. Write for the factors two binomials, the first term of each being x and the second terms the numbers found. Notice that when c is negative the second terms of the two binomial factors have unlike signs. When c is positive the second terms of the factors have the same sign as the middle term. Examples 1. Factor x 2 — 4 x — 5. Solution. The two factors of — 5 whose sum is — 4 are — 5 and + 1. ... cc 2_ 4x _5 _ (£_5)( X+ i). A variation of this type which will not cause any difficulty is seen in the following : 2. Factor a 2 + 3 ab - 18 b 2 . Solution. The two factors of — 18 b 2 whose sum is + 3 b are 6 b and — 3 b. ... a 2 + 3 a6 - 18 b 2 = (a + 6 b)(a - 3 b). 3. Factor a 2 b 2 - abc - 20 c 2 . Solution. The two factors of — 20 c 2 whose sum is — c are —5c and 4 c. .-. a 2 b 2 - abc - 20 c 2 = (ab - 5 c)(ab + 4 c). EXERCISE 225. Factor the following : 1. x 2 -lx + Vd. 5. a 2 -5a-14. 9. 2/ 2 + 3?/-l8. 2. z 2 -7a-30. 6. ra 2 + 5ra + 6. io. b 2 + 3 b + 2. 3. a 2 -7 a + 12. 7. 6 2 -96 + 20. 11. c 2 - 3 cd + 2 d 2 . 4. a 2 + 5 a— 14. 8. a 2 -fa — 42. 12. g 2 + 9g + 20. 13. a 2 - 11 03/ + 30 ?/ 2 . 18. x 2 - 27 a -f 182. 14. a 2 -f (m + n)x + raw. 19. x 2 — 28 a + 195. 15. r 2 + ar+6r-f a&. 20. a 2 - 29 a + 210. 16. # 2 +(a + &-fe>+ac+&c. 21. m 2 - 45 m + 164. 17. a 2 +(a+&+c>-+a&+ac. 22. a 2 -4a6-12 6 2 . 1 48 Factoring Factor : 23. s 2 -3.s-18. 25. m 2 7* 2 + 15wmp + 50p 2 . 24. p 2 — 5px + 6x 2 . 26. x 2 + 2xy — 3oy 2 . 27. t 2 + 2t + 1+50 + 1) + 6. 28. 7 m 2 — 14 mw + 7 w 2 — 91(m — ri) + 84. 29. 3 a 3 + 30 a 2 - 288 a. 35. a 4 -4a 2 + 4. 30. ^-^x-yV 36. b 4 + 4 6 2 c - 21 c 2 . 31. ap 2 — (3 g — 2)op— 6 aq. 37. a 2n — a" — 2. 32. r 2 -7rs-18s 2 . 38. x 2n + x n -2. 33. # 4 + 4 a 2 — 45. 39. x 2n —2x n — 15. 34. a 4 - 5 a 2 + 4. 40. a 3 - 5 a 2 x - 24 ax 2 . Case VII. Type Form ax 2 + bx+ c, — the General Quadratic Trinomial. 226. This type differs from the last type in that the coeffi- cient of x 2 is not positive 1. The expression is factored by changing the trinomial into a polynomial of four terms and then grouping. Factor 6 x 2 + 19 x + 15. Solution. 6sc 2 + 19z + 15 = 6jc 2 + 10* + 9*+ 15 = 2x(Sx + 5) +3(3 a; + 6) = (3z + 5)(2x + 3). The problem here is to change the original trinomial into the form in black-faced type. The numbers 10 and 9 which replace 19 as the coefficient of x are two factors of 90, and 90 is the product of 6 times 15. 227. To factor a general quadratic trinomial ax 2 + bx + c : 1. Find two numbers whose product is a x c, and whose sum is b. 2. Replace bx by two terms having these numbers as the coefficients of*. 3. Factor by grouping as in Case V. Factoring 149 Examples 1. Factor 6 x 2 + 7 x - 3. Solution. Here a = 6, b = 7, c = - 3. The two factors of a . c, that is of 6 • (— 3) = — 18, whose sum is 7, are — 2 and 9. Then we write 6x 2 + 7x-3 = 6x 2 -2x + 9x-3. = 2x(3a;-l) + 3(3z- 1) = (3x-l)(2z + 3). 2. Factor 3 a 2 - 11 a& + 6 b 2 . Solution. Two factors of 3 • 6 b' 2 = 18 & 2 , whose sum is — 1 1 b are — 2b and - 9 b. 3 a 2 - 11 ab + 6 6 2 = 3 a 2 - 2 ab - 9 a& + 6 & 2 = a(3a-2&)-3 6(3a-2&) = (3a-26)(a-36). EXERCISE 228. Factor the following : 1. 2z 2 + 3a;-2. 13. 7g 2 -20g-3. 2. 2x 2 - 7a- 15. 14. 2ra 2 + 9m-5. 3. 6a^-ic-15. 15. 6w 2 -5w-6. 4. 6m 2 -5m -25. 16. 9 a 2 - 9a- 10. 5. Up 2 - 39 p - 35. 17. 12 a 2 - 31 a - 15. 6. 10v 2 -29v-21. 18. 36?/ 2 -f-7^- 15a; 2 . 7. 2a 2 -4a + 2. 19. 8z2-38x + 35. 8. m 2 + 7£-12. 20. (m + 7i)2-ll( ?? i+ w )_26. 9. 6s 2 -s-12. 21. 15 #2 + 29 a -14. 10. 2b 2 -6b-8. 22. 12w 2 -31ti-15. 11. 4 + ±y-15y 2 . 23. (p 2 +p) 2 - U(p 2 +p)+2±. 12. 2z 2 -5a;-3. 24. 6a 2 -a6-356 2 . 25. (a + 6) 2 -3(a + 6)-54. 26. 5aj2 + io^ + 52/ 2 +20(a; + 2/)-105. 27. a 2 - (p -f l) 3 - i 12 5"' 11. **- - 125 in 3 n 3 . 12. 32 a J ! -1086 3 . 13. Z 2 ra 3 - jy. 14. Z> 6 - 64. (m 2 ) 3 - (n 2 ) 3 . Factor both 152 Factoring EXERCISE 233. Factor the following : 1. a 3 6 3 + 8. 2. a 3 + 27. 3. ra 3 -l. 4. m 6 + ^=(ra 2 ) 3 +( - 15 b 2 . 7. 2 a; 7 ?/ -3 ay + 5 ay. 15. 81 a 2 - 16(2 a - 3 xf. 8. z 4 + 22z 2 +169. 16. (2 a- 36) 2 - 4 b\ 17. 21 a 8 6 9 - 28 a b ¥ + 35 a 3 6 7 c. 18. (a + &) 2 +3(a + 6)-4. 19. asc — bx + ca + cm/ — by + cy. 20. 2ax — 5ay + a — 2bx + 5by — b. 21. (4a-5 6)(5c-2d)-(a + 46)(5c-2d). 22. asc-a+x-1. 33. 8 a 4 -or 2 -9. 23. a 2 - 6 2 + 2 6c - c 2 . 34. 12 a 2 - 17 a& + 6 6 2 . 24. a 2 — (c + 5) a + 5 c. 35. a6 2 c + bcx + a&i/ + xy. 25. a 2 + (a-6)a;-a6. 36. - x 2 + 2 a - 1. 26. x 2 -(n-3)x-3n. 37. a 4 + 4 6 4 . 27. 64 + 27 6 3 . 38. a 6 -6 6 . 28. 125 x*-8f. 39. a 8 -6 8 . 29. #y -81m 4 . 40. a 12 - 6 12 . 30. a 4 + 2 ajy - 15 2/ 4 . 41. 8a 4 -^ 2 -9^. 31. 3^ + 8^ + 5. 42. 12x 2 + 2xy-30y 2 . 32. 2 a 2 + 13 a + 15. 43. m 4 + 4 m 2 p 2 + 100 p 4 . Review Exercise 155 44. 2r 2 +16rs + 32s 2 . 45. (a + b) 2 - a 2 + b 2 . 46. (x — y) 2 — xrz-\- y 2 z. 47. 5 tfy - 23 x 2 y 2 + 12 xy\ 48. a6 2 c -|- 2 fcesc — afo/ — 2 a;?/. 49. (a + &) 2 +(a 2 -6 2 ,+ a + &. 50. 64 x 2 + 81 y 2 - 144 an/. 51. 18 a 4 + 72 wi 4 . (3 factors.) 52. 8 a 3 + 64 a?. 55. 125a*»-y». 53. 4a 2 -12a& + 96 2 -9. 56. 4 - 12 ab - 4 a 2 - 9 b\ 54. 18 a 2 6 2 +32 a*+48aWV 57. a - a 7 . 58. (a + 6) 3 - (a 2 - b 2 ) (a + 6) + a (a + 6) 2 . 59. (z + 2/) 4 -(x + 2/)*. 60. 36 a 4 6 2 c 2 - 24 ab b c 2 + 72 a 3 6 3 c. 61. 30 arty 3 - 45 x y + 60 an/ 5 . 62. ( a -b) 2 -(a-b). 63. z 2 -z-156. 64. 12 a&z 2 + 48 a 2 & 2 an/ -f 48 a 3 &y. 65. 6 a* - 10 a 4 a; - 18 a 2 a; 2 + 30 ar>. 66. 3 a- 3 + 3 a- 2 -36 a;. 67. 9 a- 2 - 4 y 2 + 4 yz - z 2 . 68. (a + 3 6) 2 -9(6-c) 2 . 69. a 2 (l-c)-4&2( c _i). 70. a?/ (a; — m)—ax (y — m). (First expand.) 71. ay{x — m)—ax(m — x). 72. a 2 a; -f- ab 2 x — afo/ — Wy. 73. 3 (m + n) 3 - 4 (m + n) 2 + ra + w. 74. 4(a - 6) 3 + 12 a(a - 6) 2 - 6(a - 6)a 2 . 75. 2 w(2 m - 1) - 3 rc 2 (2 m - 1) + 5(2 m - 1). 76. 2 ax -h 3 6a; -f- 4 ca; — 2 a# — 3 by — 4 cy. 77. 2 a# — 3 bx + 4 ca; + 2 ay — 3 by + 4 c#, 1 56 Factoring Factor : 78. 2 ax — 3 bx + 4 ex — 2 ay + 3 &# — 4 cy. 79. 27aaj 3 + 8ay. 80. aW -b z -a 2 + l. 81. (a — b)(x — y) — (a — y)(x — b). Expand. 82. (a + 6) 2 - 4 - 2(a + 6 - 2). 83. a 2 b + 6 2 c + c 2 a - ab 2 - be 2 - ca 2 . 84. Factor x 6 — 2/ 6 into two factors in two different ways 85. a 6 + & 6 . 93. a 4n -l. 86. a 10 -b 10 . (2 factors.) 87. a 12 -b 12 . 88. a 2n -l. 89. a 2n -b 2n . 90. a 2n -a 2 . 91. a 2n+1 — a. 92. a 2n + 2a"H-l. THE SOLUTION OF EQUATIONS BY FACTORING 237. A root of an equation has been defined as a value of the unknown quantity that satisfies the equation. Is 2 a root of x 2 — 5 x + 6 = ? is la root ? is 6 a root ? is 3 a root ? 238. It is sometimes possible to write an equation in such a form that its roots are evident. 1. Consider the equation, x 2 — 7 x -f- 10 = 0. Factoring the first member, we have the same equation in another form. (a:_5)( x _2)=0. If the product of two or more factors is zero, one of the factors must be zero. Therefore this equation is satisfied, if x — 5 = ; that is, if x — 5, or if x — 2 = ; that is, if x — 2. Therefore, 5 and 2 are roots of this equation. The roots may be verified by putting 5 and 2 for x in the equation. 94. a n+1 + 2a n b + a + 2b. 95. (a — b) m+1 —(a — b) m . 96. #3«+l _ x. 97. x 2n — y 2n — x n — y n . 98. x 2n — y 2n -f- x n + y*. 99. a 18 - 1. 100. a 32 - 1. The Solution of Equations by Factoring 157 2. Solve z 3 = 4 x. Make the second member zero by transposing 4 x. x 3 -4z = 0. x(x — 2)(x + 2) = 0. (Factoring.) The second equation is satisfied if any one of the three linear equations, x=0, z-2 = 0, z + 2 = 0, is satisfied. (Why ?) This gives as solutions of the given equation x = 0, 2, or — 2. Verify by substituting 0, 2, and — 2 for x in the original equation, x?=4x. We can solve equations by factoring if, when the second member is zero, we can factor the first member into factors of the first degree with respect to the unknown number. 239. To solve an equation by factoring : 1. Write the equation with the second member zero and the first member arranged in descending powers of the unknown number. 2. Factor the first member into linear factors with respect to the unknown number. 3. Put each factor equal to zero and solve the linear equations ob- tained. Examples 1. Solve2«(a;-l)=3a-2. Solution. 2x 2 — 2x = Sx- 2. (Multiplying.) 2z2_ 5a; _ f 2 = 0. (Transposing.) (2E-l)(Z-2)=:0. (Factoring.) 2z-l = or x = \. x-2 = or x = 2. Let the student verify the roots. 2. Solve9(z-l) = (a; + 4)(a: -i). Solution. 9(x-l)-(x + 4)(x-l)=0. (Transposing, without multiplying.) (x- l)(9-s-4)=0. (Factoring.) (z_l)(5-a-) = 0. x — 1 = or ce = l. 5 — £=0or:r:=5. Verify by substituting the roots in the equation. 158 Factoring EXERCISE 240. Solve the following equations : 1. (a; -5)0- 4)= 0. 7. x 2 -6x = T. 2. O + i)(7a;-l)=0. 8. a; 2 = 13a;-42. 3. (2x-5)(x-3)=0. 9. (a; -1)(« -2)= 12. 4. o?(3a;-7)=0. 10. 9 x 2 - 16 = 0. 5. (4-aj)(5aj + l)=0. 11. 4a; 2 -4a + 1 = 0. 6. z 2 -7a=-10. 12. ?y ( ? y_6)=72/-42. 13. (2, -11)0/ -12)= 2. 14. ( r +6)(r-4)-(2 + r)(2-r)=56. 15. 0- l) 2 +(^+ 1) 2 = 29 -(2a: -f-3) 2 . 16. 2a; 2 - 5a; = 3. 17. (x - 2) 2 - (x + 2) 2 + 7 x = 0. 18. (3a:-5)(3a: + 5)-(a;-l) 2 = 10. ♦ 19. 0+ l) 3 -3a;(a;-l)=a^-|-l. 20. (a> + 2) 3 -2(a;+2) 2 = 0. 21. x 2 — ax — bx + a& = 0. 22. a; 2 — 4 a 2 - 4 a — 1 = 0. 23. a; 3 + a; 2 =a; + l. 24. O-2) 2 +25 = 10O-2). 25. (x-7)(2a; + 5)=(3a;-l)(a;^7). 26. (a;-3)(4a;-5) = a: 2 -9. 27. x~ — 9= 8x. 28. (2 as — f) -4=(2a;-f)(5a;-ll). 29. 0_2) = 0-2)(a;-3). 30. (aj - 1)0 - 2)(a; - 3) + 6 = 0. (Find one root only.) The Solution of Equations by Factoring 159 SOLUTION OF PROBLEMS 241. 1. The larger of two numbers exceeds the smaller by 5, and their product is 84. Find the numbers. Solution. . Let x = the smaller number. Hence x + 5 = the larger number, and x(x + 5) = their product. Then x(x + 5) = 84. (By the conditions.) Hence x 2 + 5 x - 84 = 0, (Why ?) or (a; + 12) (a; - 7) = 0. (Why ?) .-. x + 12 = and x - 7 = 0. .-. x-- 12 or 7, and x + 5 = - 7 or 12. The pairs of numbers that satisfy the conditions of the problems are — 12 for the smaller and — 7 for the larger, or 7 for the smaller and 12 for the larger. To check the answers they should be put into the original problem and not into the equation. (Why ?) 2. The length of a rectangular figure is 5 inches more than its width, and its area is 84 square inches. Find its dimensions. Solution. Let x = the number of inches wide. Hence x + 5 ==' the number of inches long, and x(x + 5) = the number of square inches in area. Thenx(a- + 5) = 84. From this point the solution is exactly like that of the last problem. The answers — 1 2 and 7 as values of x have to be considered in connec- tion with the problem. The answer — 12 cannot represent the number of inches in the width of a rectangle and is to be rejected in this problem. x = 7 is evidently the answer to be used. This will make the dimensions of the rectangle 7 inches and 12 inches. 3. The product of two consecutive numbers is 72. Make the equation and solve for the numbers. 4. The product of two consecutive even numbers is 80. Make and solve the equation to find the numbers. 5. The sum of two numbers is 19 and their product is 84. Find the numbers. Hint. Let the numbers be represented by x and 19 — x. 160 Factoring 6. One of two numbers is twice as large as the other and their sum is 14. Find the numbers. Hint. Let x and 2 x represent the numbers. The equation is of first degree. 7. One of two numbers is twice the other and their product is 242. Find the numbers. 8. A rug is twice as long as it is wide. It contains 4^- square yards of material. Find its dimensions. 9. The perimeter of a rectangle is 40 inches and its area is 91 square inches. Find the dimensions. Hint. If the perimeter is 40 inches, the sum of the length and the width is 20 inches. 10. The side of one square is 4 inches more than that of another and the sum of their areas is 136 square inches. Find the side of each square. 11. If this page is 6 centimeters longer than it is wide, and its area is 216 square centimeters, find the dimensions. 12. The quotient exceeds the divisor by 8 and the dividend equals three times the sum of the divisor and the quotient. Find the divisor, the quotient, and the dividend. 13. The sum of the squares of two consecutive numbers exceeds 5 times the sum of the numbers by 6. Find the numbers. 14. A rectangle is 3 inches longer than it is wide. If both dimensions are increased by 2 inches, the area is 28 square inches. Find the original dimensions. IX, HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE HIGHEST COMMON FACTOR 242. Rational Term. A term is rational if, when reduced to its simplest form, it contains no indicated roots. Thus, Sac 2 , 3a 2 6, and Viare rational terms. 7 ay/b is not rational with respect to ft but it is rational with respect to 7 and a. Is V27 rational ? ^27? 243. Integral Term. A term is integral with respect to any set of numbers or letters if none of the numbers or letters appear in the denominator. Thus, lab is integral with respect to 7, a, and b but not with respect to 3. 4mn s pq 2 is integral with respect to all letters and numbers involved. — — is not integral with respect to c. c ... 244. A polynomial is rational and integral if all its terms are rational and integral. Thus, 3 ax + 2 by is rational and integral. 2 ft 3 ax H is rational but is not integral. y 3 ax + V2 by is integral but is not rational. An expression may be rational and integral with respect to some particular letter involved. The three examples just given are all rational and integral with respect to x. b c Also x 2 +- - x + - is rational and is integral with respect a a to x } b, and c, but is not integral with respect to a. 161 162 Highest Common Factor 245. Degree. The degree of a rational integral monomial is the sum of the exponents of its literal factors. Thus, 2 ax and a 3 are of third degree ; 4 ab 2 c is of fourth degree. Of what degree is 7 x^ ? Sax? 2 a 2 x 2 ? We are sometimes concerned with the degree of a monomial with respect to some particular letter. Thus, 3 a' 2 x is of the second degree with respect to a. It is of the first degree with respect to x. 246. The degree of a rational integral polynomial is the same as that of its term of highest degree. Thus, 3 x s + 2 x 2 y 2 is of the fourth degree. Of what degree is ax 2 + bx+c? of what degree with respect to x? with respect to a ? b? c? 247. The student should note that degree and power are not the same. The power of a term may or may not be the same as the degree of the term. Thus, 3 x 2 y 2 and a 4 are both of the fourth degree, but 3 x 2 y 2 is not a fourth power. Also (a 2 + 2) 2 is a second power, but is a fourth degree expression. ORAL EXERCISE 248. Give the degree of each of the following : 1. 3 a 2 by. ' 5. ax 2 + bx + c. 2. 4 obex. 6. axy + by 3 . 3. 2 xhf. 7. (x + y) 2 . 4. 5 axy. 8. (x + y)\ 9. (x + l) 3 + a(x + l) 2 + b(x + 1). Of what degree is each of the above with respect to a;? with respect to x and y ? 249. Common Factor. If the same factor occurs in two or more algebraic expressions, it is a common factor of the expressions. Thus, x is a common factor of 7 a; and 3 xy ; and 2 a is a common factor of 2 a, 4 a, and 6 a 8 . Highest Common Factor 163 250. Two or more expressions may have several common factors. Thus, 35 x 3 y 2 , 21 x 2 y B and 42 x s y s have what common factors of the first degree ? of the second degree ? of the third degree ? of the fourth degree ? Can you find a common factor of these expressions of higher degree than the fourth ? 251. The highest common factor (H. C. F.) of two or more monomials is the greatest common divisor of their numerical coefficients multiplied by their highest degree literal common factor. Thus, 7 x 2 y 2 is the H. C. F. of 35 x*y 2 , 21 x 2 y*, and 42 x s y*. 252. The H. C. F. in algebra corresponds to the greatest common divisor (G. C. D.) in arithmetic. The G. C. D. is the largest number that will exactly divide two or more numbers ; the H. C. F. is the highest degree algebraic expression that will divide two or more expressions. We may find the G. C. D. of 12, 18, 24 by factoring thus : 12 = 2 2 • 3, 18 = 2 • 3*, 24 = 23 . 3. Therefore the G.C.D. of 12, 18, and 24 is 2 . 3 = 6. Similarly, we may find the H. C. F. of two or more algebraic expressions. Find the H. C. F. of 12 a?bc, 18 aWc 2 , 24 a?c. Solution. 12 a 2 bc = 2 2 • 3 • a 2 bc. 18a 3 W = 2.3 2 -aW. 24 a?c = 2 3 • 3 • a s c. The H. C. F. is the G. C. D. of the numerical coefficients, 6, multiplied by their highest degree literal common factor a?c ; that is, the H. C. F. is 6 a 2 c. 253- To find the H. C. F. of two or more algebraic expressions, multiply together the lowest powers of all the prime factors common to all the expressions. In the case of monomials the H. C. F. is seen by inspection. If any of the expressions are polynomials, factor them into prime factors. 164 Highest Common Factor EXERCISE 254. Find the H. G. F. in each of the following : 1. 3a 2 &, 6aW, 9ab 2 . 2. 4a¥, 6aW, 12 ax 2 . 3. Ua 2 bW, 98ff 3 6 2 a 4 , 105 a'W 5 . 4. 45 a 6 , 18a 5 6 7 , 108 a 4 6 12 . 5. 13 x*y\ 52 a 3 ?/ 6 , 169 x*y\ 6. 3a% 3 , 9 xy, 12 a?y, 15 a% 5 . 7. 4a% 2 , 16 #y, 64 a% 4 . 8. 98 a¥, 180 arV, 300 sc 4 * 5 . 9. 15 a*bxy, 45 6 s ?/ 4 , 90 a^x 4 . 10. 14(a + b)\a - b), 10(a + &)< 11. a 2 6 - 6 3 , a 2 b - 2 ab 2 + b 3 , a*b - ab*. Solution. a?b — 6 8 = 6(a + &)(a — b). a 2 b-2ab 2 + b* = b(a-b) 2 . a*b - a6* = ab(a - b)(a 2 + ab + 6 2 ). .-. H.C.F. = 6(o-6;. . 12. 24 a 2 x 2 -f 36 a 3 x*, 9 a£ - 12 a 2 x 2 . 13. 3 aa? + 4 to 4 , ax 5 - 12 to 6 . 14. 4a 2 6 2 a; 2 -8ato 3 , 8 a 2 bx* - 12 abx 2 . 15. 18 a 2 6V - 72 a 8 , 12 ato 4 . 16. 2x 2 -17a+36,4a 2 -12a-27. 17. (a + &) 2 - c 2 , a 2 -(6 + c) 2 . 18. 9 a? 4 - 16 y 4 , 9 a 2 a 2 + 12 a 2 ?/ 2 . 19. 4a 2 +12a'?/+92/ 2 , 16a + 24#. 20. (a + b) 3 , a 2 + 2 a& + & 2 , « 2 - & 2 - 21. 48 a 4 - 12 s/ 4 , 20 x 3 - 10 a;?/ 2 . 22. a 4 - & 4 , a 3 - 5 3 , a 2 - ?> 2 . 23. a? 2 -5a + 6, 3a; 2 - 6a?, a; 2 - 6 a; + 8. 24. 3a; 2 -a;-2, 6a; 2 + 13a; + 6, 6a; 2 -5a5- 6. Lowest Common Multiple 165 25. 2 a 2 b + 2 a& 2 - 2 ad^ 3 6c 2 - 3 6 2 c - 3 abc. 26. a 2 + b 2 - c 2 + 2 a&, a 2 - 6 2 + c 2 + 2 ac. 27. a 2 — 6 2 — ac + 6c, a& + ac + 6 2 — c 2 . 28. mx — m — x + 1, m 2 — 2 m + 1. 29. 2a6- 3ac- 26 + 3c, 3a6-2ac-36+2c. 30. x 2 - x - 20, x 2 + a - 30, x* - 25. LOWEST COMMON MULTIPLE 255. A product is a multiple of any of its factors. Thus, 3 x 2 y is a multiple of as ; of xy ; of 3 x ; etc. 256. A common multiple of two or more expressions is a multiple of each of them. Thus, 6 x 2 y 3 is a common multiple of 3 x, 2 y and xy. Two or more expressions have always an infinite number of common multiples. Thus, 3x, 2y, and xy have as common multiples 6xy, 6x 2 y 2 , 6x 2 y, 12 xy, etc., indefinitely. Can you find a common multiple of these three monomials of lower degree than the second ? 257. The lowest common multiple (L. C. M.) of two or more monomials is the arithmetical least common multiple of their numerical coefficients multiplied by their lowest degree literal common multiple. 258. In arithmetic the least common multiple of two or more numbers is the smallest number which may be exactly divided by each of them. In algebra the L. C. M. of two or more ex- pressions is the lowest degree expression which may be ex- actly divided by each of them. 259. To find the L. C. M. of two or more algebraic expressions, mul- tiply together the highest powers of all the different prime factors in the expressions. 166 Lowest Common Multiple The L. C. M. of monomials is seen by inspection. If the expressions are polynomials, first factor them into prime factors. 1. Find the L. C. M. of 9 b\ 12 ac 2 , 4 abc\ Solution. 9 b 8 c = 3 2 b% 12 ac 2 = 3 . 2' 2 ac 2 , 4 abc s = 2%&c 3 . .-. L. C. M. = 3 2 • 2Wc 3 or 36 ab 8 c*. 2. Find the L. C. M. of a*-3a + 2, a 2 - 1, a 2 -4a + 4. Solution. a 2 - 3 a + 2 = (a - l)(a — 2). ^1= (a+l.)(a-l). a 2 - 4a + 4 = (a - 2) 2 . .-. L.C.M. is(a-l)(a + l)(a-2)«. EXERCISE 260. i^md* £/)(a-^) 2a(a 2 + 6 2 ) 267. In canceling a factor from the numerator and the de- nominator of a fraction the quotient 1 is not generally written. It is important to remember, however, that if all factors of either the numerator or the denominator are canceled that term of the fraction becomes 1. If the quotient is — 1 it should be written. -b m~*1 1 Thus, 3 a 2 - 8 ft 2 Z( e a^d) (a + b) 3(a + b) - 1 b — a -ft— =tt — 1 Also, a 2 - b 1 (a + b) (p^5) a + b ORAL EXERCISE 268. Reduce each fraction to its lowest terms : 1. 2. 3. X 7. rs ~r%' 13. — abc xy — a 2 c a 2 a 4 ' 8. Wed 3c ' 14. 9 % Vw 2 3uvw* ab V 9. 36 h 2 7c 2 9hk 15. 7pq 2 v 2 21 gv 2 ' 2xy 4 10. 72 mn* Smn 2 16. -xy 2 z Sc 2 d 6cd 11. — 11 a;?/ 33 a 17. — a 2 6d 2 4ad 10 pq 2 5pq 12. 14 a — ab 18. 102 Mm 51 & 2 ra 2 170 Fractions EXERCISE 269 Reduce to lowest terms : L S*£_ l4 x*-l 13. 15 xPy* (x + 1)2 147 a 3 » 2 tr a 2 -6a 15. 49aV a 2 -la + 6 3 27 a^c 4 3a 2 6-9a&2 3a6c a *-7a 2 b + 12a& 2 4 17 a 2 - 16 g - 17 51a 4 6 5 ' ' a; 2 -22 a; + 85' 35 a^c 4 lg a 3 & - ah* 42 a^c 5 ' ' a 2 6 3 -a 4 &' 6 39rg * 19 (5a-7) 2 65 rW' ' 50 a 2 -98' 7 (-2a6c) 3 2Q a« + afy 8a6 2 c3 ' ' a? + 2ajy + y 2 ' g (l&tnWpWffin) 21 7a? + 3 (14p 3 )(5m 2 (/) * * 245 a* + 210 a 3 + 45 aT 9 a 2 + ab 22 a 4 -6 4 a + a 2 a 2 + 2 a& + 6 2 10 x2 ~ l 23 g ~ Xy + * ~* y (a-1) 2 ' ' l-3y + 3y*-y*' w " 3,2 _ y2 ^ 0^ +(a + 6)fl? + aft ^ ' xz — yz' x 2 + (a + c) _ 2) = (1 - a>)(2 - x). 2. Show that (x - l)(x -2) = ~(x- 2)(1 - x). 3. Compare (x - l)(x - 2) (a? - 3) with (1 - x)(2 - x)(S - x). Reduction of Fractions 173 4. Make fractions equivalent to each of the following having the sign of the denominator changed : / \ a /j\ — b z \ m 3 * ' (-cy "' (-a)(-6) (b) L. ( e ) - (h) - W -b w -{-by V ; (-a)(-6) 2 « p* w -z^- (o p^ 5. Give a fraction equivalent to each fraction in example 4 with the signs of its numerator and denominator positive. 6. If a, b, and c are all positive numbers, what algebraic sign has the number represented by each fraction ? W (- &)c U; (_ tt -6-c) 7. What change is necessary to make the denominator the same in each of the following pairs of fractions ? ( a ) z — z> I — I' ( 6 ) a — b'b — a (x — y)(a — b)' (x — y)(b — a) (c) i , * . K ) (a-2)(a + 2)' (2-a)(2+a)' 8. Arrange the denominators of the following fractions in descending powers of x with the sign of the first term in each denominator positive. (as-2)(3 + a!) w 2-3*-a? w (3-x) 3 (6 ) "-» (d) uJLdL . (/) (1 — s)(l+a:) w (3 -a;) 2 w ' (2 - x)*(x + 2) 174 Fractions 275. An algebraic improper fraction is one whose numerator is of the same degree in some letter as the denominator, or of higher degree (§ 245). Thus, — ^-i and — - — are improper fractions. a 2 — a — 1 a 2 — 1 276. A mixed expression is an expression containing both integral and fractional parts. Thus, a 2 + - is a mixed expression. a Any integral expression may be written in fractional form by supplying the denominator 1. Thus, a + x = ®-±^. 277. Improper fractions may be reduced to integral or mixed expressions in the same way as arithmetical improper frac- tions, that is, by dividing the numerator by the denominator. Thus, ^e = aV and«=ii = l-i. xy c c ORAL EXERCISE 278. Reduce the following improper fractions to integral or mixed expressions : s 2 -P s — t ' ax + 3 a h + k h m 2 -4 • 11 a — m a — b "•?• 5. » 1J r- 6. 3 12x \ 3x 7. a 22 ^ 4. — - — • 8. 9. c — d c 10. c — d d 11. abc — d ab 12. a 2 - ab 2 Reduction of Fractions 175 279. When the numerator and the denominator are both poly- nomials, the problem is similar to that of an inexact division (§ 160). The process will be understood by studying an example. to a mixed expression. ■ ***&*-•***& « = x + 6 - -I-s (2) Why is the form (2) of the same value as (1) ? The division should be continued until the remainder is of lower degree than the divisor. EXERCISE 280. Reduce the following improper fractions to integral or mixed expressions: , x 2 + 1 p a? + b z o. Change : *. -j- x — 2 x 2 + 3 x - - 17 \x-2 x 2 — 2x \x + 5 bx — 17 5x — 10 x-1 x 2 -l x-1 & + x _ 13 x+1 Sx 2 + lx x + 1 (x 2 - x) 2 — X a 2 + b 2 a + b a? + b* 9. 10. 11. a — b x h -l x-1 6xy — ly* t xy ^-3^+7 a?-3 12 : as* + x 2 y 2 + 2y\ x 2 + xy + y 2 6. ' ■ 13. 14. 273?+ 27x 2 y + 9a;y 2 + f Sx-y a?-6a 2 b + 12ab 2 + $b z a + b a 2 + 4a6 + 46 2 176 Fractions Reduce to integral or mixed expressions: 15. *<* + ?»; yi. 9±L a + 2b x — 1 x — 1 w> 3rf + 5s-8. 18< _a^. 2Q> at a? + x+l x-1 x — 1 281. To add arithmetical fractions, we must change them to fractions having a common denominator. In reducing fractions to a common denominator we use the following principle : If the numerator and the denominator of a fraction are multiplied by the same number (not zero), the value of the fraction is not changed. ORAL EXERCISE 282. Change each of the following fractions to an equivalent fraction whose numerator or denominator is as indicated : 2x Ux a-l_ ( a + l a? ) -1 a-1 a 2 -1 1. 2. ^ = LJ. 7 . b 6 2 a + l ( ) 3 . 2s.U.. 8 . I+! = Jl+^, 5y 15xy a b ( ) ( ) 4. n = Smnx ^ 9 a +l — L_l + i. 2n 2 x ( ) a a a «r=y. io. 3 +r 5_ = P-+-2-. arx 2 b + c 6 + c & + c 11. A. + ^ = l±xll. a + 6 a - 6 a 2 - 6 2 a 2 - 6 2 (— a) 3 a& a 3 6 a?b 13 . 1+i+i.LJ+y+t^. a b c abc abc abc 14. What principle is involved in all these examples ? Reduction of Fractions 177 283. The lowest common denominator (L. C. D.) of two or more fractions is the lowest common multiple of their denominators. Thus, the L. C. D. of and is ab(a 2 -b 2 ). a(a-b) b(a + b) v J To change these fractions to a common denominator, both terms of the first fraction, , must be multiplied by b (a + b) and of the second a(a — b) fraction, by a(a — b). b(a + b) Thus, I = Ka + b) and 1 = a(a-b) a(a-b) a6(a-6)(a + 6) b(a + b) ab(a-b)(a + b) 284. To change two or more fractions to equivalent fractions with a common denominator : 1. Factor the denominators into prime factors. 2. Find the lowest common denominator (L. C. D.) of the fractions. 3. Multiply the numerator and the denominator of each fraction by all the factors of the common denominator except those factors that are in its own denominator. The multiplication in part 3 of the rule is generally indicated in the denominator and performed in the numerator. Why does step 3 of the rule not change the value of the fraction ? Example Change and to equivalent fractions having a 2 — ab ab — b 2 the L. C. D. Solution. a 2 — ab =a (a — 6). ab-b 2 = b(a-b). .-. L.C.D. = ab(a- b). (The L. C. D. is the L. C. M. of the denominators.) 1 = 1 _ b a 2 — ab a(a—b) ab(a — b) 1 1 a ab - b 2 b(a - b) ab{a - b) 178 Fractions EXERCISE 285. Change the fractions in each example to equivalent frac- tions having the L. C. D. 1 1 1 a 2 b' ab 2 ' a b c be' ca ab X — X i 1 1 2 a 2 -ab' ab + b 2 K a b c 3. x — y x 2 — y 2 y +■ x a a b x y y — x (-yY {-yf Hint. In example 6 change the denominator of the second fraction to x — y. Explain. 7 1 11 8. a? + 4^+3' aj 2_ 1 ' x+1 1 1 1 tt 3_53' a 2 + a6 + 6 2' a 2_52" Q x+y y + z z + x (y-z)( z - x Y (z-x)(x-y)' (z-y)(y-x) . Solution. First change the signs in these fractions, to avoid the repetition of a factor with opposite signs. x + y y+z z + x (y — *)(# — £)' ( z -x)(x-y)' (y-z)(x-y) Why is the change made in the last fraction permissible ? L.C.D. = {x-y){y-z){z-x): x + y x 2 — y 2 (y - z) (z - x) ~ (x - y) (y - z) (z - x) ' y + z y 2 — z 2 (z _ x) (x - y) ~ (x - y){y -z)(z-xY 10. 11. ( y - z)(x -y) (x- y) (y - z){z- x) 1 1 1 a 2 -6a + 9' 9-a 2 ' a-3* 1 1 1 a; 2 - a; -12' x 2 + 8x+15' z 2 + x-20 Addition and Subtraction of Fractions 179 3a + b a — b i/G. 6a 2 -a&-56 2 ' 18a 2 + 21 ab + 5 b* 13. a + x a — x a 2 -+- x 2 4 ax a — x' a + a' a 2 — x 2 ' a 2 + x 2 14. 5 a 16 a 2 — 17 ab b 6 6' 12a6-66 2 ' b -2a 15. 3 2 2 a + 15 2a-3' 3 + 2a' 4a 2 + 9* 1« b - a 2ab + c 2 c-2b bc-2b 2 -ac + 2a? ADDITION AND SUBTRACTION OF FRACTIONS 286. In arithmetic, only the same kind of units, or the same parts of units, can be added. Hence, if two or more fractions are to be added, unless they already have a common denom- inator, it is necessary to reduce them to equivalent fractions having a common denominator, before they can be added. Their sum is then found by adding the numerators of the fractions and dividing the result by the common denominator. Thus, I + f + f = I±|±3 = S 3 12 15 60 60 60 60 60 ™ 287. The same principle applies when the difference of two fractions is to be found. The difference of two fractions having the same denominators is the difference of their nu- merators divided by their common denominator. 'hus, -- ' 8 5_7-5_2_l ~8 8 8 4' and — 12 7 _ 15 14 _ 1 18 36 36 36 180 Fractions 288. The sum or the difference of algebraic fractions can be found in the same way. 1 a i ^ c _ a + b + c m m m m 2. Add: -J5- + -^ + _J^. 2 an 2 2 bm 2 abmn Solution. L. C. D. = 2 abm 2 n 2 . m bm 2 • m bm z 2 an 2 2 abm 2 n 2 2 abm 2 n* n an 2 • n an* + 2 6m 2 2abm 2 n 2 2 abm 2 ri 2 p _ 2 mn • p _ 2 mnp abmn 2 abm 2 n 2 2 abm 2 n 2 p _ bm 3 + an 8 + 2 mnp 2 an 2 2 6m 2 «6mw 2 abm 2 n 2 3 a & a — b a + b Solution. L. C. D. = (a - 6) (a + 6) or a 2 - b 2 . a b = a(a + b) b(a - b) = (a 2 + ab)-(ab -b 2 ) a-b a + b a 2 - b 2 a 2 - b 2 a 2 - b 2 = a 2 + ab - ab + b 2 _ a 2 + b 2 a 2 -b 2 a 2 -b 2 ' Check. Put a — 2, b — 1. Then-^ L-.£±£s«i *«#,«•«§. 2 - 1 2 + 1 2 2 - l 2 3 3' 33 Why do we not put both a and 6 = 1? 289. To add or subtract fractions : 1. Reduce the fractions, if necessary, to equivalent fractions having the L. C D. 2. For the numerator of the result, write the numerators (in paren- theses if they are polynomials), joined by the signs between the frac- tions; and for the denominator of the result write the L. C. D. 3. Remove the parentheses in the numerator, and collect the terms. 4. Reduce the result to its lowest terms. Addition and Subtraction of Fractions 181 Example 1 a — 9 a — b a + b a 2 — b 2 Solution. L. C. D. = (a + ft) (a — ft ). 1 1 ' a a + ft q — ft . a a-fc a+6 a 2 - ft 2 ~ (a + &)(«- ft) (a + 6)(a - ft) a 2 - ft 2 _ (a + ft)-(a-ft) + a (a + 6)(a^ft) a + 2& (a + &)(«- Check. Put a = 2, 6 = 1. Then 1 1 , 2 2 + 2 ' 2 -1 2 + 1 1 4-1 3.1' 1 -f + * = f;or| *) or l-i + f=f;or|=f. 290. The fraction line may be thought of as having the same effect on the numerator of the reduced fraction as a parenthesis. In writing the numerators over the L. C. D. the parenthesis is used to indicate that the whole numerator is to be added or subtracted. ORAL EXERCISE 291. Perform the indicated operations: 1. | + f. 6. 5- 2 + 5 3. - + ?• y y ft 2 6 5. J-+-L_. a + 6 a — 6 a + ft a; 2~ a; 3' a 2 a 5* .T 5 2/ 2/ a 6 s c "ft 3 1 182 Fractions EXERCISE 292. Perform the indicated operations : ~-r-~ _t q 2x 5y x 3y 8 * 15 + 12 + 5~T* Q 5a 26,3 a; 2 «?/ y 10 . l^ + 7a» 17 rf 2. a + 26 a— b 4 ' 2 2crZ c a& a 1 2 a-f b a -b a-2b 2a-5b 5 5 a; a; — 4 4 3 x — 5 6 2^ + y , x + 22/ 3 ' 3 7 ,10 12* + 21 1 1 X X X- 7 4 4. „ " • 11. 5. =■ — =-- h- -• 12. 13. 7. —X+ — X — ~X — T a> 14. 2 9 #?/ 12 xy 18 #?/ a(x + a) aj(a; + a) 3a + 5 2a + 5 a — b a + b a-\-b a — b a — b a + b 5 7 4 a; — 4 6 a; + 6 15. -^— + 1 + a; 1 — x 1 -\- x 16. -i_ + 2z-3 3-2# 2ic 2 -»-3 17 a? ~ 4 3a;-5 . 5a? + 9a; + 14 2a>-l a + 2 2a? + 3a;-2 18. -A- f-i 8_ + 3* + 7 19. x- 4- as a; 2 8a 2 x-\-2a x— 2 a a 3 — 4 a 2 # 20. 3 2 - 46 21. » 2 +^! °L-. 02 + a; 2 a 2 — aa; a; 2 — a 2 22. y + 50a;2 + 5x y -+- 5 x 25x 2 — y 2 y — 5 x 23 ^1 4 1 » 24. (a + a;) 2 a 2 — x 2 (x — a) 2 5a-2 3a + l 2a 2 -5a + 7 24a- 6 36a + 9 48a 2 -3 25. * + * * ' 26. 27. 2a; 8-4a; 8 + 4a; a; 2 - 4 1 1 1 1_ 2a;-f-a; 2 2x 2a; + 4 a;-2 a; 2 + xy + j/ 2 1 x 4 + 4 ?/ 4 a; 2 -f 2 a;y + 2y 2 ' 28 . « + * a; - 1 2 a; 2 + a; + 1 a; 2 - a; + 1 a; 4 + a; 2 + 1 Addition and Subtraction of Fractions 185 a 2_ 2 g + 3 a- 3 1 a 3 + 1 a 2 - a + 1 a + 1 30. 6a 2 + a — 2 3 a 2 -a- 2 si. l£±£ + 3 4 32. 2a; 2 -8 a; -2 2 - a; 2 2 36 a- b a + b a 2 + ab-2b* x 2 -5 x-2 3 a; 2 -12a; + 27 x 2 - 9 34 l + 3x 9-lla; u (2a;-3) 2 , * 5 + 7a; 5 -7x 25 -49a; 2 x 1/ 7a; 5x 2 -f 2xy 2a%/ \ 2a; + 2y 2^/ - x x 2 - y 2 xy + y 2 ) 36 . l_f^l_JlL__M \x + y x 2 — y 2 37. ^ --f a; 2 + 2/ 2 5 a; 2 + 3a; + 2 a- 2 + 5 a; + 6 a; 2 + 4 a; + 3 2x2-7x + 6 2a; 2 -a--6 9-4X 2 39. , ~ 2 + W , + 1 . + * (m — l) 2 wi 2 — 2 m + 1 1 — ra 40 « 2 | & 2 « 2 + & 2 a& + b 2 ab + a 2 ab 41. z ~ z+: b C (a-6)(a-c) (6- c )(6-a) (c-a)(c-b) 42. ^^1 r+ .. * . + * (a — b)(a — c) (6 — c)(6 — a) (c — a)(c — &) 43 a 2 — be b 2 — ca . c 2 — a6 (a + b)(a + c) (6 + c)(6 + a) (c + a)(c + 6) 4 . a + 6 ?> + c c + a (c — a)(c - 6) (a — 6)(a — c) (b — c)(b — a) 186 Fractions MULTIPLICATION OF FRACTIONS 295. The product of two or more arithmetical fractions is the product of their numerators divided by the product of their denominators. The result should be reduced to lowest terms. Thus, 1. 2 x 5 = L x_5 = l_0. 3 X | = 3 373x7 21 £ 2J 7 • 7 296. The product of two or more algebraic fractions can be found in the same way. a c _ac b X d~bd' To shorten the work, we usually cancel all factors common to the numerators and denominators before multiplying. a 4 ' fipc Wfe 5 c 2 5 a 2 — x 2 a 2 - b 2 (_ , ax \ a + b ax -f- a; 2 \ a — x) _ Cfl--^0 (jUt^c) (a - b) Cet^r-tf) a 2 Mlr^^O 2(a-Mr) -a — ar _a 2 (a-b) X ORAL EXERCISE 297. Multiply the following : 1 2 . O a 2 . 1 3. } .3. 4. K-i). a 5. -.0. a-b a 2 -b 2 t a + b 2 6. * • ^. o a + 6 * 6 2 6 a - 6 a 2 - b 2 '■(-©(-!> -H)-(A) Multiplication of Fractions 187 12 ( a ~ b \ 2 a + h 14 2a%3? 3b * a + b) a 2 -b 2 3 6 8 4c 4 a; 15. What is the numerator of the product when all the factors of the numerator are canceled? 16. What is the nature of the product when all the factors of the denominator are canceled ? 17. What is the product if all factors of both numerator and denominator are canceled ? 298. To multiply expressions some or all of which contain fractions : 1. Reduce all integral and mixed expressions to the fractional form. 2. Factor all polynomials that can be factored in the numerators and denominators. 3. Cancel all factors common to the numerators and denominators. 4. Multiply together the remaining factors in the numerators for the numerator of the result ; and the remaining factors in the denominator for the denominator of the result. 1. 3 4 Examples _3c 2 " 8 ' \a?~ iYi + M J\ XX vU 3 - i) 1- X 2 X 2 - f x + 1 X 2 X s x 8 -! (Changing to fractional form.) -1 _(l + aOO^ X 3 (x^T)(x2>^fl) (Factoring and canceling.) -1 " x or 1+ X 188 . Fractions EXERCISE 299. Perform the indicated operations : l xfx^. s —- — . ^ 2. 7f x ^-. &y V*z abc 3 - (f + iXA + A)- 9 (3ct*6)» ,(5c)* (4 6)* 4. 5 mn . lM . ( 5 c ^) 3 ' ( 6 «) 5 ' 4 « 3 & 5 c' 5mn 2 f_3^_ 2 Y / 2aV\s 5. 13m 2 ^.^^- 2 . 'V 4 6c 3 ; V *** 8ti 2 a 2 -6 2 a KA, U K, J.X. XI — 6 - Yc'w'ab a ~ b \ « 2 +2«6+6 2 a — 6 V a 12 wM^H 2 ( (m — y) 2 W 2 — V 2 u 2 —uv + v 2 '■ (-H)c-exf> 13 2ab , 2q + 36 4 a 2 + 12 a6 + 9 6 2 ' a - 6 14 a+5 . a; 2 + 8a; + 15 , a; 2 — a; — 12 ' x — 4 17. a 3 -&3 l g + 6 24 frjV+% (a + 6) 2 2 a 2 -6 2 \y ay 18. 2 - m2 ~P 2 ^ 3 -/> 3 , 25 . 1 /I IV 3 (m-p) 2 m+p x — y \x y) i 9 . A a _i 6 y. i ^|/i+iY-s \4 2 J a -2b 2\a bj\a- 20. _JL_.( a _6). 27. ^ a - 3 Y 77T ss\ a 2 -b 2 K J \2 J\a 2 -12a+36j 21. ^-lY^ + A 28. a!±^+A 2 .«!±l 3 . U A« / a*-a& + 6 2 a 3 - 6 3 Multiplication of Fractions 189 29 (*2±yl- x \-JL^.t=£. V y Jx-y « 3 +2/ 3 so. ( x +y + Q ^n\(?-y). V x y )\y xj 31 . (i.- 9 y.f 1 l \ 32. kl-^V +-+I \ \'2y 3xJ{ 3<* + S#-2f) \a oj a -\-o \ x — yj 35. (* + *yjL_y, \y xj\x + yj „ a 4 - 6 4 a 2 - 5 2 a 6 - ft 6 OD. • ■ * • a 3 -6 3 a 3 + 6 3 a 2 -f& 2 37 « 2 -(&-c) 2 Ca - by - c 2 C 2_(5_ a )2* (5_ c )2_ a 2* » (^)'(^r> 2x*-8x+6 ^ x* - 9 x + 20 § a? 2 -7a? r a; 2 - 5 a; + 4 ' a 2 - 10 x + 21 ' 2 a 2 - 7a?' 4 a / IV -J x 40. 41. 42. (z 2 -l) 2 V 9x*-6x 2z 2 + 3z-9 2 1 + 2/ a; + a 2 ^ 1-x^ 190 Fractions DIVISION OF FRACTIONS 300. The reciprocal of a number is 1 divided by the number. Thus, | is the reciprocal of 2 ; f is the reciprocal of § , for 1 — | = f . The reciprocal of a fraction is evidently the fraction inverted. 301. Division has been denned as the process of finding one of two factors when their product and the other factor are given. Thus, | X | « If, hence || - | = J and \ f - f = f . 1. Divide if by -f. 1. Let £| -f- f = q. (A quotient.) 2. Then if = $ x q. (By definition of division.) 3. .'. *xii = Sx$.g = ff. (Why?) 3 4 I^5 = ^ J = 3 28 " 7 2£ £ 4 4 (From 1 and 8, since each = - BH9' 10. (-«) 2 -^- 20. What is the reciprocal of - b x 21. Is - + - the reciprocal of - + -? If not, give the cor- a c b d rect reciprocal. 13. b --rC. C 14. ab s • ab 15. 2 be n -—-J- 5c. 3a 16. 1 a -. b 17. , ab abc-i cd 18. i . m n 19. 15ab . 1 cd cd ? of a + 6 ? o{a j ? 192 Fractions EXERCISE 304. 1. What is the reciprocal of - + 1? of a + 6? of x 2. Find the reciprocal of 3^- ; of a -\ 3. Find the reciprocal of - + - ; of a + 2 + i. x y a Find the quotients : * 6* c» c 3 5 «^^!. 9 27a- 2 y . 12 x* 3 ' 2* " 25z 3 ' 52/z 2 ' 6. 2i 2 ^ 3 . 10. lot* + 3* 6 6 3 6 4 3 7 2a?^_3ab t n f±a J A2/J' 6c c 2 V 9e /^ 4 12. Divide — ~^L by the reciprocal of ™ • 13. The product is — and one of the factors is — — . Find the other factor. 4 c 8 &d 2 a 14 a 2 14. The quotient is — and the dividend is — — . What is the divisor ? 15. (a-lWl-M. is. (^a-i6)+(4a-86). 16. a'-ft' ^ «-& . i9. (aJ»-6a!-7)-l>-7). (1? -6 2 a -6 a 3 + 6 3 (a + &) 2 x 2 -2x -3 cc - -3 a 2 -4a a 2 - 16 V T * ; \^5 l"*)(x + 2) aj(a> + 2) x Complex Fractions 195 Care should be taken in such complex fractions as - to in- dicate which line separates the numerator from the denomi- nator. This may be done by using a heavier line. a b a , ., a ac EXERCISE 308. Simplify the following : a + b . a 1. -""*■ *±-±^ + 2 + i- q-6 q+6 i+i 2(a 2 + 6 2 ) 2 _2_ZU_. ~^T7± ab-2cd 2 f"1 3. y + ^ « 2 6 2 - 4 c 2 d 2 aHjf f-). 3 6(a? — a) , x — b 2 6(4 a + ca;) _ q 5 a 15 & 6 a ^tHj.^-^.^ a« & 2c 3d 20 a(ft — a?) , &(c — a) = a + & _ fb a bx ex x \c b In equations 21 to 34 solve for each letter involved in terms of the others. 21. i-i=J. D d f Solution. L. C. D. = Ddf. df-Df=Dd. Solving for /, (d - D)f = Dd, . f= Dd " J d-D Solving for 2>, (- d - f)D = - df, . •. D = df d+f Solving f or d, (f-D)d = Df, ovd= -M. f-D 22. 3x-5y + 7z = %(x-y + 3z). 23. a-4 = (p4-3a)(a + 2). 24. s=?(a+i 27 - y= mx + c - 2 28. lx+ my = 1. 25. Z = a+(7i-l)d. 29. ^ + £?/ + (7 = 0. 26. S + f.l 30. i-i-li a 6 10 » a; 204 Equations Containing Fractions Solve the following equations for each letter involved : 31. A -<&+$. 33. *=*b«. z a — b 32. c=^*. 34. T = ±+t. b — a a 317. If an equation contains fractions with polynomial de- nominators, find the L. C. D., and proceed as in the preceding problems. 3 12 1. Solve the equation 1- 9 a? -|-3 3-x Arrange the denominators in descending powers of x and factor them to find the L. C. D. Solution. 3 + x 2 -9 a; + 3 x — 3 L. C. D. = (> + 3)(x-3). Multiply every term of both members of the equation by the L. C. D. to clear of fractions. 3 +(a5-3) =- 2(s + 3). 3+z-3=-2x-6. z + 2x=-3 + 3-6. 3x=-6. x=-2. Check. — ^- + 1 = -*— , or ? = ?. 4-9-2+3 3 + 2' 5 6 2. Solve Solution. l-2s 5-6z = 8(l-3z 2 ) 3 _ 4 z 7-Sz 3(21 - 52 z + 32 z 2 ) l_2g 5-6g _ 8(1- 3 gg) 3_4g 7-8* 3(3-4g)(7-8g)' The L. C. D. = 3(3 - 4 g)(7 - 8 «>. 3(1 - 2 g)(7 - 8 g) - 3(5 - 6 g)(3 - 4 g) = 8(1 - 3 g 2 ). (Why ?) 21 _ 66 g + 48 g2 - 46 + 114 g - 72 g2 = 8 - 24 #, Equations Containing Fractions 205 - 66 z + 114 z - 8 - 21 + 46. 48 z = 32. » = f Check. !=i_$rJ = »( *~ t) or -§=-§. 3-f 7-V- 3(21 -■4 A + H*) 5 ^ EXERCISE 318. /Sofae the following equations: 1. 1 7 _2 £ 16 13 .- x + 2 3x + 6 3 ' l-3a; l-3a; 2. 9 7 _13 5 3 _5 2a; + 2 3a? + 3 12 ' a? + l 2 x + 2 2 3. 7 11 9* + 7 / «-2\_ 3 8a; + 2 20x + 5~ 13 ' 6 ' 2 \j" 7 J * 7. a; + 3 x—2 3x-5 1 2 3 12 + 4 8. 60-x 5x-5_ 6 24-3a; 14 7 4 9. -, 7a; — 2 3a; + 4 Ix — 4 5a;+l 3 5 5-8 10. 3 a? + l a? „ 6a;+7 2a;-2 2a;+l x + 1 x — 1 1 — x 2 15 7 a;— 6 Hint. If some of the denominators are monomials, it is best to clear the equation of the monomial denominators first and then collect terms before clearing the equation of the polynomial denominators. In exer- cise 11 proceed as follows : 6 x + 7 - 15 ( 2 *-2) _ 6 x + 3 (Multiplying by 15.) 7x — 6 4 - 30 x — 30 = 0. (Collecting terms after transposing.) 4(7 x - 6) - (30 x - 30) = 0. (Multiplying by 7 x - 6.) 28 x - 24 - 30 x + 30 = 0. - 2 a; = - 6. a; = 3. 206 Equations Containing Fractions Solve the following equations : 12 55 79-2a; ^a; + 3 5 10-7 a; = 13 + 15 a; 3x 60-2a; a; '3a; 6-7 a; 15a; 14 8a;+5 3- 7 a; = 16a; + 15 2\ 14 6 a; + 2 28 7 " 15. 16. 8 a; + 37 7 a; - 29 = 4a; + 12 18 5a;-12~~ 9 y — 5 _ y + 5 _ 21 y 2/ + 5 y-h~ 25-y 2 ' 1 ly O . O X « 2-a? 2 + x~ x 2 -±~ 18. 5 ^L 3 1 = 1 _ 1 4 " a; + 4 4 ' a + 2 2 * a; + 6 19 . 2 x _3^-3 = 3 _l- i ^ > 5 a; + 1 10 20. _L_ 3 + I 8 — . a; - 1 1 + a; 1 - a; 2 -H) 5 V 5/ a; + 2 a; a;(a; + 2) 7a;+26 17 + 4a; = 10-a; 13 + a; ' a; + 21 21 3 7 1-a; 3-a? = 6a; + 5 l+8a; ' 3 + 5 8 a? -15 15 1 _2a ? + l a-11 =0 3a;- 15 2a;- 10 3a;-5 5a;-l a^-4 =2 5*-5 f*-7 a?-l 29. -**-:= 2- f 7a? + » V 6x + 2 \15a; + 5 3a; + V Equations Containing Fractions 207 30. 31. 32. 33. 37. 38. 40. 41. 2 4a;+l 1 2a;-l 3a? + 2 =0 x-2 3 ' x-2 5»-10 4 -2 a; 4 3 4« 2 3 6x-S 2x-\ 6a;-3 3 4a?-5 = 5 7a-3 4 ' 3a;-7 7 ' 5a;-4* *±4(3a?-ll)=3(aj-3). a?— 1 34. 6(x - 6)= 3 g ~ 14 (2 a? - 11). a; — 4 35. 3^1 + 3^+1 _3^-67_ = 5# 3a; + l »-l 3a; 2 -2a;-l 36. 2^-3 3a; + 5 a; 2 -11 _ * a;_l aj-2 a; 2 -3 a; + 2 2 a; 2 a; 2 + 7 2 x-2 a; 2 - 3 a; + 2 a? - 1 4 a? + 5 2(a? - 2) _ - 7 2 a; + 6 a; -3 6 a; 2 -54 39. -5 3_ 2a; + 2 _1 = Q Sx-6 2»-4 3a?-6» 3 2 a; + 1 2 a; - 1 _ 9 a; + 17 2 a; -16 2 a; + 12 x 2 -2 a? -48* a; — 2q a?4-2& _ 3a? — 3 a a + 6 2a + 26~~ 26 42 a * — 4 5a; , 5 2 — aa; o a 2 + 4 6 6 2 + a 43 2a; + a 3a; 2 -22q 2 =5 a? + 3 a a; 2 -9 a 2 44 2 a — a; _ 5 + x _5a -± x _x + 6 a-5 3 ~ a+2 2 ._ 1 . a + & 1 .a — 6 45. — — -\ = -H a + o a; a — o x 208 Equations Containing Fractions Solve the following equations : 46. ^=^ + ^1^4-2 = 0. b a 47. 48. a b 6 2 b &-bx 4.50-26) ^3 b 2 -5bx Sx-3b 2 6x 2 -6bx PROBLEMS LEADING TO FRACTIONAL EQUATIONS 319. 1. What number added to both terms of the fraction -| will give a fraction whose value is -| ? Solution. Let x = the required number. Then |il? = - • ( By the conditions. ) 5 + x 9 v J J .-. 18+9x = 40+8z. .*. x = 22, the required number. Check. 2 + 22 = 24 = 8 5 + 22 27 9 2. The numerator of a fraction exceeds the denominator by 20, and if 7 is added to both terms of the fraction, the value of the resulting fraction is 3. Find the original fraction. 3. What number added to both terms of the fraction f will double the value of the fraction ? 4. The sum of the numerator and the denominator of a fraction is 20. If the numerator is multiplied by 2 and the denominator diminished by 3, the resulting fraction is equal to -J-. What is the original fraction ? 5. The difference between two numbers is 16, and the quotient of the larger divided by the smaller is 2\. What are the numbers ? 6. | of what number exceeds -| of the same number by 1 ? Equations Containing Fractions 209 7. In a division the dividend exceeded the divisor by 52, the quotient was 6, and the remainder was 8. Find the divi- dend and the divisor. 8. Divide 72 into two parts such that f of one part shall exceed \ the other part by 26. 9. A man made a journey of 40 miles in 4| hours. Part of the way he traveled in an automobile at 20 miles an hour and the remaining distance he walked at the rate of 4 miles an hour. How far did he ride ? Solution. Let x = number of miles he rode. Hence 40 — x = number of miles he walked. Also — = number of hours he rode, 20 and — ^— = number of hours he walked. 4 Then — + 40 ~ x = 4*. (By the conditions.) 20 4 s v j / Solve the equation. 10. A vessel that ordinarily goes 16 miles an hour is obliged to slacken to half speed during a part of a trip of 130 miles, thereby requiring 10 hours to make the trip. For how long a distance was it traveling under reduced speed ? 11. If one man can do a piece of work in 8 days and an- other man can do the same work in 6 days, how long will it take both men working together ? Solution. Let x = number of days for both. Hence - = the part of the work both can do in one day. Theni + Ll. 8 x Let the student explain the equation and solve it. 12. A can do a piece of work in 5 days ; B works only half as fast as A. How long will it take both working together ? 210 Equations Containing Fractions 13 A can do a piece of work in 12 days, but with B's help he can do it in 8 days. How long would it take B if he worked alone ? 14. A tank has two inlet pipes. One can fill it in 40 minutes and the other in 60 minutes. How long will it take if both are running at the same time ? 15. A tank has two inlet pipes numbered 1 and 2, and two discharge pipes, 3 and 4, with the following capacities : 1 run- ning alone can fill the tank in 60 minutes ; 2 alone can fill it in 80 minutes ; 3 alone can empty it in 72 minutes, and 4 can empty it in 40 minutes. (a) Beginning with the tank empty, how long will it take land 2 to fill it? (6) Beginning with the tank full, how long will it take 3 and 4 to empty it ? (c) Beginning with the tank full and all pipes flowing, how long will it take to empty it ? (cZ) Beginning with the tank empty, how long will it take to fill it if 1, 2, and 3 are flowing ? (e) Beginning with the tank half full, will it be filled or emptied, and after how long, if 2, 3, and 4 are flowing ? 16. What amount of money drawing simple interest at 5 % will amount to $ 287.50 in 3 years ? Solution. Let x = number of dollars on interest. Hence — = number of dollars of interest per year, 100 and i— ^ = number of dollars of interest in 3 years. 100 Then x + — = 287.50. 100 . . 100 a; + 15 a; = 28750, or 115 a; = 28750. .-. x = 250. Therefore the original principal was $ 250. Equations Containing Fractions 211 17. What was the face of a note drawing 4 % simple interest if it took $132.50 to settle the note 18 months after it was given? 18. A man loaned $ 800 in two parts, one part yielding 5 % per annum and the other part yielding 6 % . The interest amounted to % 44.50 per year on the two notes. How was the money divided ? 19. A man received $ 665 for an automobile, which was 30 % below its original cost. How much did it cost ? 20. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution ? Solution. Evidently it will require the addition of water to change the solution from 5 per cent salt to 4 per cent salt. The amount of the salt is, therefore, the same in both solutions, and we may use this fact as the basis of an equation. Let x a number of pounds of water to be added. Hence 80 -f x = number of pounds of salt and water in the new solution, and yj^ (80 -f x) = number of pounds of salt in the new solution. Also jf^ • 80 = number of pounds of salt in first solution. Then T $ 5 (80 + x) = T ^ • 80. (Since there was the same amount of salt in both solutions.) .*. 4(80 + x)= 5 • 80, (Clearing the last equation of fractions.) or 320 + 4 x = 400. .-. ix = 80. . •. x = 20, the number of pounds of water required. 21. How much salt must be added to 80 pounds of a 5 % salt solution to change it to a 10 % salt solution ? Solution. Let x = number of pounds of salt added. Hence 80 + x = number of pounds of salt and water in new solution, and t$j(80 + x) = number of pounds of water in new solution. Also ^ • 80 = number of pounds of water in original solution. Then ^(80 + *)=^- 80. .-. 90(80 + x) = 95- 80, or 7200 + 90 x = 7600. .-. 90 a; = 400. .-. x = 4$, the number of pounds of salt required. Query. Why is it not sufficient merely to double the amount of salt in order to double the strength of the solution ? 212 Equations Containing Fractions 22. How much salt must be added to 100 pounds of a 10 % salt solution to change it to a 12 per cent solution ? 23. How much water must be added to change 100 pounds of 10 % salt solution to a 4 % salt solution ? 24. How much water must be added to each ounce of a 90 °/ alcohol solution to reduce it to a 60 % solution ? 25. A merchant marked an article $ 8 and gave 20 % dis- count. Another merchant marked the same article at a higher price but gave 33J % discount. Find the marking price of the second merchant, if the discounted price was the same for both. 26. If 100 pounds of sea water contain 2.6 pounds of salt, how much fresh water must be added to make a new solution 30 pounds of which shall contain .6 of a pound of salt ? 27. The sum of two numbers is 70. If 14 is subtracted from one of them and added to the other, the quotient of the numbers is inverted. What are the numbers ? 28. The population of a city increased each year 5 % of the population of the preceding year. It now has 194,481 inhabit- ants. What was the population 3 years ago ? 29. Find two numbers whose sum is s and whose quotient is*. b 30. Divide the number 144 into two parts such that one part shall be § of the other. 31. The numerator of a fraction is 35 less than its denomina- tor. If both the numerator and the denominator are increased by 2, the fraction is equal to f . Find the fraction. 32. The cost per ounce of gold in December 1914 was about 41 times that of silver. Find the cost per ounce of each if 8.5 oz. of silver and \ oz. of gold together cost $ 14.50. 33. A watch chain weighing } oz. is made of platinum and gold. How much of each metal is in the chain if the gold is Equations Containing Fractions 213 worth $ 20 an ounce and the platinum is worth $ 48 an ounce and the total value of the metal in the chain is $ 22.75 ? 34. A man invests $4500, part at 6 % and part at 5%. The total income from the two investments is $ 245. Find the amount invested at each rate. 35. A certain sum of money is invested in a 6 % mortgage and $ 500 more than this sum is invested in 4 % bonds. If the incomes from the two investments are the same, how much is invested in each ? 36. An estate of $ 12,000 is divided among three heirs. The first receives § as much as the second and the third receives $ 400 more than the second. How much does each get? 37. A man can paint a house in 6 days ; his son can paint it in 16 days. How many days would it take both working together ? 38. A football team wins a game by 14 points and the losing team scores 4 less than half as many points as the winning team. What is the score ? 39. The pressure of water at a depth of d feet on each square inch is given in pounds by the formula P (pressure) 62 5 = — -{— d. If the pressure of the air at the surface is 14 pounds per square inch, at what depth will it be 10 times as great ? 40. It is 1024 miles from Chicago to Denver. A train that usually averages 32 miles an hour is delayed 2 hours by an accident, but by running 12 miles an hour faster just makes up the lost time. How far did it run at each rate ? 41. A dairyman wishes to mix milk containing 5 % butter fat with cream containing 30 % butter fat to get a mixture containing 20% butter fat. How much of each should be taken to get 10 quarts of the mixture? 214 Equations Containing Fractions 42. Any volume of aluminum weighs -f- as much as the same volume of cast iron. When i of the cast iron of a gasoline engine is replaced by aluminum parts of the same size, the weight of the engine is 320 pounds. What was the original weight of cast iron ? REVIEW OF FRACTIONS AND FRACTIONAL EQUATIONS 320. 1. What is the rule for adding fractions ? 2. How do we " clear an equation of fractions " ? 3. What principle is involved in " clearing an equation of fractions " ? 4. How is a fraction multiplied by an integer ? 6. Simplify -l i +-l- + -l_. Note. The student should note that example 6 is not an equation, and that he is not to clear fractions. \ 7. Why do you have trouble if you try to solve the equation 1 l = 2(s + l) 9 05 + 2 X 05(8 + 2) 8. Solve (5+g(5-Q + |=05 + 12. 9. Simplify(5+|J+(5-|J+|. 10. Solve -2— L± i-=0. 05 — 9 05 — 5 05 — 4 11. Simplify—?- ?L + 05 — 1 05 — 2 05 — 3 a b c b 12. Solve 05 c 05 a Review of Fractions 215 13. Solve for t, ^=&c + -- z z 14. What number must be added to f to get the same result that would be obtained by multiplying it by % ? 15. -*-+-* §-=0. x-YI x-19 jb-18 1C flJ-8,a;-3 . oj-9 a;- 1 . a?- 13 . a;- 6 #—3 x—5 x—7 x — 3 x—5 x—7 Note. First transpose the fractions and combine each pair having the same denominator. 17 x + 2 . x + 1 x + l = x-\-9 x-3 x + 4= x + 7 x + 5 x+3 x+1 x + 5 x + 3 18. 19. q 2 -6 2 . q 2 + 2qft + & 2 a 3 _ 53 * a 2 + ab + 52 * 3? - 7 a; + 12 ^ x*-l . a^ + a? + l . x 2 — x x 2 — 4 a? a,* 2 20. Reduce to lowest terms S^ + ^ + s a?-l a 4 — a 3 — a + 1 21. Reduce q 4 + a 3 - q - 1 22 5 + 3a: 5-3a; 48-2a; 2-x 2+a; a?-4 23. Solve x- x -^ = 5§- x + 10 + g. 3 5 4 24. Simplify 12 a; 2 -f- £ -20 12 a; 2 -h 25 a; + 12 25. Solve 1.2 a; - .05 = .07 x + .3x+ 16.55. a c 26. 6 + C - a ~*~ & 1 -j- k + c 1 1 c + a a 4- 6 6 + c 216 Equations Containing Fractions 27. Prove that (*±1 - 1 Vf — _ -ii V^-i y Vs-1 2- a; s-2 x(x-\-l)(2x-S) 28. Show that ^+y^ / 2^3^ 4g-9 |! \ i^-a, [ x ^ f tf-f ) * +w q 12 10 29. Show that x = — satisfies the equation l — = 64-. 19 u a; it; — 1 3 3 °* U 2+1+ 2/ 2 A2/ *J 2-1=* 5 31. ■ — gi = 1. Solve. 32. Snnphfy t • C-3 1-JI 1-1 (Princeton.) 33. Simplify (<* + 1 + ^W»*±*? - 1 V . (Sheffield Scientific School.) a 2 + b 2 b a?-b 2 34. Simplify _ - . ^ (Yale.) _c ajj '|_c 2 a 2 _|' 35. Simplify 6 a 6 4- c 6 a 3 c 3 Verify the result by using a = 2, c = 1 in the original fraction and in the answer. (Yale.) »■ "-»H s 5 I+ i{:-3- , ^5*)} (Princeton.) Review of Fractions 217 37. Si m plif y (a 2+ -^)(a 2 + .^(^ + ^_). (Sheffield Scientific School.) 3 8 . Solve 2 (*- a ) + 8 (*-») = 6. b a 39. Simplify x + 1 + — g* 5s-4 t „n a i 7,1 23 -6,7 1 40 . Solve _ + _ = __ + _-_. 41. Show that (100a + 10y + z)^3 = 33a + 3y + a; + 3f + g , o and from this equation show that if the sum of the digits of a number of three figures is divisible by 3, the number itself is divisible by 3. Show in the same way that any number of four figures is divisible by 3 if the sum of its digits is divisible by 3. 42. Show similarly that if the sum of the digits of a number is divisible by 9, the number itself is divisible by 9. 43. Any number ending in 5 can be written as 10 a + 5, where a is the tens' figure. (10 a + 5) 2 = 100a 2 + 100 a + 25 = 100 • a(a + 1) + 25. From this we may get the squares of numbers of two figures ending in 5 by multiplying the first figure by 1 more than itself and writing the product before 25. Thus, 65 2 = 4225. (6x7 = 42.) Square all numbers of two figures that end in 5. 44. (a 4- 1) 2 = a 2 + 2 a + 1. The square of a -f 1 exceeds the square of a by 2 a + 1. This means that the square of 21, 20 + 1, exceeds the square of 20 by 2 • 20 + 1, and therefore 21 2 = 441. Square 31, 41, 51, 61, etc. XIL RATIO AND PROPORTION 321. Ratio. The quotient of one number divided by another number of the same kind is their ratio. The former number is the antecedent and the latter is the consequent. The ratio is usually written in the form of a fraction and its terms bear the same relation to each other as the numerator and the denominator of a fraction. $ 10 Thus, - — represents the ratio of $10 to $5. The value of this ratio $5 • is ^, or 2. - represents the ratio of a to b. It is usually read, the ratio b of a to 6 or a divided by b. The above ratios are also sometimes written $ 10 : 1 5, and a :b. The colon is used here as a sign of division. The value of a ratio is always an abstract number. (Why ?) ORAL EXERCISE 322. Bead the following ratioi \ an d give their values : . $6 ' $8* 5. 7 men : 21 men. q ma na 2 2a2 . Sab 6 — . my 10. 21:. 75. 3 $15 $6 7 1.1 ■• 4-2"- 11. , 3 A ft - • 10 in. 4 ii§. 8. X -: y - y x 12. 2 yd. : 2 ft, 13. If the ratio of x to 3 is equal to 5, what is the value of x ? Hint. ^ = 5. Solve. 3 14. If the ratio of a; to -J- is equal to 2, what is the value of a? 218 Ratio and Proportion 219 15. What number bears to 5 the ratio .3 ? I Solve ^ = .3. j 16. Can you express a ratio between $12 and 4 f t. ? 4 bu. and 2 qt. ? 1 rd. and 1 in. ? 10 sq. in. and 2 cu. in. ? Simplify the following ratios by treating them as fractions and reducing them to their lowest terms : 17. (ra 2 — n 2 ) : (m + n). 18. x 3 — y 3 : x — y. 19. Which ratio is the greater, -f- or -J ? f or ^ ? 323. Proportion. An equality of two ratios is a proportion. Thus, ^ = f-f is a proportion. Also - = - is a proportion, if a and 6 b d are the same kind of numbers, and c and d are also the same kind of numbers. This proportion is read, the ratio of a to b equals the ratio of c to d. The proportion is also sometimes written a : b = c : d, or a : 6 : : c : d. These proportions may be read, a is to b as c is to d. The fractional form is, however, much more commonly used. EXERCISE 324. 1. What value must be given to d, if a = 1, 6 = 2, c = 3, in the proportion - = -? b d 2. What is the value of d if a = 2, 6 = 3, c = 4 ? 3. a 2 -6 2 :a-6 = ? 4. Divide 60 into two parts that are in the ratio of 2 to 3. Hint. Let x and 60 — x be the two numbers. 325. Terms of a proportion. The four numbers, a, b, c, and d are the terms of the proportion a : b = c : d. The first and fourth terms, a and d, are the extremes, and the second and third terms, b and c, are the means. The first and third terms, a and c, are the antecedents, and the second and fourth terms, b and c?,. are the consequents. 220 Ratio and Proportion x 2 In the proportion = - , name the extremes, the means, X — 4 o the antecedents, the consequents. 326. Fourth Proportional, Third Proportional, and Mean Pro- ft Q portional. The fourth term, d, of the proportion - = - is the fourth proportional to the other three terms taken in the order a, b,c. In the proportion - = - , where the means are equal, c is a b c third proportional to a and b, and b is the mean proportional between a and c. ORAL EXERCISE 327. In the following proportions name the extremes, the means, the antecedents, the consequents, the fourth proportionals, the mean proportionals, and the third proportionals. . 2 4 . a b 1. — SB-. 4. - = -• 3 6 be 2 S 2. - = — - . 5. m : p = q : s. 3 4.5 F 3. a:b= c:d. 6. x : y = y: z. 328. A proportion may be treated as an ordinary frac- tional equation. The unknown number may be in any term of the proportion. 3 5 Solve the proportion - = - for x. i x Solution. - = -. 7 x 3x = 35. z = llf Check. Substitute llf for x in the proportion. Ratio and Proportion 221 EXERCISE Solve for x in each proportion : . x 3 = 12 51 = 68. a; 16 15 x 3. 6.3 : x = 13|- : 20. (Write in fractional form.) 4 ??=._£ 5 8 ^ = bc 95 57* ' a; \ac 6. Find the fourth proportional to (a) 3, 4, 6: (- = -"); (&) 2, 4|,9i; (c) a,b,c. V 4 ^ 7. Find the third proportional to (a) 9 and 6 ; (b) a 2 — b 2 and a — b ; (c) a and b. 8. Divide 120 into two parts which are in the ratio of 2 to 3. Hint. Let x and 120 — x represent the two parts. Why ? /» 9. Divide 182 into two parts whose ratio equals -• 7 10. What number added to both terms of the ratio - will 2 8 give a ratio whose value is - ? o 11. Find a mean proportional between 2 and 8. 2 x Solution. The equation is - = — . x 8 z 2 = 16. x 2 - 16=0, or (z-4)(x+ 4)=0. (§239.) x = 4 or — 4. 12. Find a mean proportional between : (a) 2 and 18. (c) — and -• sc a (b) 3 and 27. (d) ^L±_^. 2 an d p - q. 13. Divide $ 180 between two men so that their shares will be in the ratio of 13 to 5. Hint. See example 8, or let 13 x and 5 x represent the two shares. 222 Ratio and Proportion 14. Divide $ 180 among three men so that their shares shall bear to each other the relation 2:3:5. Hint. This notation means that the first man's share is to the second man's share as 2 is to 3. Also the first man's share is to the third man's' share as 2 is to 5. The shares may be represented by 2x, 3 a:, and 5x. 15. Solve for a?, ^=4 = f- ' x + 3 6 16. Solve for y, y — 7 : y — 3 = y — 11 : y — 9. A C B X 17 - In the figure .40 = 9 inches, 1 1 ' 1 CB = 3 inches, and BX=x. Find a if AC : CB = AX : BX. PROPERTIES OF PROPORTIONS 2 8 330. Consider the proportion - = — . Cleared of fractions 3 1— this gives 2 • 12 = 3 • 8. This illustrates the following impor- tant property of any proportion : I. If four numbers are in proportion, the product of the means is equal to the product of the extremes. Proof. Let a, b, c, and d be four numbers in proportion. Then 2 = 1- b d .-. a • d = b • c. (Clearing of fractions.) The last equation states that the product of the means in any proportion equals the product of the extremes. This is a test of the correctness of a proportion, or of the equality of two ratios. Find the value of x in : 1. 2: # = 3:6. 2. x : 4 = 3 : 6. 3. a : b = c :x 3 x = 12. 6x= 12. ax = be. » = 4. x = 2. • «!£. a 7. - : x = b : ab. a 8. ab* a 2 b 1 c 5c* 106c 9. a — x:a + x =3:7, 10. x : 1.5 = lj- : 1.8. Properties of Proportions 223 EXERCISE 331. Find the value ofx in each of the proportions 1 to 10. 1. 8 : x = 24 : 3. 2. 9:81 = ^:243. 3. 18: 7.2 = .4: a. 4. a:b = x: c. 5. a; : a = & : c. 6. x + 9 : 8 = x : 3. /State wTwcft o/ £/*e proportions 11 to 16 are correct and which are incorrect. 11. 5:6 = 15:18. 13. 3:5 = 77:112. 12. 2:3 = 5:8. 14. 5:7 = 10:11. 15. (x + y):(x- y)=(x 2 + 2xy + ?):(& - y 2 ). 5m + 3 __ 5m — 3 ^ 10m + 9~10m- 9* 17. What is a fourth proportional? Find the fourth proportional to each of the sets of three num- bers in 18 to 23. m — n(m — n) 2 m m ■+- n (m -f n) 2 n nn a 2 -b 2 1 a 1 _b 18. 5, 6, 10. 19. 8, 7, 5. 20. ra, 7i, p. 21. Ill a b c a 2 + b 2 ' b a 24. What is a third proportional ? Find the third proportional to each of the sets of two numbers in 25 to 29. 25. 9, 6 ; 16, 12. OQ m 2 Im - m 2 28. 26. (o - b) 2 , a 2 -b\ V ~ ™ 2 (! + ™) 2 27. P*-4* p-q m 29. i 9 1^!_. r r (j) -fm) 2 ' m? -\-p?' 224 , Ratio and Proportion 2 4 332. Consider the proportion, -7 = 5- Clearing of fractions 4 8 gives 4 2 = 2 • 8 or 4 = V2 • 8. This example illustrates the following property : II. A mean proportional between two numbers is equal to the square root of their product. Proof. Let a, b, and c be such numbers that a_b b~c' b 2 = ac. (Clearing of fractions.) .-. b = Vac. (Extracting the square root of both members.) Find the mean proportional between 3 and 12. Solution. 3 : x = x : 12. x 2 = 36^ x = V36, or 6. This may be verified by noting that 3 : 6 = 6 : 12 is a true proportion. (Why ?) EXERCISE 333. Find the mean proportional between each pair of numbers : 1. 25 and 36. 4. 5 a 2 and 5 6 2 . 2. 9 and 81. 5. 9 a and 4 ab 2 . 3. 4 a and ab 2 . 6. 3 a 2 b 2 and 12 c 2 . 8. *- and (™ + 5 ?. m 2 + 10m + 2o 125 9. Find a third proportional to 3 and 5. 10. Find a third proportional to x 2 — y 2 and x — y. 11. 5 a& is a mean proportional between 15 a 2 and what other number ? 12. 3« is a mean proportional between 18 and what number ? Properties of Proportions 225 334. From such an equation as 3 • 8 = 4 • 6, we may form proportions by a proper arrangement of the numbers. Thus ,3 = 6 3 = 4 8 = 6 '4 8' 6 8' 4 3 Can a proportion be made from the numbers involved in the equation 4 • 10 = 5 • 8 ? III. If the product of two numbers is equal to the product of two other numbers, the factors of either product may be made the means and the factors of the other product the extremes of a proportion. Proof. Let ad = be. Dividing both members of this equation by bd, we have - — 2. b~ d Form proportions from the equation pq — xy. Solution. & = &. qy qy .•. 2-*,orp :y = x:q. y q AlsoH = ^. px px .-. 2 = V-, or q : x = y :p. x p Let the student form proportions by dividing both members of pq = xy, (1) hjpy, (2) by qx. In writing a proportion from two equal products, if any one factor of either of the products is written as first term in a proportion, the other factor of that product becomes in every case the last term. EXERCISE 335. 1. Form proportions from ad = be by dividing both members by cd ; by ac ; by ab. 2. Form a proportion from 2 x = 3 y. Suggestion. Divide both members of the equation by 2 y. Could a proportioD be formed by dividing by 3 x ? by 6 ? 226 Ratio and Proportion 3. Form a proportion from 5u = 7 w. 4. Form a proportion from x 2 = 2 ab. 5. Form a proportion from # 2 — y 1 = a 2 — 6 2 . 6. Can the numbers 2, 9, 3, and 7 be arranged as the terms of a proportion ? Explain. Can 6, 8, 4 and 12 be so arranged ? Why? 7. Write a proportion from a = be. 8. What is the ratio of x to y in 12 x = 30 y ? 9. What is the ratio of x to y m.2>x — 2y = x~\-y? 10. Find the ratio of a to b in 2a-Sb = 2c-Sd b d 336. IV. If four numbers are in proportion, they are in proportion by inversion ; that is, the second term is to the first as the fourth is to the third. Proof. Let - = -. b d ad = bc. (Why?) .-. - = -. (Dividing by ac.) 23. Transform - = - by inversion. 6 9 J 2 3 Solution. - = -. 6 9 6 = 9 2 3 Let the student test the correctness of this last proportion. 2 4 337. If we interchange the means of the proportion -=-, o 3 3 6 we get - = -, which is another proportion. This transforma- tion is always possible, and is stated as follows : V. If four numbers are in proportion, they are in proportion by alter- nation ; that is, the first term is to the third term as the second is to the fourth. (Why ?) Properties of Proportions 227 Proof. Let b d ad = bc. (Why?) ,«4 (Why?) Transform - = — by alternation. 5 10 J Solution. - = — 5 10 f = A. (Why?) 8 10 v J } ORAL EXERCISE 338. Transform the proportions 1 to 4 by inversion. Trans- form them by alternation. 1.2:3 = 6:9. 3. 3 : - 2 = - 9 : 6. 2. x : y = a : b. 4. a : 2 a = b : 2 b. 5. Can the proportion $ 5 : $ 10 = 2 men : 4 men, be trans- formed by alternation ? Explain. 6. Can § 330, I, be applied to the proportion in the last example ? Explain. 339. Given the proportion - = — . From this we may make ., , . , n 4 + 5 8 + 10 9 18 T another proportion as follows : — ! — = — — — or - = — - . in 5 10 5 10 general this may be stated as follows : VI. If four numbers are in proportion, they are in proportion by com- position ; that is, the sum of the first two terms is to the second as the sum of the last two terms is to the fourth. Or the sum of the first two terms is to the first as the sum of the last two terms is to the third. Proof. Let - = -. b d 1 + 1*1 + 1. (Why?) a±b = c±d t (Why?) 228 Ratio and Proportion To prove 2-tA^O+A transform the proportion - = - bv a c b d J inversion and then proceed as before. Let the student do this. q f* Transform by composition - = - . Solution. i+i = °±« or I = 1* . 4 8 4 8 EXERCISE 340. 1. Given * = ™, prove that * + m = ^±^ ■ y n m n Hint. Apply first V and then VI. 2. Transform by composition x 3. Solve the equation 5 2 x — 2 x — 1 5 — x 3 — x 4. Solve the equation in example 3, first transforming by composition. 5. If - = - , prove that ^ = *+_?. (Use V and VI.) b d' * c d v J 6. U - _ -, prove that ^^+^ = - . (Use VI and V.) n y x + y y 3 9 341. Given the proportion - = — . From this we may q A Q -jo 1 3 make a proportion . = or - — = — — . We may F F 4 12 4 12 J also write ~ = — ^— ; that is, - = — . 4 12 ' ' 4 12 In general this may be stated as follows : VII. If four numbers are in proportion, they are in proportion by division ; that is, the difference between the first two terms is to the second term as the difference between the last two terms is to the fourth. Or the difference between the first two terms is to the first as the differ- ence between the last two terms is to the third. Properties of Proportions 229 Proof. Let ^=- b d Let the student complete the proof. 5 10 Transform - = — by division. 2 4 J 5 10 Solution. - = — 2 4 5 - 2 _ 10 - 4 or 3_6 2 4 ' 2 4' EXERCISE 342. 1. If - = -, prove that ^j^c-d (Apply IV b d a c and proceed as above.) 2. Apply the transformation by division to a ~*~ = c "*" • 3. Apply the transformation by composition to — - — c — d d 4. If ^L±l = & find the value of ™. no w 5. Solve £±i = ?. (Apply VII.) 343. The last two transformations are sometimes referred to as transforming a proportion by addition instead of by composition, and by subtraction instead of by division. 344. A combination of the two preceding transformations may be made. Thus, § = ^,and 3 -±^ = ^±^or-^ = *L. 5 15 3-5 9-15 -2 -6 This illustrates the following property of a proportion : 230 Ratio and Proportion VIII. If four numbers are in proportion, they are in proportion by composition and division ; that is, the sum of the first two terms is to their difference as the sum of the last two terms is to their difference. Proof. Let": b _ c ~d' a + b c + d b d a — b _c-d (Why?) Also - - = Z . (Why ?^> b d V J / Dividing the last two equations member by member, we have a -\- b __ c + d ^ a — b c — d p* 10 Transform = — - by composition and division. 6 12 Solution. 5_10 6 12* 5 + 6 _ 10 + 12 5-6 10-12' 11 _ 22 -1 -2* EXERCISE 345. 1. Transform 4 :2 = 12 : 6 by composition and division. 2. Transform — —— = — i__ by composition and division. a- 2 ra-3 J F 3. a:b = c + x: c — x. Solve for x, using § 330, I. 4. Solve the equation in 3, using § 344, VIII. 5. If 2 = - c , show that ?— b - = £^. b d a + b c + d 6. If - = -, show that ±±c = 'b±A. (Alternation and b d a— c b — d composition and division.) 346. If several fractions are equal to each other, the sum of their numerators divided by the sum of their denominators equals any one of the fractions. Properties of Proportions 231 Thus, ? = § = 1 = ?! and 2 + 6 + 8 + 14 or §5 is equal to any one ' 3 9 12 21 3+9 + 12 + 21 45 H * of these fractions. This property of equal fractions may be stated thus : IX. In a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. .~ T , a c e x Proof. Let T = - = - = - • b d f y Also let each ratio equal k. - = k, from which a = bk. (Why ?) - = k, from which c = dk. d ' -=k, from which e = fk. - = k, from which x = yk. y Then a + c + e + x = k(b + d + /+ y). a-|- c +- e + # _ a c e x k b+d+f+y b d f y rhus> l = 2 = 3^4 = l+2 + 3 + 4 Qr 10 i '2 4 6 8 2+4 + 6 + 8 20 EXERCISE 347. 1. Apply IX to ^=1 = ^. 2. Apply IX to £ = £. 3. If - = - = - = -, what is the value of a + c + e ? b d f 3' b + d+f . T o a m x i , , , a — m +- a; a 4. It - = — = - , show that ■ — = - • b n y b — n +- y b Hint. — may be replaced by ^^ • (Why?) Then apply IX. n —n 232 Ratio and Proportion 5. If ° = °=g, show that j» q + 3c + 4r = ». b d s' 26 + 3d + 4s b Hint. fl - = 2 - a . b 26 6. If = %- = , prove that each one a-\-b — c a — b + c b + c — a of these fractions is equal to ' ^ "* -» a + 6 + c 348. 1 = ? also i- 2 = - or - = ^- From ? = * we may 3 6' 3 2 6 2 9 36 3 6 J get - = — by squaring both members of the equation. 9 oo These examples illustrate the following property : X. If four numbers are in proportion, the squares (or any like powers) of these numbers are in proportion. Proof. Let - = - • b d Squaring both members of this equation, we have l2 (t)'-(S " b 2 d 2 ' The proof for other like powers is similar. Thus, - = i- • How does it follow that — = — ? '6 10 25 100 EXERCISE 349. l. If — = -, prove that — = — • n y x 2 y 2 2. If % = - , show that ~-^ = %' ( x and IX -) b s' b 2 + s 2 b 2 v ' a 2 c 2 3. In the proof of X we produced the equation — = — a c from - = -• Is — = -? Explain. b a o l o Summary of the Properties of Proportions 233 SUMMARY OF THE PROPERTIES OF PROPORTIONS 350. Following are statements, in algebraic symbols, of the properties of proportions : II. If a: b= b:c, then b = Vac. III. If ad = be, then a.b=c:d etc IV. If a: b = c: d, then b:a = d: c. V. If a: b = c: d, then a:c= b:d. VI. If a: b = c:d, then a + b: b = c+ did. Or a + b:a = c + d: c. VII. If a: b=c:d, then a - fc: & = c - d: rf. or a — b:a= c — d: c. VIII. If a: b = c: d, then a + b: a - b = c + d: c - d. IX. If a:b= c:d= e:f, then a + c+ e: b + d + f= a:b. X. If a : & = c : d, then d 2 : b 2 = c 2 : d 2 , or a" : b" = V : d". EXERCISE 351. 1. What is meant by transforming a proportion by inversion ? by alternation ? by composition ? by division ? by composition and division ? 2. If - = -, show that —=^; also that— = —• b d' 5b 5d 2b 2d 3. If « = £, show that 2q + 56 = 2c + 5cl. 6 d' 2a-5b 2c-5d 4. Apply I to see if 13 : 17 = 19 : 24. 5. Given « = £, show that «+26 = c + 2d. 6 rf a c 6. Find a mean proportional between SB-L. and 2-^JL. p-q P + <1 7. If £=4=4 show that ^t_ C =^; also that *±± = *. 234 Ratio and Proportion 8. If^ = ^ = e ,showthat a J-- c = ^±- e = c +^. b d f b+d b+f d+f 9. (a) Find a third proportional to — and -• lb b (b) Find a fourth proportional to a 3 — b 3 , a 2 — b 2 , a — 6. 10. If £ = £ and g- ^, show that ™' = ^. 6 d b' d' bb' dd' 11. Transform so that x shall occur only once, - = c -i-®. b x Solution. <* = 10 20 .30 40 50 Age oo 70 6. The following table shows the annual premium per $ 1000 at different ages for life insurance. Age 21 25 30 35 40 45 50 55 60 Premium $ 18.40 $ 20.14 $ 22.85 $ 26.35 $30.94 $37.08 $45.45 $ 56.93 $72.83 Construct a curve showing the relation between the age and the premium. Measure the ages along the horizontal line and the premium on the vertical line. Note. The pupil should use cross-section paper for this work. From the curve estimate the premium for a person at the age of 28, 37, 42,64. 7. The following temperatures were taken from the weather reports at a certain city for January and February. 240 Graphs Day 1 2 3 4 5 6 7 8 9 10 11 January February 30° 24° 31° 28° 32° 36° 32° 24° 26° 22° 26° 32° 31° 23° 34° 4° 34° 6° 25° 11° 20° 7° Day 12 13 14 15 16 17 18 19 20 21 22 January February 16° 0° 9° 6° 20° 16° 34° 13° 38° 14° 28° 18° 26° 22° 32° 20° 36° 16° 24° 18° 20° 17° Day 23 24 25 26 27 28 29 30 31 January February 34° 4° 34° 7° 18° 14° 33° 25° 42° 34° 42° 36° 44° 30° 28° This temperature record is shown graphically in the follow- ing figure : Iff j inuar y »S , F ■bi ua ry } I ^C 7 / \ <, ft > i \ / \ / I k ¥ \ I 1"/ \ i ^ \ ( i / \ V s J 1 H n ° \ T \ / ', : 3 1 ■ 1 ) 1 1 1 1 2 1 ?, i i l j l ; i 7 1 8 1 'J i I 1 2 2 2 3 2 L 2 r ) 2 1 2 7 2 8 2 i S o a i Observe that time, or dates, are represented on the horizontal line using 1 space for one day ; the temperatures are measured in the direction of the vertical line, using 1 space for 5°. Graphs 241 8. Two trains leave Chicago going east on parallel lines. One starts at noon and runs at the average rate of 30 miles an hour ; the other starts at 1 o'clock and runs 40 miles an hour. How far from Chicago, and at what time, will the fast train overtake the slow train ? Let the spaces on OY represent the number of miles traveled as indicated, and the spaces on OX represent the time. y y / / S jt $ y a t p< 0D p* 80 p-, n \ 40 D ... ... ... fl Ft h n\s X o 12 i i i i Hours At 1 o'clock the slow train will have run 30 miles. Measure the 30 miles along OY as OB. Measure the time, 1 hour, along OX. Draw the rectangle ODP\P b . Pi, by its distance from OX, represents the distance traveled, and, by its distance from OF, represents the time. Similarly P 2 represents the distance and the time after 2 hours; P 3 , after 3 hours, etc. The points O, Pi, P 2 , P 3 ..-lie in a straight line. Draw this line and call it h. If any point is taken on this line, it will be found that a distance and the corresponding time can be read at once from the figure. In a similar way draw Z 2 through P 6 P 6 P 7 P4, representing the progress of the fast train. It is evident that the intersection of the lines h and l 2 will indicate 242 Graphs the time of the day and the distance traveled when the distances are equal ; that is, when the fast train overtakes the slow train. From the figure it appears that this occurs at 4 o'clock when the trains are 120 miles east of Chicago. Determine from the figure how far the fast train is behind the slow train at 3 o'clock. When will the fast train pass the point where the slow train was at 2 o'clock ? 354. In representing statistics and data graphically, first look over the numbers involved so as to choose convenient units. In general, if the numbers are large, select small units. EXERCISE 355. 1. If a person saves 10^ a day and deposits it in a savings bank which pays 3 % interest, the balances, to the nearest dollar, at the end of certain years are as follows : Year 1 2 3 5 8 10 14 17 20 Balance $37 $75 $115 $197 $ 330 $425 $635 $809 $999 Using two spaces on the horizontal line OX for one year, and four spaces on O Y to represent $ 100, draw a smooth curve through the points located from the table and estimate the balances for the years omitted. 2. The table below gives the expense and receipts of a certain newspaper for various numbers of copies. Numbers of copies 1000 2000 3000 4000 Expense in dollars 425 550 675 800 Receipts in dollars 300 495 690 885 Graphs 243 Construct a graph showing the relation between the number of copies produced and the expenses. On the same diagram show the relation between the receipts and the number of copies. From the diagram estimate as nearly as possible the smallest number of copies that can be produced to make the paper pay. Use 1 inch on the horizontal line for 1000 copies, and \ inch on the vertical line to represent $ 100. 3. The following table shows the distances in miles of certain railway stations from Chicago, and the time of two trains, one to and one from Chicago. If each run is to be made at a constant speed, show graphically the progress of each train. Going West Miles Chicago Miles Going East 9 :00 a.m. 284 6:00 12 : 40 p.m. 127 arrive Bloomington leave 157 1 :30 12:45 leave arrive 1 :25 p.m. 2:25 185 arrive Springfield leave 99 12 : 00 m. 2:35 leave arrive 11 : 55 4:45 258 Alton 26 10:00 5:45 284 St. Louis 9 : 00 a.m. At what point do the trains pass and how far is each from Chicago ? Let the horizontal line represent the time, using one half inch for one hour and one inch on the vertical line for 100 miles. 4. The following table gives the length of the circumferences of a circle for given radii : Radius 1 2 4 6 8 10 Circumference 6.28 12.56 25.12 37.7 50.24 62.8 Measure the circumferences along the vertical axis, us- ing one space for 2 units. Use two spaces for 1 unit in 244 Graphs measuring radii on the horizontal axis. Locate all the points tabulated and draw a smooth curve through them, (a) Estimate from the figure the circumference of a circle with radius 2-j- units, (b) What is the approximate radius of a circle whose circumference is 44 units ? 5. The following table shows the areas of circles for certain radii : Radius 1 2 3 4 5 AY-ea 3.14 12.56 28.26 50.24 78.5 Locate the points, using the same units as in the last example. Draw a smooth curve through the points. (a) Estimate the area of a circle with radius 2\ inches ; 6 inches. (6) Estimate the radius of a circle with an area of 40 square inches ; of 70 square inches. 356. Axes and Coordinates. If we draw two straight lines at right angles to each other as in the figure, we divide a plane surface into four quadrants. The lines are the axes. The hori- zontal line XX' is the X-axis, and the verti- cal line 77' is the Y-axis. The quadrants are the first quadrant, the second quadrant, the third quadrant, and the fourth quadrant, as indicatedbytheBoman notation. We name the spaces along these axes as shown in the figure. If we select any point in the plane, as P lt we can describe its position completely by telling how far it is to the right of Y I \X + 3 J IX V + -j- 1 J ,( -I 3) 9 n Pi [(3, 2) I .Y' Pi X ( t I 1 1 2 i I i \ -1 Ft H -O ] ii X- y- IV f, + -8 1 *< -2 i-3 ) Ps -4 Y' Graphs 245 the Y-axis and how far it is above the X-axis. These distances are, for the point P x , 3 and 2 respectively, and they are the coordinates of P x . The coordinate measured in the direction of the X-axis is the abscissa, usually designated by x, and the coordinate measured in the direction of the Y-axis is the ordinate, designated by y. The coordinates of a point are written in the form (x, y), the abscissa always being written first, followed by the ordinate. The signs -f and — indicate the direction to be measured. EXERCISE 357. 1. The coordinates of the point P 2 are (— 1, 3) ; of P 3 , (-2, -3). What are the coordinates of P 4 ? of P 6 ? of the intersection of the axes ? 2. What are the coordinates of P 6 ? of P 7 ? of P 8 ? 3. Locate the points (- 1, 1), (- 3, 0), (0, - 3), (0, 0). 4. Where are all the points which have abscissa 1 ? 5. Where are all the points which have ordinate — 2 ? 6. Give the signs of the coordinates for each quadrant. 7. How many points may have 3 and 4 as the absolute values of the coordinates? 8. Locate the points (3, 4), (3, 2), (3, 0), (3, - 1), (3, - 4). Draw a line through these points. What kind of line does this give? 9. If the abscissa is zero, where must the point be located ? 10. If the ordinate is zero, where must the point be located ? 358. Function. When one quantity depends upon another for its value, the first quantity is a function of the second. If a train travels at a uniform rate, the distance traveled is a function of the time. The cost of 10 yards of cloth is a 246 Graphs function of the price per yard. The area of a square is a function of its side ; the area of a circle is a function of its radius. The algebraic expression 2 x -f- 3 is a function of x. In the equation y = 2x + 3, y is a function of x. 359. Graph of a Function, equation y = 2 x + 3 may be pictured by means of a graph. Tabulating sets of values of x and y that sat- isfy the equation (allow- ing two squares for each unit) we have the follow- ing : The values of x and y in the X y 3 1 5 2 7 -1 1 -2 -1 -3 -3 Y 7 M o B - R 1 A'' A' -4 -. 1 1 / l (J l z ! 5 -1 -a / / -8 / N / Y 1 When all the pairs of values of x and y are used as coordinates we have a series of points that appear to lie in a straight line. If fractional values of x are taken, other points between these will be found. The line MNj if extended indefinitely in both directions, is the graph of the function of x, 2 x + 3, or of the equation y — 2 x + 3- By this we mean that : 1. Every pair of values of x and y that satisfies the equation will, if used as coordinates of a point, give a point on this line MN. 2. The coordinates of any point on this line satisfy the equation. Graphs 247 EXERCISE 360. Answer questions 1 to A by referring to the figure of § 359, and verify the answers by seeing if they satisfy the equation y = 2 x + 3. 1. What is the value of x f or y = ? for y = — 4 ? 2. What is the value of y for x = 2\ ? for x = — %? 3. Does the point (— 3.5, — 4) lie on the line MN? 4. Do the values x = \, y = 4 satisfy the equation ? 5. W^hen sugar is 6 ^ a pound, the cost, c, is a function of the number of pounds, p. The equation connecting them is c = 6p. Construct the graph for finding the cost, using the axis OX for the cost and O Y for the weight. y 9 8 H3 6 &* 3 2 1 t 1 1 5 2 2 5 3 3 5 1 4 B 5 5 5 G J P c 1 6 3 18 6 36 10 60 Cents Note. When no negative numbers are to be used, the points are all in the first quadrant and the bottom line and the line at the left side may be used Determine from the figure the cost of 5 pounds of sugar ; 9 pounds. How many pounds can be bought for 48y? for 55^? 248 Graphs 6. Construct a graph to show the cost of eggs at 28^ a dozen. Extend the graph to 8 dozen and estimate from it the cost of 2\ dozen ; of 5 dozen. Use 4 spaces for 1 dozen eggs on the X-axis, and 1 space for 7 ^ on the F-axis. 7. A train travels uniformly 45 miles an hour. Construct a graph and determine the distance it covers in 12 minutes, and the time it takes to go 24 miles. Hint. The equation is d = 45 t where d represents the distance in miles and t the time in hours. Use 1 space for six minutes on the X-axis and 1 space for 5 miles on the F-axis. 8. To change Fahrenheit temperatures to centigrade, the equation C= -J(F — 32) is used. In this equation F represents the number of degrees Fahrenheit and G the same tempera- ture measured by a centigrade thermometer. Thus, 50° Fahrenheit is changed into centigrade by substituting 50 for F - C= f(50-82) = | • 18 = 10. Plot the graph for C = ^(F — 32) for the following Fahrenheit tem- peratures: - 10°, - 20°, 32°, 40°, 50°, 60°, 70°, 80°, 90°. What, temperature Fahrenheit will correspond to 20° centi- grade ? to 25° ? 15° F. corresponds to what temperature centigrade? 72° F. ? 9. Knowing that 1 kilogram = 2.2 pounds, construct a graph that will enable you to convert pounds into kilograms or kilograms into pounds. The equation is K=2.2p. From this graph determine the number of kilograms in 11 pounds ; in 14.3 pounds; in 22 pounds. Determine the number of pounds in 3 kilograms ; in 5 kilograms ; in 8 kilograms. 10. Given that 1 inch = 2.54 centimeters, construct a graph by means of which inches can be converted into centimeters and centimeters into inches. The equation is i = 2.54 c. 11. At noon a boy begins to walk along a road at 4 miles an hour, and at 2 p.m. a cyclist rides after him at 10 miles an hour. Show in a graph the distance traveled in any time by Graphs 249 the boy and by the cyclist and use the graph to find when the cyclist overtakes the boy. (See example 8, § 353.) 12. A newsboy sells papers at 1 cent each and makes ^ cent profit on each paper. Represent graphically his sales and profits up to 50 sales. Hint. For locating points, use numbers of sales that are multiples of 3, as 6, 12, 18, etc. The equation is p = | s. 13. Another newsboy sells papers at 1 cent each and gets -J- cent profit on each paper. He has 6 cents carfare to pay. Represent on the same axes and to the same scale as used for the last exercise the sales and profits up to 50 sales. Hint. The equation is p = I j — 6. For what number of sales will the profits of the two boys be the same ? When will the first boy make more than the second ? When will the second make more than the first ? XIV. LINEAR SIMULTANEOUS EQUATIONS WITH TWO UNKNOWN NUMBERS 361. Consider the equation x + y = 5, (l) where both x and y are unknown numbers. There is an indefinite number of pairs of values of x and y that satisfy the equation. Thus x = 1, y = 4 is a solution, since 1 + 4 = 6. Also x = 2, y = 3 is a solution, since 2+3 = 5, and x= —4, y = 9 is a solution, since — 4 + 9 = 5. Tabulating some of the values of x and y that satisfy the equation, we have the following : This tabulation could be continued indefinitely in both positive and negative numbers, and also in fractions. This means that there is an indefinitely large number of pairs of values of x and y which satisfy the equation. The equation is therefore indeterminate. X y x + y 1 4 5 2 3 5 3 2 5 4 1 5 5 6 6 - 1 6 - 1 6 5 -2 7 5 Tabulating the values of x — y = 3, we have the following : (2) X y x — y 1 -2 3 2 -1 3 3 3 4 1 3 6 3 3 -3 3 - 1 -4 3 -2 -5 3 -3 6 3 This equation is indeterminate. It is seen, however, that the set of values, x = 4, y = I, occurs in both tables. That is, x = 4, y = 1 will satisfy both equations. Thus the two equations considered together become a determinate system, since they determine a definite set of values of x and y ; that is, x = 4, y = 1. The two equations, however, are each satisfied by sets of values of the unknown numbers which do not satisfy the other. They are therefore independent equations. 250 Linear Simultaneous Equations 251 362. Independent Equations. Two or more equations contain- ing two or more unknown numbers, and expressing different relations between the unknowns, are independent equations. Thus, x + y = 5 and x — y = 3 are independent. (Why ?) x — y = 3 and 2 x — 2 y = 6 are not independent since the second can be reduced to the first by dividing both members by 2. Any solution of one is a solu- tion of the other. 363. Simultaneous Equations. Two or more independent equations containing two or more unknowns which are satisfied by the same set of values of the unknowns are simultaneous equations. Thus, x + y = 5 and x — y = 3 are simultaneous equations. (Why ?) 364. Principles used in Solving Simultaneous Equations. (a) If equal numbers are added to equal numbers, the resulting num- bers are equal. (6) If equal numbers are subtracted from equal numbers, the resulting numbers are equal. Illustration. 3 + 5 = 7+1 (1) 2 + 3=5 (2) 3 + 5 + 2 + 3 = 7+ 1 + 5. (Adding equations (1) and (2) .) Let the student subtract (2) from (1) and note the result. 365. Tabulating values to find a set common to a system of simultaneous equations is too long a process. The following method is much shorter. Solve the system of equations x + y = 5, x-y = 3. Solution. 2x = 8. (Adding the equations.) x = 4. (Why?) Substituting this value of x in the first equation, we have 4 +y = 5 y = 1. (Why ?) By adding the equations we get rid of one of the unknowns ; this process is known as elimination. 252 Linear Simultaneous Equations 366. Elimination. The process of combining a system of equations so that one of the unknown numbers disap- pears is called elimination. ORAL EXERCISE 367. Eliminate x in the following exercise and solve for the remaining letter : 1. x + y = 10, 5. M- * = 9, 9. 2x + y = 7, x — 3y = 2. x = 7. 2x = 0. 2. 2x + y = 5, 6. x-z = 3, 10. 2a; + 3 #=10, 2x-y = l. x-2z = l. 32/-2a=-4. 3. y-x = 7, 7. x — w — 5, 11. £ + 3a = 4, y + x = 9. x -f- w = 4. t-3x = -2. 4. x-2y = 3, 8. x - y = a, 12. 2x-2a = $, y - x = 5. x + y = b. 2x-a = S. 368. Without attempting a complete discussion it may be stated that, in order to solve a system of linear equations with two or more unknowns, three conditions are necessary : 1. There must be as many equations as there are unknown numbers. 2. The equations must be simultaneous ; that is, there must be a set of values of the unknowns that will satisfy all the equations. 8. The equations must be independent ; that is, they must express different relations between the unknown numbers. 369. 1 The preceding discussion and definitions may be made clear by the use of the graphs of the equations. 1 Section 369 may be omitted, if desired. Linear Simultaneous Equations 253 V ~ 5 S/D _ 2 ^L nz ^7 ^^ x ' / V -5-4-3-2-1 1 2 /S 4 g\ -l z 2 =2 _^y = 2a, x -2yL-2a-b-2c. a*x-Vy = a> + V. 31. 5aj + 3y = 4a + 6, 38 « a>4-w# = -l, 3*+5y = 4a-6. y «■ n(* + 1). 32. a + 2/ = 10 a -3 6, 39. a + 1 = a?/, 2 a — y = 2 a + 3 b. y-bx = b. 33. az+&2/ = a, 4Q 3 a* + &2/ = a 2 + 1, ?*-y=2 6 y b *x-y = -. (>x-2b*y = 2a-2b. 34. 10 + 7 y 4- 4=0, 41. 3a + 2 y = 8 a- 7 6, 6 a 4- 5 2/ 4- 2 = 0. ax + by = 2a*-2b 2 . Elimination by Substitution 259 ELIMINATION BY SUBSTITUTION 374. Principle of Substitution. Any number may be substi- tuted for its equal. 375. 1. Solve the system 2 x + y = 25, (1) 3x-2y = 6. (2) Solution. y = 25 — 2x. (From equation (1).) (3) Sx - 2(25 - 2x) = 6. (Substituting in (2).) 3x-60 + 4x = 6. 7s = 56. 85=8. y = 25 - 2 • 8 = 9. (From (3).) Compare this with example 2, § 370. 2. Solve the system x + y= 15, (1) x:y = 2:Z. (2) Solution. 3x = 2y. (From (2).) (Why?) (3) x = \y. (4) | y + y = 15. (Substituting in (1) .) | y =15. 2/ =9. x = 6. (Substituting in (4).) Let the student check mentally. 3. Solve the system 3 x — 4 y = 8, Solution. (1) (2) * =§J 3^' (^"WO (3) 4 /8+^jA + 3 y = _ 6 (Substituting in (2).) Check mentally. 32 + 16 ^ + S y =-6. 32+ 16y + 9y = — 18. 25 ?/ = - 50. y=-2. = 8+4.(-2) =Q 3 (From 3).) 260 Linear Simultaneous Equations 376. To solve a system of simultaneous equations by substitution : 1. Find the value of either of the unknowns in terms of the other unknown and known numbers from one of the equations. 2. Substitute the value of this unknown for the same unknown in the other equation. 3. Solve the resulting equation. 4. Substitute the value of the unknown that has been found in one of the preceding equations to find the other unknown. 377. The method of solving simultaneous equations by substitution is especially convenient when one of the unknowns can readily be expressed in terms of the other. It is also much used in later work and should be well understood. EXERCISE 378. Solve the following systems by the method of substitution : 1. 2x-lly = -95, 9. 7x - y - 6a = 126, x — 3 y = 0. x = y. 2. B*-2yW21, l0 ax + by = c, x : y = 5 : 2. 3. fc-2d = l, x = 2y. ie-d = 0. «' ^-2^ = 69, 4. x = 3y -19, 2x + y = 78. y = 3x-23. 15 M * 12. m + — = 13, 5. 2w-%y = 4=, n 3w-$y = 0. m + ^ = 16. n 6. 5u- 4.9v = l, 3w-2.9 r = ?. X 2/ t- -3 p u q f ? = ?. t u 24. - + - = m, x y l_l =ri » 2/ 379. Most of the equations thus far solved have been given in the form ax -f- by = c. In the following exercises the equa- tions should be simplified before applying the rule. EXERCISE 380. Solve by either method : 1. 4(3a>-5)-2(y-3)=2 > 2(5x-y)-3y=5. 2 7 - 1 Ax — y x — y 2(x-y)=y-S. 3. 4(*-3y)=8, 262 Linear Simultaneous Equations Solve by either method : 4. x+y = lO, 3x + 5y 5x + 3y _ x + 5y 5 4~ 11 11 *2* 5. 5x-(3y-i)=.75, 4 + oj-2(2/-1)=0. 6. a: (b + y)=b:(3a + x) } ax + 2 by = b 2 . 7. x:y = 3:4, x-l:y + 2 = l:2. 8. icH-l:i/ + l:a; + 2/ = 3:4:5. 9. x — 5:y + 9:x + y + 9=:l:2:8. 10 2a?+y — 1 = 1 5 a; — 3y + 4 _3 t 3a + 22/ + ll~2' 6a-3?/-r-3~~4' 11. bcx = cy-2b, 6V + a(c3 ~ 63) = — + eta. 6c c 12. (a-6)x+(a + % = a + J a + 6 a — 6 a-f-6 13. "±!-«i £±±=&. X a 2 — 6 2 a 2 — 6 2 (Do not clear of fractions.) 16. (a + 6)* + (a - 6)y = 2(a 2 + V), (a - 6)* + (a + b)y = 2(a 2 - 6 2 ). (First add the equations, then subtract them.) Linear Simultaneous Equations 263 17. s + y + l = a +1 a? + y + l = 6-t-l T a — y + 1 a — 1' a; — I/ — 1 6 — 1* (Use composition and division on both equations.) 18. (z-f5)(y + 7) = (a; + l)(y-9)+112, 2a; + 10 = 3y + l. 19. 2.60 a; - .41 y - 4.28 + 2.50 x = 0, .50 a; + 3.6 y + 3.23 + .5 y = 11.93. 20 g + 1 -.y+g-20 g -y) 3 4 ~5 ' a;-3 y-3 rt 21 3 a; — 2 y 5 a; — 3 y = * + l, 2a,--3y.4a;-3y 22. 4 (a ._ 2/) __i ffa ,__i T _ 2/=14) f(z-14)-^(y + 12) = -2. 23 10a;-2y + 22 7a?-3y = a? + y-l 56 14 8 9 i8 - 7 *-^- 24. 2(a; + y-c)=2(a;-c)+a; + 3y-c, a; + 7 y = 15 c. 25. (»-l)(5y-3)=s3(3a? + l)+5ay, (s-l)(4y + 3)=3(7y-l)+4a>y. 26. -4-+-JL. = « + &, a + b a — b X I V o a 6 264 Linear Simultaneous Equations Solve by either method : 27. (a -f- c)x — (a — c)y=2 ab, (a + b)y — (a — b)x = 2ac. (Add the equations and simplify.) 28. 5x -y= Sx - 5 y= y -3. 11 4 * (Put each of the first two equal to the last expression.) 29. ? = !«» + ? + = 22 in (3).) Let the student check the results. 383. To solve a system of linear simultaneous equations with three unknowns : 1. Transpose, if necessary, so that the unknowns will all be on the left-hand side of the equations and collect like terms. 2. Examine the equations to see what unknown can be most easily eliminated. 3. Eliminate that unknown, using two of the equations. 4. Eliminate the same unknown, using the third equation and one of the others. 5. Solve the resulting system for the two remaining unknowns. 6. Substitute the values of these two unknowns in one of the original equations to find the third unknown. 384. In the solution of a system of simultaneous equations involving three unknowns the following suggestions will be found useful : 1. If one of the three equations contains but two of the three unknowns, eliminate the third unknown from the other equations. There will then be two equations containing two unknowns which can be solved, according to § 371. (See ex- ample 2, § 382.) 2. In general, eliminate first the unknown that can be most easily eliminated. If the coefficients of this unknown are not the same in two of the equations, make them the same by using the smallest multipliers possible, Three Unknown Numbers 273 3. Addition or subtraction is usually the simplest method of elimination. 4. Study the model solutions and the suggestions given for special methods of shortening the work. 385. To solve a system of four equations with four unknowns, eliminate one of the unknowns by using the four equations in pairs three times, thus deriving three equations with three unknowns. Con- tinue with these equations as indicated in § 383. EXERCISE 386. 1. How can x be eliminated from the system x+y + z = 12, 4sc + 3y + 5z = 49, 5x-2y + z = 12? 2. How can y be eliminated from the above system ? How can z be eliminated ? 3. Which letter will be most easily eliminated ? 4. Which unknown should be eliminated first in solving the system 2x-3y + 7z = 6, Sx + 4y + llz = 18, y -3z = -2? 5. Solve the system of equations given in example 1. 6. Solve the system of equations given in example 4. 7. Can you find a definite solution of a system of two equa- tions containing three unknowns ? Solve the following : 8. 2x + 5y-3z = 13, 10. 2x 4- 3y = 12, 6 x - 3 y + 4 z = 16, 3x + 2z = ll, 5x + 3y-6z = W. 3 y + 4 z = 10. 9. 3x-y + z = 7, 11. 2u-7v = 9, x + 2y — 4z = - 8, u + 4:v = 12, 2x — 2y + z = 2. u + v + 2 w = 14. 274 Linear Simultaneous Equations Solve the following : 12. x+2y-.7z = 21, 3x + .2y-z = 24:, .9x -\-7y-2z=27. 21. X = ^- 7 10 13. lx-iy = 0, \x — \z = l, \z-\y = 2. 14. p + q + r = 36, 4p = 3?, 2p = 3r. (Try substitution.) 15. x + y + z = 100, y = .7x-4, z = .3 x 4 4. 16. » + 2/ 4 2 = 26, x : % = 11 : T, 2/:z = 14:9. 17. r + » 4- * = 99, r : s : * = 5 : 3 : 1. 18. a? + y + 2 = «, x _y _z a b c 19. x+y = c, y + z = a, z 4 x = b. 20. 2jc + y + 3 = a, a + 2?/ + z = a, (Add all the equations and di- vide by 4.) z 5' 2x+3y = 88. 22. 1 + ^ + ^+1 = 0, 3 6 9 i + ^ + .ZL+1-O 6 + 9 + 12 +1 ~ U ' i + ^ + iL + i =0 . 9^12^15^ 23. - + - = a, x y -+- = &, x z -+-=c. 2/ ^ 24. a + b + c = m, & 4- c 4- d as n, c + d 4- a = ' v, d + a 4- 6 = q. (First add all equations, then divide by 3.) 25. x 4 y 4 z 4 w; = a, # — 2/4z — w = b, x + y — z — w = c f x — y — z-\-w — d. 26. ?-A +2= a^, a? Sy _L + i + 2z = 6H, 4 a; ?/ 6z y ** Three Unknown Numbers 275 27. x + 2y = % 31. x + 3y=±£> 3 y + 4 z = 14, a; -f- 5 ?/ = 3 z, 7z + u = 5, 10y-3z + 2 = 0. 2u + 5z = 8. 28. a; + 2?/ -z = 4.6, 32 . 3(a + 2?/)- 2 = 2, 2/ + 2 z - a = 10.1, x + 2 y = i z, 2 + 2 a - J/ = 5.7. x + y + 2 — 5|. 29. 3a; + 22/ + 3z = 110, 5a + 2/ = 4 Z , 33 fr^-y-l, 2-0 + *-** a? + 2g = 2 30. z = 2/+l, 3 3' •-• + * 2/-- = 17. 2/ = 2z -10. y 6 - M Sx + y-z _ Sx + 3y-A = 2x + y _ Zf 3 6 20 + y + » — 6. 35. a + 26+3c = 32, 2a+36 + c = 42, 3a + 6+2c = 40. 36. + y + z =3, 2x + 4y + 82 = 13, 3a + 9# + 27z = 34. 37. a + ?/ + 2 2 = 34, x + 2 y + 2 = 33, 2 a; + y + 2 = 32. 38. x=2%y-6, y = 3\z-l,z = l\x -8. PROBLEMS SOLVED WITH THREE UNKNOWNS 387. 1. The sum of three numbers is 100. If the first is divided by the second, the quotient is 5 and the remainder 1. The second divided by the third gives the same result. What are the numbers ? 2. Find three numbers whose sum is 999, and which are to each other as 2 : 3 : 4. Solve, using three unknowns. 276 Linear Simultaneous Equations 3. Solve problem 2 using only one unknown. 4. A number equals the sum of two other numbers. The largest number diminished by 2 equals three times the smallest. The largest increased by 2 equals twice the re- sult of diminishing the middle number by 2. Find the numbers. 5. Find three numbers such that if the sum of each two is diminished by the other the results are respectively 0, 4, and 8. 6. Three men, A, B, and C, working together can do a piece of work in 5^ days. A and B together can do the work in 6^ days. A and C can do it in 9f days. How long will it take each one working alone ? Suggestion. One equation is - -\ \-- = — x y z 51 7. Suppose they all work together and receive % 96 for doing the work, how much should each one receive? 8. In the figure the circles touch each other. The sides of the triangle are AB = 7 inches, AC = 7 inches, BC = 5 inches. Find the radii of the circles. 9. In the figure AQ= AP, BQ= BR, CR = CP. Find AP, BQ, and CR, know- ing that AB = 6 inches, BO = 8 inches, CA — 9 inches. 10. The expression ax* + bx 2 + ex + 5 is equal to 10 when x=l, to 15 when x = 2, and to 20 when x = 3. Find the values of a, 6, and c. Hint. Substitute 1, 2, and 3 successively for x. The resulting equa- tions are a + b + c + 5 = 10. 8a + 4fc-f2c- + 6 = 16. 27 a +0 b + 8 c + 5 = 20. Problems with Three Unknown Numbers 277 11. Find the values of I, m, and n in the equation - + - -f l m —=s 1, if it is satisfied by x = 1, y = 2, 2 = 3; x = 2, y = — 1, n z = 3, and 6 asty + a^ 2 -80 is l ar g er tDan the remainder 12. Put a zero in the root and bring 403 1 12 09 down another period. The trial divisor now be- 1 12 ' comes 400, and the next figure in the root is 3. The square roots are ± 203. 286 Square Root 3. Find the square root of 2 to three decimal places. 2.'00'00'00| 1.414+ After pointing off into periods, the decimal point may be neglected. How will the num- ber of figures to the left of the decimal point in the answer compare with the number of periods to the left of the decimal point in the number ? 604 EXERCISE 402. 1. In extracting the square root of a number, why do we separate the number into periods of two figures each ? 2. Will the division of the remainder by the trial divisor always give the next figure of the root ? Explain your answer. 3. Square the result in example 3, § 401, and add the remainder ; that is, 1.414 2 + .000604. 24 100 96 281 400 281 2824 11900 11296 Extract the square roots of the following 4. 4096. 5. 6241. 6. 161.29. 7. 2.3716. 8. 61504. 9. 1108.89. 10. 277729. 11. 13456. 12. 30276. 13. 119025. 14. .093025. 15. .007569. 16. .098596. 17. 12.8881. 18. 11669056. 19. 6504.4225. 20. .83064996. 21. 95121009. 22. 101062809. 23. .00917764. 24. 1400.2564. 25. .00762129. 26. .0009979281. 27. 100020001. 28. 29495761. 29. 64128064. 30. 44105040144. Find the square roots of the following, to two decimal places : 31. 2.2. 35. 7. 39. 3.666. 32. 3. 36. 8. 40. 27.1917. 33. 5. 37. 3.1416. 41. 391. 34. 6. 38. 210. 42. 10.004. Square Root of Arithmetical Numbers 287 43. 40.003. 45. V_. 47. 80. 44. J/- = 6.5. 46. 5|. 48. 82. By reducing to a decimal find the square roots to three deci- mal places in examples 49 to 53 : 49. f. 50. 2f. 51. 5f 52. 16|. 53. f By first making the denominator a perfect square find the square roots in examples 54 to 61 to three decimal places : 54. f /5 J30 = V30 = 5.477 '36 6 6 to three decimal places. Suggestion. -v-=A/— = — : — = -913. This result is correct 55. T V 56. f 57. f. 58.11 5 9. #. 60 . V4 + V2 61 . fe+VO 2 * X 2 62. To find the approximate square root of \, why is it better to use \ than \ ? 403. It is sometimes desired to find the square root of an algebraic expression that is not a perfect square correct to a certain number of terms. The process does not differ from that used when the expression is a perfect square. Find the square root of 1 -f- x, in the ascending powers of x to three terms. 1 + x 1 2 + ix l + ±x-±x 2 x x + \x* I- 2 1/^3 1 r/A To check, square the result and add the last remainder. 288 Square Root EXERCISE 404. Find the square roots of : 1. 1 — a to three terms and check. 2. 1 — a 2 to three terms and check. 3. 4 + x 2 to four terms. 4. 1 -f x + x 2 to three terms and check. 5. a 2 + x to three terms. 6. 1 -f 4 a 2 to three terms. 7. 1 + x — x 2 to three terms. 8. 13 x 2 — 3 x 3 + 4 x 4 — 12 x + 4 to three terms and check. 9. a 6 + 4a 5 6-2a 4 & 2 -12a 3 6 3 + 9a 2 & 4 . 10. 16 a 6 - 24 a 5 ?/ + 65 a 4 ?/ 2 -42 a 3 ?/ 3 -f 49 a 2 ?/ 4 . 11. a 2 + 4a& — 2ac+4& 2 — 46c-f c 2 . 12. 4 a 4 - 12 a 3 6 2 - 4 a 2 6 3 + 9 a 2 6 4 + 6 ab b + 6 6 . 13. a 4 -2a 3 + 3a 2 -2a + l. 14. 1 4- a 2 — 2 a 4 + a 6 4- 2 x — 2 a 8 . 15. 2 mw + £> 2 -j- 2 np + a' 2 + 2 mp + m 2 . 16. l + 4?/ 2 + a 2 -4?/ + 2 a — 4 ay. 17. 4 + 13 a 2 + 9 a 4 -4 a -6 a 3 . 18. 18a 2 +a 4 + l-8a 3 -8a. 19. a 2 6 2 +2a 2 6 + a 2 -2a& 2 -2a& + & 2 . m 4 ?i 6 m 2 /) 4 m 2 n z p* n 3 p 4 20 - t + 9 + ^ — r + «r 6 01 a 2 4az ■ 4z 2 , 6ga , 9g 2 12^z y 2 uy u 2 vy v l uv Square Root of Arithmetical Numbers 289 Find the square root oj each of the following polynomials to three terms : 23. 1 + x 2 . 26. 1 + x + x 2 + a 3 + x 4 . 24. 1 — 4 a 2 . 27. 1 — a + a 2 — X s + x A . 25. a^ + x 2 . 28. 1 + a? + x 4 + a 6 + «*• i^md £/ie fourth root of: 29. -^ to two decimal places. Suggestion. Take the square root of the square root 30. aj* + 4B 3 + 6 2. x* + 5x + 6 = 0. 2 6a? 6 6. 3« 2 -5a;=10+2a; 2 -2ic. 3. 2*2 + *_3 = 0. 7 (^ + 5)2=2(^ + 3)2-17. 4. m 2 - 2m -24 = 0. 8. aj(aj - 1) = 380. 9. (2jp - 8) 2 = 4(3 p + 25) + 12. 10 z + 2 _ 36 j^ 14. 4(?- 2 -l)+r + l = 0. 2 + 3 (z + 3) 2 15 . 5(a 2 -4)-(a-2)=0. 11. Kv + 5)=3(-2/-5). 16. S 2_ 8 = 7s-14. & = x_x. 1 17 - 3w 2 + w = 10. '632 18. (5x-2)(6x*-x-2)=0. 13. (2m) 2 -5(2m)-6 = 0. 19. x^-5x + A = 0. 20. What is a root of an equation ? 418. Completing the Square. In § 204 we learned that any one of the three terms of a perfect trinomial square can be sup- plied if we know the other two terms. Complete Quadratic Equations 295 ORAL EXERCISE 419. In each of the following supply the proper number in the parenthesis to make a perfect trinomial square, and find the square root of the trinomial : 1. x 2 + 2ax + ( ). 8. x 2 +6x + ( ). 2. 4# 2 + 4a+( ). 9. x 2_$ x +( y 3. 4a 4 + 4a; 2 +( ). 10. x 2 + x + ( ). 4. 4a 4 +( )+4# 2 . 11. a 2 -3ar+( ). 5. 9 + 6z+( ). 12. a 2 -?+( ). 6. ( ) + 6* + l. 2 * 7 . a 2 + 2z + ( ). 13 ' x2 ~a +( } * 14. State a rule for completing the square in expressions of the form x 2 -f- px. 420. The >form. Every quadratic equation with one un- known number can be written in the form ax 2 + bx -f c = 0. This can be further simplified by dividing through by a, giving be b c x 2 -f- -x -f- — = 0. By putting - = p and = q, the equation a a a a assumes the form x 2 -f px = g. For convenience we shall call this the p-form. It requires that the coefficient of x 2 be -f- 1, and that the absolute term be in the second member of the equation, p and q may be any positive or negative numbers, integers, fractions, monomials, or polynomials. Examples Examples 1 to 5 below are in the p-fovm. 1. x 2 -7x = 10-, p = -7, and q = 10. 2. aj 2 + | = -9;p==i,andg = -9. 3. x 2 = 90 ; p = 0, and q = 90. 4. x 2 + (a -f- 6)a; = ; p = a + 6 and g = 0. 5. x 2 — - = (& - c) 2 ; p = - -, and q = b 2 - 2 be + c 2 . ft ft 296 Quadratic Equations EXERCISE 421. Change the following into the p-form, and determine the value ofp and q for each : 1. x 2 + bx -f- c = 0. 2. 2z 2 -b 15.9 = 13.6 a;. 3. (a? — 7)(aj — 5)=0. 4. (x-iy = a(x*-l). 5. c(a-xy+(x-by = a 2 + b 2 . 6. ?L^_^+1 = 0. 4-x 4 422. The Solution of the Complete Quadratic Equation by Com- pleting the Square. The solution of a quadratic equation by factoring fails when the factors cannot be found. The method about to be given will solve in all cases. 1. Solve the equation x 2 -f- 9x = 10. Solution, x 2 + 9 x = 10. x 2 + 9 x + V = if 1 . (Adding *£ to both members.) x + | = ± V- (Extracting square roots.) z = 1 or - 10. Check. 1 2 + 9.1 = 10. (_ 10)2 + 9 (_ 10) = 100 - 90 = 10. How do you determine that y is to be added ? Why do you add it to both members ? Why do you use the double sign in the second member ? Why do you not use the double sign in both members ? The solution of example 1 illustrates the method of solving complete quadratic equations by " completing the square." This equation was in the j>form at first. The steps required to reduce any quadratic equation to the p-form are already- familiar to the student. Complete Quadratic Equations 297 2. Solve, getting the answers correct to two decimal places, X 2_ f _( ;r + 2) 2 = 180. Solution. x 2 + x 2 + 4x + 4 = 180. (Why?) 2 x 2 + 4 x = 176. (Why ?) x 2 + 2x = 88. (Why?) x 2 + 2x + l = 89. (Why?) x + i = ± 9.43+. (Why ?) x = 8.43+ or - 10.43+. These roots can be* obtained to any required degree of accuracy by finding the square root of 89 correct to more decimal figures. 423« To solve a complete quadratic equation : 1. Reduce the equation to the p-iorm. 2. Complete the square of the first member by adding to both mem- bers the square of one half the coefficient of x. 3. Extract the square root of each member of the equation and solve the resulting linear equations. Examples 1. Solve 6 x 2 = x + 15. Solution. 6 x 2 - x = 15. (Why?) x*-\x = \. (Why?) *-A=±H- (Why?) Check. 6 • (f ) 2 = \ + 15 or ^° = *£. Let the student check the other root. 2. Solve x 2 + ax = ac + ex. Solution. x 2 + ax — ex = ac. x 2 + (a — c)x = ac. *+(._,), + («-«)' = «, + («-«)■ or «* + 2f + ig. 4 4 4 2 2 _ _ a - c , a + c X — -^ ± 2 ' x = c or — a. Let the student check mentally. 298 Quadratic Equations EXERCISE 424. Solve the following equations by completing the square, finding all roots correct to two decimal places: 1. a? 2 + 2a;=3. 24. (2a;-15)(3a;+8)=-154. 2. x 2 - 10 x = 200. 25. 8a; 2 + 2a;-15 = 0. 3. tz + t = 12. 26. 20 a; 2 + 2a;- 7 = 0. 4. x 2 -x = 12. 27. 6(a; 2 + l)=13a;. 5. a? + 3a; = 10. 28. 3s 2 -16 = 7s. 6. w 2 + 3w = 108. 29. a? + ia; = 2. 7. a 2 + 17 a; = 30. 30. x 2 + 6.51 = 5.2 a. 8. a; 2 -8a; + 15 = 0. 31. ?/ 2 + .2 y - .15 = 0. 9. a 2 -40a + 111 = 0. 32. x 2 + 6a? - 2 b 2 = 0. 33. (a? — 7) (a? — 5) =40. 35. a; 2 + 22(a; + 5)=0. 36. (4a;-l)(a; + l)=75. 37. p(p — 6)=7p-42. 38. 4ar i + (a;-l) 2 -3a;=31. 39. q(q-2)=67. An „ ,1 4a; + 7 40. 7s + T = T ■ 4 16 a; 41. a?(aj + l)=-^. „ 3 a,/ 4\ 19 42 ' T^-3) = 24- 43. a; 2 -8 a; -14 = 0. 44. (x + 4) (a? + 5) = 2(s + 2)(a> + 4). 45. (3-2 a>)(l - 3 a;) (2 - aj)=* x(l — 6 a>)(» - 2). 46. (a; + 6) (a? - 4) + (x + 2)(a? - 2) = 56. 10. x 2 -2Ax + .$ = 0. 11. x 2 -2ax = 3 a 2 . 12. c 2 + 2c = -l. 13. a; 2 -36a; + }& 2 = 0. 14. a; 2 -32 a; = 32. 15. 2v 2 + 3v = 108. 16. 3a; 2 -5a; = 2. 17. 6 x 2 + 1 = 5 a;. 18. 5 ra — m 2 = — 50. 19. 15 a; 2 + 8a; = 3.75. 20. 9a; 2 + 17 a; = 310. 21. Ja^ + ^Of-A-O. 22. &(7-5)=6. 23. far 5 -11a; -15 = 0. Complete Quadratic Equations 299 47. (a-l) 2 +(a + l) + (2a + 3) 2 = 29. 48. 4c 2 -3c=31-(c-l) 2 . 49. (4-d)(5d + l)-d(4-d)=0. 50. (7 - a)a =(l - -V a? + 8). 51. x(x + l)(a> + 3)-(a> + £)(a + -J) (a? + ±)= 0. 52. 3a(a + l)-(7 + 2a)=0. 53. 118z-2i2 2 = 20. 54. a 2 4- a& = a(a + 6). 55. (a-3 6)(a + 2&)=6& 2 . 56. (2y-«)(2e-y) + (5y + 2e)e = 0. 57. r(r + 1.25) = .75(r + 1.25). 58. (r + 3) 2 +(r + 5) 2 = 514. 59. £ 2 = l-£. 60- A^ + tJ^^-A— - 61. a; (7a ; -l )+ ^- 20 ^ + 3 )+ 8 = 0. e2 4a? ~ 7 2a + 3 _23 2 94 62 ' ~X~ + ~~9~"-45 a; ~45' 63. M^*l 2 3==()> 2a-l 2a- 1 64. J^-6-|^ = 0. 69. *:2(*-3Wa?-3:»-l a + 2 3 a; 65. ^ + lY-?a; = 3. 70. 10:Z=Z:10-^. 66 . ?0a + l = i9. 71 . __3 L_ = l. 3 a? 3 2(a?-l) 4 (a? + 1) 8 a? + 11 2a? + l = 72 4a g + 3 ' =1 a + 3 a + 5 a — 1 a; 67. 68. -i§ l^_ + 5 = 0. 73. a 2 -a + 3 = a + 5 2 + 3 2 + 10 aj*-4a> + 5 a-1 300 Quadratic Equations Solve the following equations by completing the square, finding all roots correct to two decimal places : 74 M-2_2(y-2)_ 2=a 75 J_ + ^ 1_ = . y y+1 x + 1 x + 2 x + 3 76 x-5 x + 8 80 , = 1 ' s + 3 + x _r9-^ 2 77 2 * + 7 3s-2 =5 2 a; — 3 05 + 1 425. 1. In solving any problem by means of an algebraic equation, the student should first carefully read the problem so that he can correctly translate the verbal language into the algebraic language of the equation. 2. He should then solve the equation in the most direct way possible. 3. He should check and interpret the results of the solution. It should be noted that the conditions of the problem, with all their restrictions, cannot always be translated into an algebraic equation, so that the solution of the equation may give roots that do not satisfy the conditions of the problem. See example 4 following. PROBLEMS 426. 1. The area of a circle is ttjR 2 where ir == 3.1416 and R is the radius. Find the radius of the circle whose area is 78.54 square inches. 2. The area of a circle is 100 square inches. Find the radius correct to two decimal places. 3. Find two consecutive integers if the sum of their squares is 25. Solution. Let x = the smaller number. Hence * 4- 1 = the larger number. (Why ?) Then « 2 + (as + l) 2 = 25, (By the conditions.) or 2x 2 + 2x + 1 = 25. (Why ?) .•.x 2 + z = 12. (Why?) Complete Quadratic Equations 301 & + x + i = - 4 ? 9 -- (Why?) x = 3 or — 4 = the smaller number, and z + l = 4 or — 3 = the larger number. The answers are 3 and 4, or — 4 and — 3, either pair of numbers satisfying the conditions. 4. The square upon the longest side of a right-angled triangle is equal to the sum of the squares upon the other two sides. In a certain right-angled triangle one of the sides about the right angle is 1 inch longer than the other and the hypote- nuse is 5 inches long. Find the two sides about the right angle. Solution. Let x = number of inches in one of the sides. Hence x + 1 = number of inches in other. Then z 2 + (x -f l) 2 = 25. ( Why ?) The solution from this point on is exactly the same as in problem 3, but the negative answers that were satisfactory in problem 3 have to be rejected. The sides of the triangle are 3 inches and 4 inches. Note. In the solution of applied problems, careful attention must be given to the interpretation of the answers obtained. Sometimes one, sometimes both answers satisfy the conditions of the problem. It may happen that neither of the answers will satisfy the conditions. (Why ?) 5. By solving as in problem 3, find out if there are two consecutive integers the sum of whose squares is 32. 6. Separate 360 into two factors whose difference is 9. (This problem can be solved by any one of the three equations (a) x(x - 9) = 360 ; (6) x{x + 9) =360; (c) x - — = 9. Explain and solve each equation.) x 7. The sum of a number and its reciprocal is -f . What is the number ? Do both roots of the equation satisfy the con- ditions ? 8. The area of a rectangle is 720 square inches. The dif- ference of its two unequal sides is 12 inches. Find the dimensions. 9. How long is each side of a square if the diagonal is 10 inches long? (See problem 4.) 302 Quadratic Equations 10. The two unequal sides of a rectangle are in the ratio of 5 to 12, and the diagonal is 6.5 inches long. Find the dimensions. Suggestion. Let the number of inches in the sides be 5 x and 12 x . and see problem 4. 11. The area of a rectangle is 2400 square inches. The ratio of its two unequal sides is 5 to 12. Find its dimensions. 12. The sum of the areas of two squares is 233 square inches ; the sum of their sides is 21 inches. Find the side of each square. 13. In the accompanying figure the shaded area is equal to 21.46 square inches. Find the radius of the circle. (The side of the square equals twice the radius of the circle, and the difference in their areas is the shaded part. See also the first prob- lem of this set.) 14. Find two numbers, one of which is double the other, such that the sum of then- squares exceeds the sum of the numbers by 68. 15. Find two numbers, one of which is double the other, if the square of their sum exceeds the sum of their squares by 100. 16. Find two consecutive numbers if the sum of their squares exceeds the product of the numbers by 43. 17. If 18 is divided by a certain number, the quotient is greater by 1^ than if the divisor were increased by 2. Find the first divisor. 18. Find two consecutive even numbers the sum of whose reciprocals is ^. 19. A train makes a run of 280 miles in 1 hour and 45 min- utes less time than another train whose rate is 8 miles an hour less. Find the rate of each train. Suggestion. Remember that distance -4- rate = time. Complete Quadratic Equations 303 20. A woman buys cloth for $ 8. Had she paid 40 ^ more per yard she would have received one yard less for the same amount. How much per yard did the cloth cost ? 21. A man bought a flock of sheep for $ 75. If he had paid the same sum for a flock containing 3 more sheep, they would have cost $ 1.25 less per head. How many did he buy, and at what price per head ? 22. S = \gt 2 -f v t is a formula much used in physics. Find t when S = 520, g = 32, and v = 24. 23. Find the value of t in S = ±gt 2 + v t when £ = 100, g = 32, and v = 0. 24. m : n = x 2 : (a — x) 2 is a relation used in the study of light. Find the value of x when m = 4, n = 3, and a = 150 cm. 25. A rope 100 feet long is stretched around four posts set at the corners of a rectangle whose area is 576 square feet. Find the dimensions of the rectangle. 26. The sum of the two unequal sides of a rectangle is 20 feet and the diagonal is 16 feet long. Find the lengths of the sides correct to 2 decimal places. 27. A farmer bought some sheep for $134.40. If each sheep had cost him 80^ less, he could have bought 3 more for the same amount. How many sheep did he buy ? 28. A traveler made a journey of 630 miles. He would have required 4 days less to make the journey had he gone 10 miles farther each day. How many days did the journey require, and how many miles did he travel each day ? 29. A traveler made a journey of 630 miles. He would have required 4 days more to make the journey had he traveled 10 miles less each day. How many days did the journey require, and how far did he travel each day ? 30. Solve V= i h(S 2 + s 2 + Ss) for S where V = 252, h =12, and s = 3. 304 Quadratic Equations 31. The sides of a triangle are 18 inches, 16 inches, and 9 inches. By how much may the sides be equally shortened so that they may form the sides of a right-angled triangle ? 32. Divide a straight line 8 inches long into two segments such that double the square on one segment shall equal the rec- tangle whose base and altitude are respectively the whole line and the other segment. 33. Solve the equation ax 2 + bx + c = when a = 5, b = 20, c = 16. REVIEW QUESTIONS 427. 1. What is a quadratic equation ? 2. Illustrate each of the three forms of quadratic equations. 3. What is a complete quadratic equation ? an incomplete quadratic ? 4. Give the rules for solving incomplete quadratics. 5. In what form must an equation be written if it is to be solved by factoring ? 6. Give at sight six roots of the equation (x 2 + 2x)(x 2 — 1) (x 2 — 5 x + 6) = 0. Can you give at sight any roots of (x 2 + 2 x){x 2 - 1) = 37 ? (Explain.) 7. What is the p-i orm of the quadratic equation ? How is the quadratic in one unknown reduced to the p-i orm ? Why is the j>form used when solving by completing the square ? 8. Reduce 7 x 2 — 3 x + 2 = 5(3 — x) to the p-form and give the value of the absolute term when in the p-form. 9. Given the equation x 2 + 2 x + 1 = 9. In solving this equation the next step gives x -f 1 = ± 3. Why is it not ±(a + l)=±3? 10. Can you solve a quadratic equation that lacks the abso- lute term by completing the square ? XVIL SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS 428. One equation of the first degree and the other of the second degree. 1. Of what degree in a; is ax 2 + bx + c = ? of what degree in a ? (See §§ 245, 246.) Of what degree in x and y is 2 z+ y =10 ? Of what degree in x and y is 3 xy = 1 ? 2. What is the principle of substitution ? (See § 374.) 3. Explain the solution of simultaneous equations of the first degree by the method of substitution. 429. Solve the simultaneous quadratic system, x + y = 6, x 2 + 3y=l6. y = 6 — x. (From the first equation.) X 2 + 3(6 - x) = 16. (Substituting.) x 2 - 3 x = - 2. x 2 - 3 x + f = I x = 2 or 1. y =4 or 5. (Substituting in the first equation.) There are two sets of answers, x = 2, y = 4 will satisfy both equations. Also x = 1, y = 5 will satisfy both equations. 430. To solve a system of simultaneous equations when one equation is of the first degree and the other of the second degree : 1. Find the value of one of the unknown numbers in terms of the other unknown and known numbers from the first degree equation. 2. Substitute the value of the unknown thus found in the second de- gree equation and solve the resulting quadratic. 3. Substitute each value of the unknown already found in the original linear equation and solve for the other unknown. 4. Arrange the answers in pairs as found. 305 306 Simultaneous Equations Involving Quadratics EXERCISE 431. Solve the following systems of simultaneous quadratic equations. Find results involving decimals correct to two decimal places : 1. x — y = 2, (Why is it better, in this example, to substi- x 2 -f xy = 40. tute * — 2 than to use x = y + 2 ?) 2. 2 x 2 - y 2 = 7, 12. 15(a 2 - ?/2) = 16 xy, 2x — y = S. x — y — 2. 3. 3 x — y = 5, 13. ic -f ?/ = 15, xy — x = 0. x 2 + y 2 = 150. 4. % 2 + 'u 2 = 40, 14. ^ 2 + ? 2 = 25, w = 3v. 3p + 4 g = 24. 5. 5 ic 2 + y = 3 xy, 15. x 2 -\-2xy — y 2 =7(x—y) } 2x — y — 0. 2x — y = 5. 6. (z + y)(a; — 2y)=7, 16. r : s = 9 : 4, a; - y = 3. r : 12 = 12 : s. 7. an/ = 135, 17 ^ 2 + y + l = 3 aj = 3 i/ 2 + ^ + l 2' 2/5 a: - ?/ = 1. 8. x 2 -y 2 = 240, 18. a?y = 360, a? — y = 6. a? — y = 9. 9. x + y = 37, 19. a=2&, a 2 + 2/ 2 = 949. (a +6) 2 -(a 2 +6 2 ) = 100. 10. m 2 4-^ 2 = 130, 20 a! 2 H-2/ 2 = a; + 2/ + 2 = 5 m + n:m — n = 8 : 1. # + 3/ 3 3 11. a; 2 + 2/ 2 + «2/ = 147, (B + 2/ = 13. 21. — 4- — = 13, (Regard - and - the unknowns.) x 2 y 2 x y i+l-* a; v Simultaneous Equations Involving Quadratics 307 22. 2/-z = 8, 1 + 1 = 1. 2/2 = 240. a b 20' 23. ^ = 12. i + i= 41 x — y x*-y* = 48. a 2 6 2 400 432. Many of the problems in § 426 conld have been solved by using two unknown numbers instead of one. In general, the student will find it easier to state such problems alge- braically by using two unknowns than by using one unknown. 433. Problems Involving Simultaneous Quadratics. 1. The difference of two numbers is 4 and the sum of their squares is 106. Find the numbers. The equations required are evidently x — y = 4, ar*-f ?/ 2 =106. Let the student solve the system. 2. The sum of two sides about the 3* right angle in a right-angled triangle is 17 inches, and the hypotenuse is 13 y inches long. Find the sides about the right angle. Solution. Let X = the number of inches in one of the sides, and y = the number of inches in the other side. Then x + y = 17, (By the first condition.) and x 2 -f y 2 = 169 (By the second condition.) x = 17 - y. (17-y) 2 + y 2 = 169. 289 - 34 y + y* + y* = 169. (Why ?) y 2 _ 17 y = _ 60. (Why?) yl _ 17 y + i|9 = _4^. (Why ?) y-V = ±h .-. y = 12 or 5, and x = 5 or 12. (Why ?) Therefore the sides about the right angle are 12 inches and 5 inches. 3. The sum of two numbers is 21 and their product is 68. What are the numbers ? 4. The perimeter of a rectangle is 27 feet and the area is 44 square feet. What are the dimensions ? 308 Simultaneous Equations Involving Quadratics 5. The perimeter of a rectangle is 34 inches, and the diag- onal is 13 inches. What are the dimensions ? 6. Two fields of unequal size are both square. Their total area is 50 acres and it takes 1^ miles of fence to inclose them. Find the dimensions of the fields. 7. The sum of the areas of two circles is 13,273.26 square yards and the sum of the radii is 79 yards. Find the lengths of the radii. 8. The product of the sum and the difference of two num- bers is a and the quotient of the sum divided by the difference is b. Find the two numbers. 9. The area of a rectangle is 1224 square feet and the unequal sides are in the ratio of 3 to 5. Find the dimensions. 10. A line AB, 10 inches long, is divided at P into two y parts, x and y, so that a; is a mean proportional between AB and y. Find the lengths of x and y. 11. In a right-angled triangle the hypotenuse is 20 inches long and the sum of the other sides is 28 inches. Find the other sides. 12. The hypotenuse of a right tri- angle is 10 inches and the perimeter is 24 inches. Find the length of the two sides about the right angle. 13. The area of a right-angled triangle equals one half the product of the sides about the right angle. If the area of a right-angled triangle is 30 square inches and the sum of the sides about the right angle is 20 inches, find the length of these sides correct to two decimal places. 14. The perimeter of a rectangle is 26 inches and one of the diagonals is 10 inches long. Find the lengths of the sides. General Review 309 GENERAL REVIEW 434. 1. Factor (a) x 2 - 6 ax - 9 b 2 - 18 ab. (6) 24 x 2 + 6 ay - 18 ?/ 2 . (Princeton.) 2. Find the L. C. M. and the H. C. F. of O 3 + a 3 )(» 2 + a 2 ), (x 2 + az + a 2 )(3 x - a), 3 x 2 4- 2 aa - a 2 . (Harvard.) 3. Simplify (_^ t _i I ) + (-l t --A_) > (Regents.) 4. A number multiplied by 17 is increased by 1056. What is the number? 5. In 1912 a father's age was three times that of his son who was born in 1890. When will the son's age equal one half the father's age ? 6. Solve the system by addition or subtraction : a-3 ^6 x + 5 = 7 y-4: 7' y + 1 6* 7. Solve by substitution : 9* = 13 y> X -l = l- y 5 35 8. Divide x* - 3 x 3 - 36 x 2 - 71 x + & by x 2 - 8 x - 3. 9. For what value of A: in example 8 will the division be exact ? 10. A and B start from the same place, A traveling due north and B due west. B travels one mile an hour faster than A and at the end of 3 hours they are 15 miles apart. What is the rate of each ? 11. Resolve into factors : (a) ^-3^+2. (6) tf-y*. ( c ) 9-6C + C 2 . y 2 x 2 12. Simplify — 5 £±4 - t^ — x+1 x-1 1-x 2 > x+ 1 x 2 x + 1 x — 1 1 — x 2 13. Solve— L_-£±i — = 0. 310 General Review 12 3 1 ° YG 2x-l~x + 2~ 2x + 2 2^ + Sx-2 = 15. Simplify a? ~ 4 _3a>-5 5* + 9s + U * J 2x-l x + 2 2x 2 + %x-2 16. Find the value of 8 in 8 = | gt 2 4- v Q t when t = 3, a = 32, and -y = 0. 17. Find g in example 16 if 8 = 277.6, t = 4, v == 5. 18. Find £ in example 16, if 8 = 450, a = 32, v = 10. 19. Solve 3 x 2 — 7 a; — 2 = 0, finding the values of x correct to two decimal figures. 20. Divide x 4 4- a? 4- ax 2 + 6a; — 3 by a? 4- 2 a? — 3, and find what values a and b must have in order that there shall be no remainder. oi rci™ « c __ 2a — 3 6 2c — 3d 21. When - = _, prove _^- = ^_. 22. Solve 5s-g(^ = 3 ^- 5 ). a; 3 2 23. Find VO to 3 decimal figures. 24. A room is one yard longer than it is wide. At $ 1.75 a square yard a carpet for the floor costs $ 52.50. Find the dimensions of the room. 25. Solve ax — by = 0, x — y m c. 26. Factor (a) 27 a? -64; (6) 16a-25a& 2 ; (c) 16 a; 2 4- 25 ?/ 2 4- 40 a#; (d) a*4-2/ 6 ; (e) a; — 1 4- x 3 — x 2 . (Eegents.) 27. Simplify _b rf-y /a + 6 q-i\,/ a , 6 \ 1_1 a» + b* \a-b^a + b) \a + b^a-b) b a (Regents.) General Review 311 28. Find the square root of 5 x 2 - 23 x A + 12 x + 8 x h - 22 x* + 16 x 6 + 4. 29. Solve the system x+ y + z = 4, 2x4-31/ — z = 1, 3 a* = a 12 . .-. ( a *)» = a 4 * 3 . In general, (a™)* 1 = a™* 1 . Power of a Power. 2. To find a power of a product : (ab) 3 = ab-ab-ab (Definition of Exponent.) = a- a- a- &•&•&. (§73.) Also (abc) 2 = abc • abc = a- a-b-b • c • c = a 2 6 2 c 2 . In general, (a&) n = a n b n . Power of a Product. An integral exponent of a product can be distributed to the factors of the product. 315 316 Exponents » 3. To find a power of a quotient : (ay « . a § a = a? t (why9) \bj b b b 63 ^ vv r'J In general, ( ~ ) =^. Power of a Quotient. \bj b n An integral exponent of an indicated division can be distributed to the dividend and the divisor. ORAL EXERCISE 437. Perform the operations indicated : 1. a 5 • a 4 . 12. ra p+2 • m p_3 . 23. (c 5 ) 4 . 2. a; 10 • a 7 . 13. t 2v+1 • p-» 24. (ic 7 ) 9 . 3. yt-tf. 14. a 10 -^- 6 . 4. m 2 • ra°. 15. a 11 -r- a 7 . 5. & c • b\ 16. m 5 -r- m 2 . 6. a 2n -a n >a. 17. c* -^- c 2 . 8. ct*a. 19. 6 c45 -f-6 4 . ' Uv 9. c n_2 c 2 . 20. 2/ 2x+1 -r- 2/* _1 . 28. (x n y p y. 10. d** 1 .^- 1 . 21. a 2n " 3 -^a n+1 . 29. (t"" 1 • s n " 2 ) w . 11. aj^+V" 1 . 22. (a 2 ) 5 . ■ 30. (a 2 6 3 ) 4 . 438. According to the definition of an exponent (§64), such expressions as a~*, a , a~ 5 have no meaning, since it is impossible to use a two thirds times, or zero times, or — 5 times, as a factor. It is convenient, however, to use fractions, zero, and negative numbers as exponents and to define them in such a way that the laws for positive integral exponents shall hold for these exponents. 439. Fractional Exponents. Assuming the law of multiplica- tion to hold for fractional exponents, we have a* > a? = a * + 2 = a. ,\ a* = Va, since a 2 is one of the two equal factors of a. 25. 26. {-ft- Exponents 317 Similarly, oft • eft • a? = a*" 1 ^** = a. .-. a*=^a. (Why?) l In general, a** = Va. Again, a* > a* - eft = eft**** = a 2 . .-. a *=%l (Why?) Similarly, a* . a* • a* • a I , a* - «*W+I+W = a s. .-.<** =#5 (Why?) In general, ai = a/op. 440. Stated in words we have : The numerator of a fractional exponent indicates a power and the de- nominator indicates a root. 1 q Thus, 2a* = 2#a»; 3o'6 S = S^aV6?; 8$ = y/¥ or (#8)* = 32. EXERCISE 441. Write with radical signs, noting carefully ivhat is the base for each fractional exponent : a*. 3 r«. \Sx) ft. 6. a + 2 - a • . _ . 6 i 10. 5r). 15 . V25-5-1. 5 - («°) n - 16. (a + b° + c°)-\ 6. 1».(-1) 3 . 17# 64 . 2 -6. 7. (x-yy>.5-\ 18# 4-2_j_2" 4 . 8. (25 i + 8*)- 1 - 19 - 64- 1 -2 6 . 9. 3 a" 1 ; S^a. 20. a* - or 1 - or 2 . 0. 9*. 3" 2 ; 9 +3-2. 11. 9-*-3" 2 ; 9-! + 3- 10. 9.3-; 9+3-2. ^ a0+ I a EXERCISE 447. Simplify as much as possible : 1. (25* + 8*)- 2 - 4 ' 9 ' 4 - 5_1 - 2. 8-2-321 5. 5-2+(-i) 3 . 3. 22 + 2-i.4i 6. 25-5-1. 320 Exponents Simplify as much as possible : 7. a 3 -K-a) 2 ; (-ay + a 2 . 11. 3 . 3" 1 + 4 -^ 4rK 8. 100* + 100* + 100-i 12. (2« + 3 ) 3 ; (2 3 + 3 3 ) . 9. 3-2 + 27* 13. -^8. 8^; 2~ 2 + 8~l 10. 3" 2 + 3.9l 14. 16* + 2-2. 8*. 448. Negative Exponents in Fractions. ab' 1 a 7_i a 1 a /tt. i • i . \ 1. = - • b l = - . - = — (Explain each step.) c c c b be ab^ab ; 1 *£ ; (Explain.) 2c 2 c 2 v r ' 3. -=a l >b- = -•& — d = — cd~ x c d~ l a c ac A /a\-2 1 1 b 2 fbV « ,. v 4. f _ ] = = — = — =[-] • (Explain.) \bj fa\ 2 a 2 a 2 [a K ^ ' a\ 2 a2 a 2 V 6 2 449. These examples illustrate the following principles : . 1. Any factor of the numerator of a fraction may be made a factor of the denominator, or any factor of the denominator may be made a factor of the numerator, if the sign of its exponent is changed. 2. Any fraction affected by a negative exponent is equal to the re- ciprocal of that fraction with the sign of its exponent changed. The student should carefully note that factors, not terms, can be changed from the numerator to the denominator or from the denominator to the numerator by changing the sign of the exponent. EXERCISE 450. In the following examples make the exponents positive and simplify the expressions as much as possible : 1. 2 a;" 3 . 4. a- l b~ x . 2 - 2a ~' x ' 5. a-i + b-\ 3 -fL+°L. x~ A x-* 6. (a + 6)" 1 . Exponents 321 7 a- l + b'\ 11. («)-'• x- 1 + y 1 8. (-3) 3 . 3-3. 9. a - + h -. b 2 a~ 2 12. 13. (-i)- 3 - 10 - aJ £ ar A b 14. 2a- 1 6" 1 a~2 - 6-2 Write examples 15 to 26 in in tegral j brm, using negative expo- nents when Jiecessary 15. 1. a 19. X 23. i-i- X- y 2 ,. 1 # 1 16. — • 20. -=— • 24. a 2 — z 3 x -f ?/ „. (!Y. 21 . *L. 25 . ttl. \a 2 J xy 2 x — y 18. «1. 22 . 1 . 26. £i±£. a _m 2 _1 #*/ Find the numerical values in examples 27 to 34 : 27. !._*_. 29. A. 2- 32 ' 48 ' 10 - 5 - 50 C_4)-3 3-2 o. 1 * 1 33. 5 + 17 • 10-3. 28 X 2-4 3 °- 100,5 ' Mm 4- 2 * 31. 1000-5-2. 34. 2135- 10- 7 . 35. Write with positive exponents and simplify the result : a" 3 + b~ 3 a-2-6-2' Solution. _1_ J_ ft 8 -f a 3 a 3 6 3 a 3 6 3 6 s + a 3 a 2 6 2 1 1 ~ &2 _ CT 2 ~ a 8 & 3 6 2 -a 2 a 2 6 2 a 2 6 2 _ b 2 -ba + a 2 ab(b — a) 322 Exponents Write examples 36 to 43 with positive exponents, and simplify the results : 36. "-""' a + a -1 a~ 3 b~ m • (ir + & 38. 39. x-*y~ n 3 a- 1 3+ a" 1 ' 40. 2 a;" 1 - 6 2a- 6" 1 41. a" 1 + 6" 1 a~ 2 + 6- 2 42. ^•(* +«•>-• 43. 2(a + 6)- 1 + 2(a-6)- 1 . 451. We shall assume that all the laws of exponents that have been established for positive integral exponents hold for the other exponents that have been defined. For convenience, we repeat here the four definitions and the five laws of ex- ponents in algebraic symbols. Definitions of Exponents 1. an = a • a - a ••• to n factors when n is a positive integer. p 2. aq = yaP. Fractional Exponent. 3. a = 1. Zero Exponent. 4. a-w = Negative Exponent. Laws of Exponents 1. a™ >a n - a rn+n . Multiplication Law. 2. a 1 * 1 -=- a n = a™-* 1 . Division Law. 3. (a rn ) n s= a nin . Power of a Power. 4. (ab) n = a n b n . Power of a Product. Power of a Quotient. \b) bn The definitions of exponents and the laws of exponents should be thoroughly committed to memory. Exponents 323 ORAL EXERCISE 452. Apply law 1 to each of the following : 1. a^a~\ 11. («+&)(« + b)~ 2. a 5 • a" 5 . 12. (a + b)\a + b)~\ 3. y m • y°. 13. (a -f b)~\a + 6)°. 4. x m -x~ n . 14. (a; + ?/) n+1 (a; -f y)2-«. 5. r m • r. 15. 5 aic -6 • 5~ 2 abx 7 . 6. r m+3 • r m-3 . 16. (a — »)" 3 (a; — a)" 2 . 7. 6 n+2 • 6 2 ~ n . 17 - ( a ~ x )~ 4 (v — a) 6 . 8. a"" 1 • a • a\ 18- ^/a^-y/a- aK 9. a n_1 • a n_1 . 19. Va~ x * (T*, 10. a" 3 • 6~ 2 • 6 5 • a 3 . 20. d* . d^Vd. Appty l aw % t0 examples 21 to 41 : 21. a 8 + a\ 32. 3a°6" -s-(— &»- 2 ). 22. a 8 ^- a" 3 . 33. (a; - y)" 1 ■+ (x - y)~\ 23. a~ h +a h . 34. ( X -yf^{y-x). 24. a 5 -f- a -5 . 25. b n ~ z + b\ 26. or 2 & 3 -5- cr 3 6 2 . 27. a~ 3 '--(-a)- 5 . 28. or 3 -=-(- a 2 ). 29. a" 3 ^-(- a 2 ). 30. 2 n ~ 3 -j- 2"" 4 . 31. 2ar n ^a n - 2 . Apply law 3 to examples 42 £o 47 : 42. (a" 2 )" 3 ; (-a 3 ) 2 . 45. (a 4 )<>; (a*>)-i. 43. (-a 2 ) 3 ; (a 2 )" 3 . 46. (- &-*)-«; (- 6" 2 )- 44. (a~ 3 )- 4 ; (a 3 )" 4 . 47. (*-"-•)*-*. 35. 2 -1 a? s-a s . 36. a^ -j- a$. 37. a-f-2a~?< 38. (2x)o-r-(2a) _ i 39. Scrfo^-a^c . 40. ( a + 6)^_( a + 6)-i. 41. 25a"t-5-iai 324 Exponents Apply law 3 to examples 48 to 61 : 48. (a x+v ) x ~ v . 55. (m*- v ) x+v . 49. [(2a)" 3 ]-i. 56. Vxyz+(xyz)t. 50. [(x + y)-^. 57 - C(-«) 4 ] 3 - 51. (^-i)«+i. 58 * [(«"")"'?• 52. (2e)f; [( -2)-]-t. 59. [(~f)J 53. (a 2 )"; (a») 3 . 60 . ( a -i). ' 54. {x n ~ i y. 61. (2~ 2 )-3. Apply laws 4 and 5 in ^e following : 62- (2a<6°)'. 72 ^ 3 M-. 6 63. (-a&) 2 ; (a.&) n ~ 2 . VV a 64. (-a&) 3 ; (-a&) 4 ; (-a&)«. 73. [(x - y)(y - x yj. 65. (#•#)". 74. [(a; - y)\y - a?)" 2 ] 3 . 66. (3a&«- 2 ) 3 . 75. (i)- 3 (|). 67 /flW\» 76. [2(a>-2,)-i]- 2 . ' V2c/. 77 / 81 U /*.y V« 2 + 2a6 + 6V* ^2 68. 69. (^X\ 78. /« 2 + L 2a6 + 6 2 4 79. (6° • 25* • a 4 )*. 70. [(x-y)(x + y)-*]-+. 80. (8"i • a" 3 6 6 )-*. 71. f?r \xj ' * w ' ' V * / EXERCISE 453. 1. State the four definitions of exponents in words. 2. State the five laws of exponents in words. In the following examples use definitions and laws to simplify : 3. 9° • 9-i • 9* • 9"*. 5. (-64)"*. 4. (9" 2 )-i. 6. (64)"*. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. .008-2 5 • 2~ 2 . 7.7-1; 7 + 7-i •I" 2 ; (--I) 2 - (a°+&°) 2 ; (aPbo)\ (a 2 + 6 2 )o. 4, (a -by. tt)-<-16<> (-l)o-f(-l) 2 . (-l)o+(-l) 2 +(-l) 3 2 9( a +& 0_f_ c 0)-2- _3 a * -=- 2 a~ l . Exponents 26. (a-i + fc-^-j-afc. a; ^ajj 325 ic 5 -r- a? -5 . 23. al 24. 25. 27. - 28. a-!-j-a°. 29. & 8 h-& 8 "*. 30. g "^~ 1 . a m b n 31. (o*6*)i 32. 9 2 • 3-*. 33. (-z) 3 -=-(-a;) 2 . 34. [(i)-3]->. 35. 1-4- 2 . 36. l-r-(-3)" 3 . 37. .2-1; .5" 2 ; 1.5- 2 . 38. 5-2-i; 8-10-5. 39. (a**)-*. 1 1 40. 41. (_3)- 2 ' (-3) -3 x n ~* + x\ 42. (a" 2 )- 3 ; (a" 3 ) 2 ; (er 3 )- 43. (-a 2 )" 5 ; (-a 5 )" 2 ; (-a" 5 )" 2 . 44. (-a 3 )" 4 ; (-«~ 3 ) 4 ; (-a -4 )- 3 . 45. (-a 3 )" 2 "; (-a 2 «)- 3 ; (_ <*-*»)-• 46. (-a 2n "i) 2 . -2 48. 49. 47. erW\- x~y) a 4 6~ 6 a%- 8 \a%- 2 50. (a* + &*) (a* - &i). 51. (a* + ^) 2 . 326 Exponents In the following examples use definitions and laws to simplify : 52. (a* + &"*) (at - aV* + &""*> 53. Va — 2aM + 6. 54. (a* 4- a"^) 2 - (a^ - a - *) 2 . 55. (a* - 2) (a* - 3). 56. (a — x)(a 2 — x 2 )~ l . 58. Vcriar 1 * =(afar*)*, etc 68. -\/xfyx. 69. 2Vi*Va*. 70. 2 • 4* • 8" 1 . 71. 3-^3. 72. V3-a/3. 73. Simplify lt 2.3. 4. 5.6. 7 ( a va?) 6 (- a?Vo) 7 . 74. Simplify ( V2^) 6 + 5 • ( V2^/) 5 ( - V2^) + 10 ( V2^) 4 (- V2xy) 2 . 76. Show that 2 n+l - 2 n = 2\ (Factor.) 77. Show that 2" +2 n+2 = 5-2". 78. Show that 5" + 5" +1 = 6-5-. 79. Is 3-3 2 = 9 2 ? Is 3 .3 n = 9 B ? 80. Show that 2" • 4* = 2 3 \ 81. Show that 10 + 2 ~* = 3. 1° - 2" 1 82. (a + or 1 + or 2 ) a 2 . 83. Compare the value 2 3 ' with (2 3 ) 4 without actually per- forming the indicated operations. 57. (a 2 &""M)- u . six? *--l). 16. (1 + 8 a" 1 + 15 a" 2 ) -*- (1 + 3 a" 1 ). 17. (a* + 12 at - 48 + 52 a* - 17 a) ■*■ (a* - 2 + a*). 18. (m* - 36 - 21 m"^ - 3 m*-71 t»-*)+(l-3 m^-8 m"*). 19. (a; + a^ + l)-5-(a^ -a;* + l). 20. (a" 1 + 27)--(a~t-3a"3+9). 21. (a? - * + y"t) -s- (art + y~*). 22. Find the square root of a} + 12 a* + 36. 23. Var 2 — 2 x~ l y + ?/ 2 . 24. (a;- 1 - 22 x^y~* + 121 fty. 25. V2 + a*ar 2 + a~*x*. 26. (a?" 4 - 6 a;- 3 + 13 x~ 2 - 12 or 1 + 4)*. 27. (2a + 2ar 1 + 3 + a* + ar*)K 28. (cr* + 6 a- 1 +9)~*. Exponents 329 Expand by applying type forms, examples 29 to 36 : 29. ( *-&*)(a*+6*). 33. (a* + 2 a*) 2 . 30. (x$-y~ty. 34. (a;* + »"*)(#*— ar*). 31. (a^-affy. 35. (a* - 2) (a* + 7). 32. (a*-&ty. 36. (a" 1 + 3)(a- 1 -5). 37. Factor a - 3 a* + 2 ; 2ar 2 + 33T 1 - 2. 38. Factor a — b into two factors, one of which is a 2 — 6*. Hint, a - b = (a*) 2 - (6^) 2 . 39. Factor a — 6 into two factors, one of which is a* — b 3 . 40. Factor (a) x - 20 &* + 100. (6) a — 4 Va — 5. (c) a 2 c^ + ac + aM + a^c 2 . (<£)# — aM — 62x2 + a^fti. (e) 2a + 5a^3. 41. a* • of? • v^v^ • a" 1 . 45. (4) 2p • (-f) 2p • (|) 2p . 42. f^Y i (?]~ 3 fS\~ A . Hint. 4 2 * = 2*p. (Why?) 43. (a 2 &')-^(a-3&-*)3. ^ ^—j +(_J . 44. (a- 2 -6- 2 )-^---J. 47 2«(2»- 1 )»-s-(2"+ 1 .2»- 1 ). 48. Show that (2»+ 4 - 2.2" +1 ) . 2 -*-« = 3. 49. (xyz) x+v+z -^r(x y+x y x+z ^ +y )' 50. (^ + e -) 2 -2. Y-3^2 .KinA 55 - W*^"*)" 1 - L wJ ■ 56 - ( + y° + g P)-» > W^VV 7 (27a^r 7 a "V 4(o° + g 67. 2 64. (9arV/9, y/— 27 are rational numbers. All other real numbers are irrational numbers. Thus, V5, V 9 + V4 are irrational. The irrational numbers, so far as we shall be concerned with them in elementary algebra, are indicated roots that can be obtained only approximately. 458. Radical. The indicated root of a number is a radical. Thus, V2, V9, y/a + 6, V^i are radicals. 331 332 Radicals A radical may be a rational number, an irrational number, or an imaginary number. Thus, a/4 is rational, V5 is irrational, and V^~4 is imaginary. Radical Expression. An expression that contains a radical is a radical expression. Thus, Va, 3 + V2, (a -f Vb) 2 are radical expressions. 459. Order of Radicals. Indicated square roots are radicals of the second order; indicated cube roots are radicals of the third order, etc. Thus, V3 is of the second order, Va is of the third order, y/a 2 + b 3 is of the fourth order. 460. Index of a Radical. The index of a radical is the num- ber placed to the left and above the radical sign to indicate the order of the radical. The index of a square root is omitted. Radicand. The expression under the radical sign is the radicand. Thus, in 2v / 3a 3 6, 3 cfib is the radicand, and 5 is the index of the radical. 461. Surd. An irrational number which is the indicated root of a rational number is a surd. Thus, V2, Va are surds, but V9 and V 2 + V3 are not surds. Quadratic Surd. A surd of the second order is a quadratic surd. 462. Principal Root. It has been seen that VI = ± 2. From this we should infer that x -f- V4 = x ± 2. However, in dealing with radicals and expressions containing radicals it is customary to use only the positive root. Thus, x + VI = x + 2, and x - V4 = x - 2. The positive square root of a number is its principal square root. Reduction of Radicals 333 463. Principle 1. The square root of a product equals the product of the square roots of its factors. In symbols, Vab = VaVb. This principle follows immediately from the fourth law of exponents. Thus, Vab = (ab)$ = ah^ = Va • Vb. Principle 2. The square root of the quotient of two numbers equals the quotient of their square roots. In symbols, a/- =— =• **> Vb This follows from the fifth law of exponents. 'a\i _a% _ Va b \ Vb Thus 'VR? These two principles may be stated for any root. V * Vb REDUCTION OF RADICALS 464. Case I. To remove a factor from under the radical sign. Vo2& = Va 2 Vb = a Vb. (§ 463) ytfb = VtfVb=aVb. (Why?) 465. If any factor of the radicand is a perfect power of the same de- gree as the radical index, it may be removed from under the radical sign by extracting the required root of the factor and multiplying the result by the coefficient of the radical. Examples 1. 6^54 =6^27^2 = 18^2. 2. 2 V2W = 2 V(a 4 6 2 )2 ab = 2 a 2 bV2ab. 3. V(a 2 - b 2 )(a + b) = V(a + b)\a - 6) = (a + 6) Va - b. 334 Radicals 466 sign : . Whenever possible, VIS. 3. V4 ab\ 4. EXERCISE remove factors from under the radical 1. 2. V9a 4 & 2 c Va 3 ^ 3 . 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 5. Va 3 + 6 3 . 6. fV27 6 5 . 7. Va 2 + a 2 b\ Va 6 + n. 8. V5x 3 -20a 2 + 20a;. -\/6a 2 & 2 . aVaM>. aVa^~\ V2(a 3 -3a 2 6+3a& 2 -& 3 ). 9. v/(s - 2/) TO . 10. V(a - 6) 2 (a 2 - 6 2 ). 11. 12. V(o 3 - 6 3 )(a - 6) 2 . Vm 2 * +1 . 13. 2V(a 2 -& 2 )(a-6). Va 2 - b\ &Va 6 6 8+ °. 14. V(# 2 — 2/ 2 ) 3 . 15. Vm 2 — 2 mw + w 2 ' Va- 2 " +1 . 16. VC^-S^ + ^aj-lX 2 . 17. V(a> - 2/) 2 " 1 . Va 4 & 4+ *. 18. Vm 3 — n 3 )(m — n). Va 4 + b\ When the radicand is negative and the root index is an odd number, the negative sign should always be removed from under the radical sign. Thus, ^"16=^- 8 • 2=-2v/2; V^a = ^=. •y/9 a 2 6 6 . ^343. 7. A / c -2n . C 14n ( 8. V# n • x n+1 . 473. Simplest Form of a Radical. A radical is in its simplest form if : 1. The radicand is integral. 2. The radicand is as small as possible ; that is, contains no factor that can be removed from under the radical sign. 3. The radical index is as small as possible. 474. Corresponding to the three parts of § 473 radicals may be simplified as follows : 1. If the radicand is a fraction, simplify by Case II. 2. If the radicand contains a factor that may be removed, simplify by Case I. 3. If the radical index can be reduced, simplify by Case III. 475. Real numbers occur in five different radical forms. 1. Radicals that are rational numbers, Vl6 = 4. 2. Radicals in their simplest form, \/7, y/£. 3. Radicals with fractional radicands, \/jt = iT v/ ^* 4. Radicals of which a factor of the radicand can be removed from under the radical sign, \/28 = 2V7. 5. Radicals of which the radical index may be reduced, y/a 2 = Va. 338 Radicals EXERCISE 476. In the following list of radical expressions, each of the five forms given in § 475 is included. Examine each, telling to which class it belongs, and simplify all that are not simple. 16. ^^648. 29. ^ieR. 17. -y/ax 2 — bx 2 . /o^iY 30. \- 18. ab< <* + * ^ 2n 1. Va 3 . 2. Va 4 . 3. -i)(a> - x- 1 ). (d) (a* + a?)(a? - a*). 2. 4"^ +(£r 2 + 21° + 9*, 3. 4 a +(4 a) + 4^0°. Review of Exponents and Radicals 359 4. Find the value of V# 2 — y 2 when x = a? + a~* and y = a i _ a ~\ 5. Find the value of (x 2 — y 2 ) when x = 3 and ?/ = 1. 6. 5V24-V54 + 3V96. 7. (a - 6) -s- (a 1 + a*&* + &^). 8. Find the square root of : x -i + y i + 2 a;~y — 2x~hj — x~ x y%. (Yale.) 9. Simplify, writing the result with rational denominator : =— (Mass. Institute of Technology.) #+Va 2 + a 2 10. (a) [(tf + &fy*+(a* -**)*]■ J (6) [(a* + &i)2+(a*-&*) 2 ] 2 . 11# fl? "^+y~^ . (a>Vty- (Princeton.) Vai + Vy 12. 2^ • 23 -f- 54~^. (Yale.) 13. -s 2 (9-s 2 )^+V9^ 2 + 3 . (Yale.) 3 L fx^ Vi- 1 o 14. Simplify x/^Y'.Y—') • (£\ . (Princeton.) 15 - 3^ + V40+J|--4=- (Princeton.) 16. Find to 3 decimal places — — . 4 + V3 17. V17 + 12 V2. 18. .M±5_ A §El + _ 2 jLv?=7. (Yale.) V*-0 Mz + j/ a?-i/ 2 * v y 360 Radicals 1Q / /64a 2 6 6 . 3/ a^y n . 19 - \ ^ — r~» + \/ * • ^ lve answer with positive ex- \ * 81 m 6 n 2 \ m~V ponents. 20. Find a mean proportional between V6 — V2 and V6+V2. 21. Find a fourth proportional to 1, 2 + V7, 2 — Vf. 22. ^ ^i — 4 a£ -f- 2 cc^ + 4 a; — 4 x* + x%. 23. (a6" 2 c 2 )*(a 3 6 2 c- 3 )^ + ^j-- 24. Simplify (a) Vl4 + 6^5. (b) |Vl - x + a(l - a;)"*. ^/2 2a/5 (c) . Give answer in simplest radical form. V3 + V5 (d) SVl + SV^-Wi. (Sheffield.) 25. (f)-i -2- 3 +(V 6 ) i + 128-^-(7V5)°. 27. Show that 4"" 2 . 8 2 "" . 2» = 4. *. Simplify (^(%^)-*. (Yale.) 29. Simplify — — — , and compute the value correct to V2-VI2 two decimal places. 30. Simplify (a) £^£. (&) 3V| + 2V3:-4V^. (c) ^2^W8^. (d) 2,^(1 + 4^-4^(1+4,)^ (Sheffield) (1 + 4*)* Review of Exponents and Radicals 361 31. Simplify : («) (A)* x (iF)i (c) J^+Ji*-J™. * X "ft * o (b) 2(1 - 2 X)"* + 3i/l-2x. (d) V\ -f- VJ. (Sheffield.) 32. Simplify ^=H 1_ a- 2 1-V2* 1+V2a l-2x 33. Simplify: (a) V2-VI. 2V3-2a;-a;(3-2a;)-^ , (6) 2^-^+V3 XX. RADICAL EQUATIONS 508. A radical equation, or an irrational equation, is an equa- tion in which the unknown number is involved in a radicand. Thus, 3Vx = 5, Vic + 3 = 5 are radical equations, but xV3 = 5 is not a radical equation. (Why ?) Is a?* = 10 a radical equation ? To solve a radical equation it is generally necessary to ra- tionalize the equation. The following examples will illus- trate the method : 1. V3»-5 = 0. Solution. V3x = 5. (Why.) 3 x = 25. (Squaring both members.) x = 8f Check. VS • 8J - 5 = V25 -5 = 0. 2. 14+V2^=16. Solution. V2x = 2. 2x = 4. x = 2. Check mentally. V. 3. £ + 40 — 3 = 7 — Vx. Solution. Vx + 40 = 10 - Vx. (Why?) ar + 40 = 100-20Vx + x. 20Vz = 60. Vx = 3. x = 9. Check mentally. Radical Equations 363 4. (a) aj-4-V»-hl6 = 0. (b) x — 4 + -y/x +- 16 = 0. Solution. x — 4 = Vac + 16. 3 — 4 = — Vx + 16. 3 2 -83 + 16 = x + 16. x 2 -8x + 16 = 3 + 16. x 2 - 9 x = 0. 3 2 - 9 x = 0. a(a; - 9) =0. 3(3 - 9) =0. 3 = or 9. 3 = or 9. Check. When 3 = 0. Check. When 3 = 0. 0_4-V0 + 16=-4-4 = -8. 0-4 + V0 + 16=-4 + 4 = 0. . •. is not a root. .-. is a root. Check. When 3=9. Check. When 3 = 9. 9-4-V9+16 = 9-4-5 = 0. 9-4 + V9 + 16 = 5 + 5 = 10. . \ 9 is a root. . •. 9 is not a root. If the principal square root of the radical is taken, 3 = 9 satisfies (a) but not (b). Also, 3 = satisfies (6) but not (a). 509. Extraneous Root. From example 4, (a) and (b) it is seen that in solving a radical equation, roots are sometimes found that do not satisfy the equation. Such roots are extrane- ous roots. The student will notice that in step 2, under both (a) and (6) the equations are the same since the squares of V# +-16 and — Vx +-16 are the same, x +- 16. It is here that the extrane- ous root is introduced. 510. To solve a radical equation : 1. Arrange the terms of the equation so that a radical is alone in one member. 2. Raise both members of the equation to a power corresponding to the order of the radical. 3. Solve the resulting linear or quadratic equation by the usual methods. When more than one radical occurs in the equation it may be necessary to repeat steps 1 and 2 of the rule one or more times. 364 Radical Equations Example Solve: Vx + 60 = 2 V# + 5 +V-l)(2a> + 3)=2a-l. 15. V32 + a; = 4+V». 19. 2V# — V2a; = 2. 16. 5V5-7 = 3V«-1. 20. 2 V^ + l = 2V^ + 3 17. Vx + ^/2x = l. SVx - 2 3 Va; - 5 21 V^4-29 = Va + 37 18. V^ + V3« = 2. VaJ + 5 V» + 7 22. V^ + 4 = V^ + 8 _ V# + 2 Va + 5 Radical Equations 365 23. (18 - VlO - V3(^ - 3))* = 2. 24. (2V« + 3)(2V^-3)=2. 25. 26. VI + 16 x 4- 2 V14 + 4 a; = 11. 5x-9 V5^-3 . - = f- 5. V5 a; - 3 2 27. V7a>4-2- 5x + 6 • V7 * + 2 o 28 V14 a? + Vll a?- Vll-a; 29. ^ + 12 a: 2 _ x + 4. 30. Va; 2 + 2 a; - 14 = Va; 2 - 5 - 1. 31. V5a;-4=V2a: + l + l. 32. V(a; + 2)(aj-5)=2. 33. 2^^0=V37 34. Va; — 6+Va;- l=Va;— 9 4-Va7+6. Solve, and determine whether any of the roots of equations 35 to 43 are extraneous : 35. V10 + a+V10-a;=6. 36. VlO + x - VlO - x = 6. 37. Vl2 a; + 109 = 2 a; + 3. 38. V3a>-5+VaT+"6 = 0. 39. Va; -f 5 = x — 1. 40. 3aj-4V»-7=2(a;-f 2). 41. Vl + a; + a; 2 +Vl — x + x 2 = V6. 42 - \/3 + \/^^ + Va = 2. 43. VaT+~3+V2a;-3 = 6. 366 Radical Equations Solve : V3x 2 + 4+V2x 2 + l 7 Hint. Use composition and division. 45. a; + 4a;V4a; + 5=(4a;+ l)V4a + 5 — 2. 46. V3a? 2 -1+V3^^ = a V3a 2 -1-V3^¥ 2 & 47. Find a number which added to its square root gives 56. 48. g + g = «. a + V2-a 2 a;-V2-x 2 49. 5 n-1 aal V5n-1 V5^ + l " 2 50. 51. VJ — z Vl + Z 2-VI+2 2+Vl^ 1 1 Jlx 52. V2 8 - 3 = 8 - 3. 53. V2«-5 + 6 = ic4-2. 54. V2 a; + 1 + 2 VaJ 21 V2oj + l t _ Vm . 3 — Vm 5 55. — -\ — — = — 3— Vm Vm • 56. -v3a 2 -6a + f ==V5a-2a; 2 -- 2 ir 2 -. 57. V(a-l)(3a>-6)=a-2. 58. V2p Vp 2 - 9 Vp + 11 59. VIOT^ V*-2 +1 VlO - a? V10 - a; XXL IMAGINARY NUMBERS 512. Consider the equation x 2 + 4 = 0, or x 2 = — 4. This equation asks the question : What is the number whose square is — 4 ? There is no rational or irrational number that will answer this question, for all real numbers are positive or negative, and their squares are positive num- bers. Hence the square root of a negative number has no meaning. A similar difficulty arises if we attempt to solve the quadratic x 2 + 2 x 4- 2 = 0. In order, then, to make the solution of the quadratic equation general, that is, always possible, we require a different number from any we have previously studied. If we solve the equation x 2 -f 4 = 0, or x 2 = — 4, by the method of § 410 we get x = ± V— 4. In order that the result should represent the solution of the equation, ± V — 4 must be such a number that ( ± V — 4) 2 = — 4. We define ± V— a as such a number that (±V^) 2 =-a. 513. Imaginary Number. An even root of a negative num- ber is an imaginary number. In the present chapter we shall deal only with the square root of the negative number. 514. The student must not fall into the error of thinking that V— aV— a=V(— a)(— a)= a, as would be the case in the multiplication of ordinary radicals. We are now dealing with a new kind of number which does not always obey the laws of radicals and which is defined as such a number that ( ± V — tt) 2 = — a. This number is wholly different from an 367 368 Imaginary Numbers ordinary square root. Thus, V9 = ± 3, and V7 = ± 2.645 — correct to three decimal places, but it is not possible to find exactly or approximately in real numbers the value of V — 4. We shall always deal with the imaginary number as the product of two factors, one real and the other the imag- inary unit, V— 1. For example, V — 4 = 2 V — 1, V— a = VaV— 1. To further facilitate the work, we introduce the symbol i for the imaginary unit and write V — 4 = 2 i and V — a = i Va. 515. Powers of the Imaginary Unit. By definition, .( V^) 2 = — 1 or i 2 = — 1. From this we get the following : i=i, fi = i } & = ? i» = ? i 2 = -l, i 6 =-l, i 10 = ? ; 14 = ? * = - j, ^ 7 = - i, i n = ? i 15 = ? *«1, t*=sl, l 12 =? l 16 = ? Any power of the imaginary unit may therefore be reduced to one of the four numbers, i, — 1, — i, 1. What is the value of i 4 + * 6 ? i 7 -f i" ? 516. Complex Number. The sum of a real number and an imaginary number is a complex number. Thus, 2 + i, and 3 + 2 i are complex numbers. a -f fo* is the general form of the complex number, hi is the general form of the pure imaginary. In either of these a and b may have any real values. 517. Operations with Imaginary Numbers. All operations with imaginary numbers can be performed by first writing the numbers in the general form and then proceeding as with real numbers, using the symbol i as we should use any other letter. In case higher powers of / occur at any time in the course of the work, they should be reduced as indicated in § 515. Po not leave 1 in any denominator; that is, rationalize the denominator. Imaginary Numbers 369 Examples 518. 1. Reduction to General Form. (a) V^6 = V- 1 • b = V^IVb = i VS. (6) 3-V :r 2 = 3-iV2. (c) 2+V^4 = 2 + 2l 2. Addition and Subtraction of Imaginary Numbers. (a) V-9+V^16 + V-25 = 3* + 4t + 5t = 12i. (6) V^2 + \/^18 + V^200 - -v^64 = i V2 + 3 i V2 + 10 iV2 + 3^2 = 3^2 + 14 i V2. ( C ) V^44+V^99-V^176+V^275 = 2i vii + 3i vii - 4^ vn + 5i vn = 6i vn. 3. Multiplication of Imaginary Numbers, (a) 1* = *; i 10 = -l. (6) (1 + t)(l - = 1 - * = 1 - (- 1) = 2. (Explain.) (c) V r=: 12V^3 = iVi2.iV3 = i 2 V36 = 6t 2 = -6. (Explain.) (d) V^3 . V27 = iV3 • V27 = tVST = 9i. 4. Division of Imaginary Numbers. (a) V' = n*^V^2 = 2*V2--iV2 = 2. V75 5V3 5 5i 5i W 2V^3 2tV5 2» 2* 2 (c) v ^ [ = ^ V2(2-tV2) 2 + V^2 2 + * V2 (2 + t V2) (2 - t V2) == 2V^-2i = 2V2 -2f = V2- 4-2*2 6 3 370 Imaginary Numbers EXERCISE 519. Simplify the following expressions, according to the gen- eral rule given in § 517, and the illustrative examples of § 518. 1. Write in general form : (a) 3V^3. (d) V3+V^3. (b) 7-V-5. (e) 5+V-a 2 . (c) 3+V-4. (/) V-16. 2. V-9 + V- 16-V-36+V-81. 3. V-16+V-4-V-9+V-144. 4. V := ^ + V^ r 9 + 3?;4-Vl6. 5. V-2+V-72-V-32. 6. V-63+V-700-^-64. 7 . V- 45 -3V20 + 4V- 80 +->!/- 125. 8 . V^84 + V^"f o 13. 14. 15. 9. 3&V-a 3 c + -V- a b c*. c 10. i -f- V + $ + i*. 11. (V-2)»; (V-3) 4 ; (-V-4).2 12. (V-l) 100 ; (V-l) 101 ; (V-l) 102 . V^-V-32. 17. (-V-a)V-6. V^3 • V^4. 18. VHf ' V^5 • V25. V^-V-5. 19. V-12-V3. 16. (_V-5)V-125). 20. V-25-V4. 21. (_V-3)(V-12)(V^4). 22 . (_V^|)(-V"^2).(V^3). 23. (V :: ^3+V7 + 3V :r 5)2V^3. 24. (_3V : ^5 + 4V8-3V' =r 7)(-4V^3). 25. (_l-}-V^3) 2 . 26. (-1+V^3) 3 . Imaginary Numbers 371 27. (l+O 3 ; (l + O 4 . 28. (4 + 3 V :r 2)(4 - 3V^2). 29. (Vl2 + 2 V^8)( V12 - 2V^8). 30. (V^+V^)-KV^2-V=3). 31. 3V^8+(-2V^2); 6 V^^ -r- 4 V^~3. 32. 5V28^-3V^7; — ^=- -V-16 33. 42-(3-2V^3). 34. (l + *)-(l-i). 35. (1+V=2+V3) + (1 + V=3). 36. 42-r-(2 2'V3 + 3tV6). 37. Find the value of x 2 + x + 1 (a) when x (b) when cc = — — -l+V-3 -3 38. Find the value of x + - when x x l + i. 39. i + i 2 + i 3 --- i 8 . 40. t * i* • i 3 ••• i 8 . 41. (x*-cHy. 42. (ra 3 + atf) 2 . 47. (V5 + 48. (Vn 2 -V-w 2 ) 2 . 49. (a* — bi) z . 50. (a + fo') 3 -(a 2 43. (4-V^i) 2 . 44. (T+V^) 2 . 45. (_4 + 2fo') 2 . 46. (3 + 5 0(4-7i). "6)(V6-V^8). 1-2 i V3 6i) 5 51 52. 3+V^2 2/ 3-f-2iV :r 6 53. 54. 55. 56. 1 + 2 tV3 i-£ l i-i l + l + i l + i l + i 1 — * 372 Imaginary Numbers Simplify the following expressions : 57 3 + 2i + S-2i 7-24* 3-2i 3 + 2»" ' 4-3i* x — iVy _ E 5 — 29iV5 bo. — . 7-3 i V5 1+33 i V3 4 + 3 1 V 3 ' V3 + 1 V2 V3 - ?V2* 68. /H , 1 . N + 66. 67. 4 + 3* ' 56 + 33i 12-5*' 1- 20 iV5 69. 63. ^. 70. (l + f)2 (i_ f -)i 1 1 (i + o 4 (i - ¥ ■y/x — y + Vy — a; 7—2 t V5 Vx — y — Vy — a; 71. Va? + V-y Vy+V^# Va? — V — 2/ V.y ~" V — » #oZve £/*e following equations, writing imaginary answers in the general form : 72. a 2 + 2 # + 2 = 0. Solution. x 2 + 2x=—2. Z 2 +2Z+1:=-1 a; + 1 = ± i. x = — 1 ± i. Check. (- 1 ± £)2 + 2(- 1 ± *) + 2 = + 2 i - 2 ± 2 i + 2 = 0. 73. # 2 +4oj + 6 = 0. 78. x 2 + x + 1 = 0. 74. # 2 -4# + 8 = 0. 79. 3# 2 + 4 = 2#. 75. # 2 -2aa? + 4a 2 = 0. 80. # 2 + 5 = 4a\ 76. a; 2 -4 # + 7 = 0. 81. # 2 + 2# + 4 = 0. 77. 2# 2 + 5a? + 4 = 0. 82. 3a; 2 - 10a; + 10 = 0. XXIL QUADRATIC EQUATIONS {Continued from Chapter XVI) 520. Equations of the forms x 2 = k, and ax 2 + bx = 0. 1. What is a quadratic equation ? (§ 405.) 2. What is an incomplete quadratic ? (§ 407.) 3. What are the two forms of incomplete quadratics ? 521. In the solution of the following examples, irrational answers may be left in the simplest radical form, and imagi- nary answers in the general form. EXERCISE 1. Give the rule for solving the quadratic in which the first degree term is missing. (§ 410.) Solve the following : 2. #2 = 10.24. 5. 2a) = 0. 15. 0+V6)(a;-V6) = -2. 373 16. 5 4 7 2 # 2 3~4a 2 ' 17. 1 4- a # + 25_ 1 — # a; — 25 18. a+x x+ b * a—x x—b 19. ax 2 = a 2 (a +4 b) + 4 a& 2 . 20. 4a; 2 + a;V— 1 = 0. 374 Quadratic Equations Solve the following : 21. *=-!. 24. aa 2 -frP + c = £ t a; mx 2 — nx+p p 22. (x + 4)2 = 23. a;. ^ ^+2__3x-J = () 23. a 2 (b 2 -x) = b 2 (a-x) 2 . 3 a; + 4 x - 2 26. (2 x + 7)(5 a- 9) + (2 »- 7)(5 a; + 9) = 1874. 27. (1 + x)(2 + a>)(3 +*) + (!- a>)(2 - x)(S -x) = 120. — x 1 — bx 28. 1 — ax b — x 29. *= + 1 _*S. 30. V# + 4 — V5 a; — 24 = Va; + 4 31. 2V5 + 2x-Vl3-6a; = y37-6a;. 32. If a quadratic equation lacks the absolute term, one root is zero. Why ? 33. The roots of a quadratic in the form x 2 = k are equal in absolute value but of opposite sign. Why ? COMPLETE QUADRATICS 522. Solution by Completing the Square. What is a complete quadratic ? (§ 408.) We may solve complete quadratics by three different methods ; namely, by completing the square, by formula, and by factoring. 1. Which of these methods have already been treated in Chapter XVI? 2. What is meant by the p-form of the quadratic ? (§ 420.) 3. How is an equation reduced to the p-i orm ? 4. Change ax 2 + bx 4- c = to the ^p-form. 5. What must be added to complete the square ? (a)tf-3s+( ); (b)x 2 + x + ( ) ; (c) ±x* + 5x+ ( ). Complete Quadratics 375 523. To solve a complete quadratic equation : 1. Reduce the equation to the />-form. 2. Complete the square of the first member by adding to both mem- bers the square of one half the coefficient of x. 3. Extract the square root of each member, using both roots in the second member of the equation, and solve the resulting linear equations. Unless otherwise suggested, the irrational answers may be left in simplest radical form. Example x-\-l = 2x — 1 3 x — 1 x + 1 Solution. L. C. D. = x l — 1. x 2 + 2 x + 1 = 2 x 2 - 3 x + 1 - 3 x 2 + 3. (Why ?) 2x 2 + 5x = 3. (Why?) x 2 + $ a = f . (The p-form.) *+f*+H=f|. ( Wh y ? ) ,-. x=±|-f =|or-3. EXERCISE 524. Solve the following by completing the square: 1. 2x 2 — 5x = S. 2. z 2 - fa + 1 = 0. 3. 9«* = a?+f. 4. 5^ 2 =39+2y. + i=l? = 2.5. 6. = x — 1. z + l 7. z-1 1 x — 13 a; 8. (*-5)(*-3)+a*-15=sG. 9-z 9. / f ? + 5V9aj-l) = (3 + 5ar).4 10. (x — p)(4 x — 5 p) = x 2 — p 2 . 11. g*C2»-fi) + _2_,3. 2z-l l-2z 12. a 2 -6a; + 4 = 0. 13. x 2 -2ax + b = 0. 376 Quadratic Equations Solve the following by completing the square : 14. ax 2 -2bx = c. 18. 2 a; 2 + 15.9 = 13.6 x. 15. 2x* -f 5 x + 4 = 0. 19. a 2 + 2a = 0. 16. a 2 + a; + l = 0. 20. 2 a; 2 - .21 a? + .001 = 0. 17. Verify 16. 21 5^-l + 3^-l = 2 + !K _ 1< 9 5 a; 22. 7 -. 8 4a? " 5 = 2. ll_2a; l-3a; 5 + a 8-3»_ 2a; 2o. 3 — x x x — 2 6 x + 4 15-2a; _ 7(a;-l) 5 a?- 3 5 (Multiply through by 5 and transpose.) M » + l , 12 _a>-4 , 17 25 ' "9~" l "aqT4"~4~ + "6"' 2 a; + 2 3 16 a; 29. 30. 5ar(2a?-l) x -4a; 2 4-l a; + 2 7a;-2 6a; 2 4-9a; + 5 ^ 3a>-2 3a> + 2 4-9a; 2 a? + a | a; 4- & . a; 2 — ab _ q a b ab x -\- a — b , a;— a — 2 b , a 4- 5 31. — | r T a o x 3« _ 3 a; -20 = 2 3 a; 2 - 80 2 18 -2a; 2(a; - 1) ' 33. V2a;4- 1 -r 8 «* $• 34. V10 a;- 34 4-2 Va; 4-4= V2(3 x 4- 35). Complete Quadratics 377 35. j° + Vf=3- 36. Va + 2 + V2 a; 2 H- a; = 2. Ill 37. x - 1 a? - 3 35 38. ^ lm 4 10 - a; x-7 39. (a?-2)(3oj + l)=10+(2aj + l)(aj-3). 40 . Sirl+S-i.-i^Sf^l 9 x 5 In examples 41 to 44 ,/md £/ie roots correct to two decimal places : 41. a* + 2a;-2 = 0. 43. a; 2 + Sx- 11 = 0. 42. 2L+! + ?L±i = 0. 44. aj + 10+- = 0. #— 3 a?-f- 2 x 525. Solution by Factoring. The equation (x — a)(x — b) = is satisfied when x = a or x = 6 and not otherwise. For suppose a; = a, We then have (a — a)(a—b) = 0, or • (a - 6) = 0. Therefore the equation is satisfied when x = a. When x = b f we have (& _ a ) (b - 6) = or (b - a) = Therefore the equation is satisfied when x—b. To prove that the equation is not satisfied when x has any- other value : Suppose x= c, where c is different in value from both a and h .-. (c-a)(c-6)=0. We should now have the product of two factors, neither one of which is 0, equal to 0, and this would be absurd. 378 Quadratic Equations 526. Section 525 furnishes us the basis of the factoring method of solving equations. To solve an equation by factoring : 1. Write the equation in order of powers of the unknown number, and with the second member zero. 2. Factor the first member into linear factors if possible. 3. Put each factor equal to zero, and solve the resulting equations. 527. This method of solving equations applies equally well to equations of higher degree than the second if the factoring can be accomplished. Also, it is not necessary to factor into linear factors, for factors of the second degree with respect to the un- known may be put equal to zero and solved by the preceding methods of solving quadratics. Example Solve (3 a + 2)(2 x + 3)=(a> - 3)(2* - 4) Solution. 6 x 2 + 13 x + 6 = 2 x 2 - 10 x + 12. (Why ?) 4z 2 +23x-6:=0. (Why?) (4z-l)(z + 6)=0. (Why?) 4s-i=0andx + 6 = 0. (Why ?) x = | or — 6. EXERCISE 528. Solve the first 30 examples of the following set orally. The student should review Cases VI and VII of Factoring. Solve : 1. (a;-4)(»-5)=0. 9. a; 2 + 13 a; + 40 = 0. 2. (a? + 4)(a?-7)=0. 10. x 2 + 19a? + 90 = 0. 3. (2a?-5)(a:-3)=0. 11. x* - 21 x + 20 = 0. 4. (a; + |)(7aj-l)=0. 12. a* - 7a; + 12 = 0. 5. x 2 -13a + 30 = 0. 13. a 2 - 7a + 6 = 0. 6. a 2 + 13 a + 30 = 0. 14. x 2 + 16a + 48 = 0. 7. a 2 -13 a -30 = 0. 15. a 2 - a -42 = 0. 8. a 2 + 13a -30 = 0. 16. a 2 + a - 42 = 0. Complete Quadratics 379 17. x 2 -2x = 63. 27. 3a 2 -2a-l = 0. 18. a 2 + 2 # = 35. 28. 4z 2 - 4# + 1 = 0. 19. a 2 -JO a = 39. 29. (x - 1.25) (a + .75)= 0. 20. a 2 -10a; = 0. 30. a 2 - f a + i = 0. 21. x*- 16 = 0. n. 52±I + -li-.-« + L 22. ax 2 — a = 0. 23. (ar-a)(a;-a)=0. 32. -^ - -A_ , 2. 24. a 2 -6a + 9 = 0. 25. a (a -5) =36. 9 2#+3 a? — 3 a — 2 26. *-7 = *2- 34 1 =J 2 *~ 3 * * - 1 V » - 1 35. Vl + 2/ + Vl - V = V3 - y. 33 36. V4 — a? -h Vl + « = V9 + a;. _2a-3 = 40 3r-2 (x-1) 2 x-1 r r + 1 37. -i ^-3 = a 4Q> 3r-2_2r-i = 2> 38 ^ 2 _?_^ + / 2 = 41 - x 2 + 3ax = abx + 3a?b. P 9 9 2 1 - o 42. (a 2 -9)(a; + 3)=0. 39 4- - 1 = 2 a;-3 x- 2 2' 43. 3aa + 12a 2 Va; = 15a3 a_l ^-2 1 44. = - • x + 1 a 3 45. (x-a)(«+«)= 4 - a2 - 46. (x — a)(x — b)=ab — a—b + l. 47. (x - l) 2 = a (a 2 - 1). 48. a 2 (6 - a) 2 - 6 2 (a - x) 2 = 0. 49. a 2 - a 2 - (a - x) (b + c - x) = 0. 50. a 2 (a-a) 2 = 6 2 (5 _ ^ 52 . gp+l-a+l. x a 51. a*-(a + &)* + a&=:0. 53. aa;2 + bx + c = *■ a-^x 2 + ^a; -|- c x Cj 54. (a-7)(2aj + 5)-(3a;-l)(a:-7)=0. 380 Quadratic Equations Solve : 55. U--\(2x-3)=4,x-l. x 56. (7-x)x=ll--\(3x + 8). Kft 3x 3a-20 „ . 3a 2 -8 57. = ZH • 2 18-2a> 2a-2 529. General Form. The general form of the quadratic equa- tion is ax 2 + bx + c = 0, where a, b, and c are any numbers. 1. 3x 2 — 7 a? + 5 = is in the general form. In this equa- tion a = 3, b = — 7, c = 5. 2. (a 2 -f-& 2 )a; 2 +(a-Z>)a; + 3 = 0. Here a = a 2 + 6 2 , 6 = a- b, and c = 3. 3. Change to the general form (ax — b)(c — d) — (a — 6) (esc — d) x. acx — adx — be + bd= acx 2 — bcx 2 — adx + bdx. (Expanding.) — acx 2 + bcx 2 + acx — bdx — be + bd = 0. (Transposing. ) (6c - ac) x 2 + (ac -bd)x- (be - bd) = 0. (Collecting.) .•. a = be — ac, b =(ac — bd), c = — (be — bd) . 530. Any quadratic equation may be changed to the general form ax 2 + bx + c = 0. The steps are, in general, clearing of fractions, expanding, transposing, and collecting terms. EXERCISE 531. Reduce each of the following equations to the general form ax 2 -f bx -f c =0 and determine the values of a, b, and c, in each case : 1. 9a; 2 -f 4a = 325. 7. ax 2 -bx=c. 2. 17 a; 2 = 418. 8. ax 2 - (a 2 + l)x + a = 0. 3. x 2 -2aa; + & = 0. 9. (a - x) 2 = 0. 4. x 2 -ax = 0. 10. (x + a + b) 2 =0. 5. a 2 — a; = 2 + V2. 11. abx 2 — a 2 x — b 2 x = ab. 6. x 2 + l=%x. 12. Va; + 16 = x - 3. Complete Quadratics 381 532. Formula for Solving Quadratic Equations. We have seen that every quadratic equation can be reduced to the form ax 2 + bx + c = 0. The solution of this equation leads to a formula that can be used for solving any quadratic equation. ax 2 + bx + c=0. ax 2 + bx = — c. (Why ?) a 2 + -a; = --. (Why?) a a . *+£„, + £=£_£ (Why?) a 4 a 2 4 a 2 a " 4ac (Why?) 4 a- ^2a 2a _-6±V6 2 -4ac 37 — — • 2a The last result gives the two roots of the equation whatever the values of a, 6, and c may be. The roots of any quadratic equation can therefore be found by substituting the values of a, b, and c in any particular equation in the formula for the roots ; that is, in — b ± V& 2 — 4 ac 2a This formula should be carefully committed to memory. 533. To solve a quadratic equation by the formula : 1. Change the equation into the form ax 2 + bx + c = 0. 2. Determine the values of a, b, and c for the given equation. 3. Substitute the values of a, b. and c in the formula and simplify the results. 382 Quadratic Equations l. Solve Solution. Examples 6 a; 2 + x = 15. 6x 2 ±x- 15 = 0. (General form.) a = 6, b = 1, c =— 15. g = -l±Vl-4.6.(-15) 2-6 12 2 3 Check. 6 (f) 2 + f = V + f = 15. 2. Solve mx 2 — ra 2 a; — a; + m = 0. Solution, mx* -(m 2 + 1)* + m =0. a = m, b = — (m 2 + 1), c = m. (m 2 + 1) ± Vw 4 + 2 m 2 + 1 - 4 m 2 2m _ (m 2 + l)j:(w 2 -1) 2m = m or — m 3. Solve a*- 7a -30 = 0. Solution. » = L±^»±i20 = 7±18 =10op-8b 2 2 EXERCISE 534. /Sofae 6y £/ie formula : 1. a; 2 -6»- 7 = 0. 2. a 2 +8a? = 20. 3. a 2 -6a; -16 = 0. 4. a;(aj + 5) = 84. 5. a; 2 + 7 a; = 30. 6. a; 2 -13a; + 42=0. 7. x 2 — 19a = 0. 8. x 2 + 4 x = 1. 9. 2a 2 4- 5a; + 4 = 0. 10. 9a; 2 + 5 = 12a;. 11. 36a; 2 + 6 x -5 = 0. 12. x 2 + 8 a; + 16 = 0. 13. 4a; 2 - 4a; + 1 = 0. 14. a; 2 -4 = 0. 15. a; 2 + | a; = 40. 16. 3a; 2 + 17a; + 70 = 0. 17. a 2 + T V^ = TV 18. 20a; 2 -2a; =6. Complete Quadratics 383 19. y i2-ic 2 -2« + 12 = 0. 23. ax 2 + 2bx + c=0. 20. 2 x 2 — f x = 1 24. cm; 2 4- &x = c. 21. (2x4-l) 2 = 0. 25. x 2 - ax = 0. 22. x 2 + ax = b. 26. (x — l) 2 — ax 2 4- a = 0. 27. (x-6)(x-5)+(x-7)(x-4) = 10. 28. (2x-5) 2 -(x-6) 2 =80. 29. 2x + ^ = 3. 31. ^4- — = 3. x 4 x 30. 5-21=^=1. 32. *" 8 X " 1 4 4-x x + 2 2(x + 5) 33. (a — x)(x — b) + a& = 0. 34. (a - x) 2 +{x - b) 2 = a 2 + & 2 . 35. x 2 - xV3 + 1=0. 36. Verify 35. 37 . *=!-**=* + *=* = 0. x x— 1 x — 9 38. x(a + b — x) = c(a 4- b — c). 39. (n — p)x 2 +(p — m)x + (m — n) = 0. .. ax 2 — bx -f c 40. 4— = c - x 2 — x 4- 1 41. (a — x) 2 — (a - x)(x — b) — (x - 6) 2 = (a — 6) 2 . 42. (a - x) 3 + (x - &) 3 = (a - 6) 3 . Tri ^e following equations find the roots correct to two decimal places : 43. 3x 2 -2x = 40. 44. x 2 -x-l=0. 45. x 2 -30 = -. 3 46. Given S = ±gt 2 + v t. When S = 200, g = 32, and v = 10, find t. 47. x 2 + 1.92 x- 3.83 = 0. 49. 3x 2 -4x-10 = 0. 48. x 2 4- 3.14 x 4- 2.45 = 0. 50. 5(x 2 - 1)4- 10 x = 7x 2 - 15. 384 Quadratic Equations THEORY OF QUADRATIC EQUATIONS 535. Relations between the Roots and the Coefficients of a Quadratic Equation. Consider the equation ax 2 + bx + c = 0. If we let r x and r 2 represent the roots, we may write, _ — b + V& 2 — 4 ac Tl ~ 2a ' 2a — 5+Vfr 2 — 4ac . — 6 — V& 2 — 4ac 6 .-. r x + r 2 = 1 = — - , 2a 2a a andrir2 = -6 + V& 2 -4ao.-6-V^-4 a c = c. 2a 2a a The student should work out these results in detail. Therefore we have : The sum- of the roots of a quadratic equation in the form ax 2 + bx -f c b c — is — , and the product is - • a a Examples 1. 3a; 2 -7a; + 2 = 0. The sum of the roots is | and the product is f . 2. 3a 2 -3a: = 7. 3 x 2 - 3 x - 7 = 0. . \ n + r 2 = 1,. and nr 2 = -f EXERCISE 536. Find, without solving, the sum and the product of the roots of the following : 1. a; 2 -21 a; + 20 = 0. 4. 4a; 2 - 8a; - 3 = 0. 2. a; 2 + 7 a: + 12 = 0. 5. x 2 -\x = \. 3. a; 2 + 16 a; + 48 = 0. 6. 3a; 2 -x = 24. Theory of Quadratic Equations 385 7. lx 2 + x = 50. 12. (x-3)(x-5)=0. 8. 2x 2 -14 x + 23 = 0. 13. 3z 2 + ll = 5a\ 9. 5z 2 + 13 = 14a>. 14. 5x 2 = 12. 10. ax 2 — 2bx = c. 15. 3 a; 2 = 5 a;. 11. 6a 2 + 7a; = 3. 16. x 2 + px = q. 17. Show that, if an equation is in the form x 2 + mx + n = 0, r 1 -\-r 2 = — m, and r x • r 2 = n. 18. What is the sum of the roots of an incomplete quadratic equation of the form x 2 = k ? 19. One root of x 2 -f- 4 x — 45 = is 5. Determine the other in two ways, without using any of the usual methods of solving. 20. One root of 2 x 2 — 7 x — 15 = is 5. Find the other, in two ways, as in example 19. 537. Nature of the Roots of ax 2 + bx + c = 0. The roots of the equation a# 2 + bx + c = have been found in § 532 to be _& + y& 2 - - 4ac 2a _6_V6 2 - - 4ac and r 2 = 2 a* Whether the roots ^ and r 2 are real or imaginary (§ 456), and if real whether they are rational or irrational (§ 457), depends upon the expression V& 2 — 4 ac. (Why ?) 1. If 6 2 — 4 ac = 0, the roots are real, rational, and equal. (Why?) 2. If b 2 — 4 ac is a negative number, the roots are imaginary. (Why?) 3. If b 2 — 4 ac is positive, the roots are real and unequal, and they are rational or irrational according as b 2 — 4 ac is or is not a perfect square. 386 Quadratic Equations 538. Discriminant. The expression b 2 — 4 ac is the dis- criminant, since, by means of it, we determine the nature of the roots. 1. Determine the nature of the roots of x 2 + 5 x — 6 = 0. Solution. The discriminant is 25 — 4 • 1 • (— 6) = 49. .-. The roots are real, rational, unequal. Let the student determine the roots. 2. Determine the nature of the roots of 9 x 2 + 5 = 12 x. Solution. 9x 2 — 12x + 5 = 0. The discriminant is — 36. (Why ?) .-. The roots are imaginary. 3. For what value of k are the roots of x 2 — 6x + k = equal to each other ? Solution. The discriminant is 36 — 4 k. If k has a value that satisfies the equation 36 — 4 k = 0, the roots will be equal. This gives k = 9. Therefore k = 9 is the value required to make the roots equal to each other. If k < 9, 36 — 4 k is a positive number, and therefore the roots will be real. If k > 9, 36 — 4 k is negative and the roots will be imaginary. Let the student determine the roots when k — 9 ; when k = 10 ; when k = S. EXERCISE 539. Determine, without actually solving, the nature of the roots of the following equations : 1. a 2 -7a + 10 = 0. 8. 3z 2 + 2z + 5 = 0. 2. 12a 2 -x-l = 0. 9. z 2 -5a; = 50. 3. 3 x 2 - 12 x + 5 = 0. 10. x 2 - 5 x + 50 = 0. 4. 3z 2 -8;r + 7 = 0. 11. 2a 2 - 7x + 30 = 0. 5. 2x 2 -5a-9 = 0. 12. 2z 2 -7z-30 = 0. 6. 4.t 2 -13z + 3 = 0. 13. -7a 2 + 22z = 3. 7. 25a 2 - 10 z + l = 0. 14. 2x 2 + 3 = 5a. Theory of Quadratic Equations 387 15. x 2 —3x+k=0 when k=2\. 17. Answer 15 when k > 2\. 1 5 16. Answer 15 when k < 2\. 18. x -j- - = - • X £ 19. x + - = k, when k lies between 2 and — 2 in value. x 20. What value of c will give equal roots in2# 2 4-4a;+3c=0? 21. Verify 20. 22. For what value of k is one root three times the other in X 2 _ jcx +. 75 = 3 ? Solution. Here n = 3 r 2 , or *±^MZM = 8 * - VF=800 Solve for A;. 2 2 23. For what value of A: does one root of x 2 — kx + 40 = exceed the other by 3 ? 24. Find k if r, = 7 r 2 in a 2 — foe + 63 = 0. 25. Find & if r, = 2 r 2 in 4 x 2 - 9 a + k = 0. Verify. 540. To form an Equation with Given Roots. We have seen (§ 525) that (x — a)(x — b) = has the roots a and b and no other roots. Similarly (x — a)(x — b)(x — c)= has the roots a, 6, and c and no other roots. Thus, we can make an equation with any required roots. 1. Make an equation whose roots are 2 and 3. Solution, (x — 2)(x — 3) = 0, or x' 2 — 5 x + 6 = 0, is the required equation. 2. Form an equation whose roots are 2 and — 5. Solution, (x - 2)(x + 5) = x 2 + 3 x — 10 = 0. Explain the factor £ + 5. 3. Form an equation whose roots are f and -f. The result is indicated by the equation (x — f )(z — I) = 0- F° r con- venience, multiply by 15 in the form of the two factors, 3 • 5 ; thus, 3(x-!)5(z-f)=0. (3 x - 2)(5 x - 3) = or 15 x 2 - 19 x + 6 = 0. 388 Quadratic Equations 4. Form an equation whose roots are 1 ± V2. Solution, (as — 1 — V2)(x - 1 + V2) = 0, or x 2 — 2 x— 1 = 0. 5. Form an equation whose roots are 1, — 1, 2. Solution, (x — 1) (a + 1) (x - 2) = 0, or x 3 - 2 x 2 - x + 2 = 0. EXERCISE 541. Jfafte equations whose roots are as indicated: 1. 2,3. 6. -a, -b. 11. ±V3. 2. 4,5. 7. 2,2. 12. 1±V5. 3. 7, - 1. 8. ± 2. 13. 1 ± i, 1. 4. 0,6. 9. a, 2 a. 14. 2, 1±2V3. 5. - 3, - 2. 10. 6, - 2 6. 15. ± i, 2. 16. 2+V2, 3-V2. EQUATIONS IN THE FORM OF QUADRATICS 542. Quadratic Form. An equation is in the form of a quad- ratic if it contains two powers of the unknown, one of which is the square of the other. x* — 5 x 2 + 4 = 0, x + x* — 6 = 0, and x~$ + x~^ — 6 = are examples of quadratic forms. These equations may be solved for a; 2 , x^ y and x* respectively, and the results so found can then be solved for x. An equation may be in quadratic form with respect to some polynomial containing x. (x 2 — 2)-f- V# 2 — 2 — 6 = is such an equation. It may be solved for V# 2 — 2 just as z 2 -f- z — 6 = may be solved for z. The method of solving such equations will be understood by examples. 1. x 4 - 5 # 2 + 4 = 0. (x 2 — 4) (x 2 - 1 ) = 0. (Solving for x 2 by factoring,) x 2 = 4 or 1. x = ± 2 or ± 1. Equations in the Form of Quadratics 389 2. x + x* — 6 = 0. x \ _ — 1 ± v 1 + 24 _ _ 3 or 2. (Solving by the formula for x^.) x = 9 or 4. This is a radical equation; 4 is a root and 9 is an extra- neous root. 3. aT* + afs _ 6 = 0. x~s = — 3 or 2. (Solving as a quadratic for x" 1 .) -I- = -3 or 2. (Why?) X s x i = _ i or |. (Why ?) X=-^jOT^. This is a radical equation, and both roots satisfy the equa- tion. 4. (a? - 2) + Va; 2 -2-6 = 0. Here we regard Vx 2 — 2 as the unknown, ^— - _ _ i ± Vl + 24 = _ 3 or 2 . (By formula. ) 2 je 2 - 2 = 9 or 4. z 2 = 11 or 6. x = ± y/TL or ± V6. Do the roots satisfy the equation? In the solution of such an equation as example 4, it is some- times convenient to substitute a single letter for the expression we are regarding as the unknown. For example, we might have put z for V# 2 — 2 and then we should have z 2 + z - 6 = 0. , = -l±V26 = _ 8or8 , Vx 2 -2 = - 3 or 2, etc. 390 Quadratic Equations a — x x — b a 2 + &2 5. x — b a — x ab To . a — x x — b Then the equation takes the form * + l = «i+-L 2 . z ab abz 2 -(a 2 + V 2 )z + ab = 0. (az-b)(bz-a)=Q. whence z = - or -. a b ^f = f. Also « = *=», x-b b x -b a x=™?-. (Explain.) x = <*±*. a + b * a +b EXERCISE 543. Solve the folloiving equations as quadratics, substituting a single letter for an expression containing the unknown ivhen desirable : 1. x* -13a; 2 + 36 = 0. 8. x 4 - 21 a; 2 = 100. 2. x\x*- 90) +729 = 0. 9- (a; 2 - 10) (a; 2 - 3)= 78. 3. a;-7V* + 12 = 0. 10 - (* 2 -5) 2 +(* 2 -l) 2 =40. 11. ar 6 -7ar 3 = 8. 4. x — 4 = 3a;i 12. 9a* = 9 + 2a;*. - a; 2 5 26 5 + 5» T" 13 ' 2^+Va^-3 = 0. 6. a*-9a? + 8 = 0. 14 ' 2(Va;-3) 2 -3=V*. 7 _2_ 5 =2 !5- x^ + Sx^ = 9x. a; 2 + 3 x 2 ' 16. 2x^ - 3x% + a; = 0. 17. (a; 2 + 2 a;)2+ 3(a; 2 + 2 a?) = 10. 18. fx + -\ 2 -sfx+-\^-7 = 0. 19. ^±^+2^^? + 3=0. a; 2 -3 a; 2 + 3 Equations in the Form of Quadratics 391 20. 60 — Wx 2 + x + 6 = x 2 + a + 6. Let z = Vx 2 + x + 6. 21. a 2 -2a;+6Vx 2 --2a; + 5 = ll. Hint. This equation may be written O 2 -2 x + 5) + 6>/a:- 2 - 2 » + 5 = 16. 22. x 2 4- 5 = 8 a; + 2 V# 2 — 8 a; + 40. 23. 2.» 2 + 3Va; 2 -a; + l = 2a,'-t-3. 24. (2z 2 -3a; + l) 2 =22a; 2 -33a;+l. 25. Ac-h*Y + 4a; + - = 12. \ xj x 26 * + s + 5 * + s-5 =ia tf + x-2 a 2 + a; - 4 Put x 2 + a; = z. 27. 2(a + 3)(z + 4) = (^ + 7z)(a 2 + 7a;-3). 28. x 2 = 8 Va 2 + 16 - 32. 29. a? 2 "+ V5 « -f a; 2 = 42 — 5 ». 30. x 2 -9x-9Vx 2 -9x- 3 3 31. 11 =-9. — =4. x + V5 — x 2 a; — V5 — a; 2 32. 2 a; 10 = 3 a 6 -a* 33. 2a; 2 + 3a;-5V2a; 2 + 3a; + 9 + 3 = 0. 34. (x 2 - 2)* - 4(a? - 2)* = 5. 35. 9ar 4 +4ar 2 = 5. 36. ar J -4Va^^2=-l. 37. V2~a7- 7x = — 52. 38. ar 2 -7a; + Var J -7a; + 18 = 24. 392 Quadratic Equations Solve the following equations as quadratics: 40. ^H-lY+2^+i^ = 15. PROBLEMS LEADING TO QUADRATIC EQUATIONS 544. 1. The sum of two numbers is 17 and their product is 42 ; find the numbers. 2. The sum of two numbers is 17 and the sum of their squares is 185 ; find the numbers. 3. Find the sides of a rectangle, knowing that its perimeter is 52 feet and its area is 160 square feet. 4. In a right-angled triangle the measures of the two sides about the right angle and the hypotenuse are three consecu- tive integers. Find them. 5. Same as problem 4, if the sides are consecutive even numbers. 6. The sum of the two sides about the right angle of a right-angled triangle is 21 inches, and the hypotenuse is 16 inches. Find the sides correct to two decimal places. 7. A certain rectangle contains 216 square feet. If both dimensions are increased by 2, the area is increased by 64 square feet. Find the dimensions of the rectangle. 8. The sum of the roots of a quadratic equation is 3 and their product is If. Find the roots. 9. Same as problem 8, if the sum of the roots is 2 and their product is 4. 10. Two numbers differ by 2.1, and the square of their sum is 25. Find the numbers. 11. Determine the positive value of x correct to two decimal places for the equation x 2 -f y 2 = 36, knowing that 5 y = 27. Equations in the Form of Quadratics 393 800' 12. Determine the positive value of b to two decimal places, in a 2 + b 2 = c 2 , if a = 2.1 and c = 4.3. 13. Determine the larger root of x 2 = .100 — .200 x correct to three significant figures. (Harvard.) 14. A rectangular box is 3 inches deep, and is 2 inches longer than it is wide. Find its length and breadth, if its volume is 105 cubic inches. 15. A rectangular plot of land is 600 feet by 800 feet. It is divided into four rectangular blocks by two streets of equal width running through it. Find the width . of the streets if to- gether they cover an area of 67,500 square feet. 16. What is the width of the streets in problem 15 if together they cover one fourth the area of the plot ? 17. How wide a strip must be cut around the outside of a lawn 60 feet by 80 feet so that the strip cut may contain half the plot ? 18. A rectangular tin box is 2 inches deep, and is 2 inches longer than it is wide. Find the length and breadth if it requires 88 square inches of tin to make the box, including the cover, making no allowance for waste. 19. If from a certain number the square root of half that number is subtracted, the result is 25 ; find the number. (Regents.) 20. The numerator of a fraction exceeds the denominator by 2. If both terms of the fraction are increased by 2, the value of the fraction is diminished by -^. Find the fraction. 21. The units' digit of a number exceeds the tens' digit by 1. The product of the digits equals -§- of the number. What is the number ? 22. In an automobile race of 462 miles the winning car runs 2 miles an hour faster than the losing car and wins the race by •i- hour. What is the winner's rate and what is the time ? 394 Quadratic Equations 23. A broker buys a certain number of shares of stock for $960. Later the price falls $20 a share and he finds that he might have bought 4 shares more for the same money. How many shares did he buy ? 24. If q is the area of a rectangle and p is the perimeter, show that x 2 —^-x -\-q — is an equation for finding the dimensions. 25. A man buys apples for $ 12. If the price had been 20^ less per bushel, he could have bought 5 bushels more for the same money. Find the number of bushels bought and the price per bushel. 26. How wide a strip must be plowed around a field 60 rods long and 40 rods wide to have the field half plowed ? 27. A stream flows at the rate of 4 miles an hour. A man can row up the stream 10 miles and back to the starting point in 6 hours. Find the rate at which the man would row in still water. Note. It is to be assumed in this problem that the rate at which the man rows upstream is equal to his rate in still water minus the rate of the current, and that in going downstream his rate is that of his rowing in still water plus the rate of the stream. 28. A motor boat goes 12 miles up a river and returns to the starting point in 4| hours. Find the rate of the current if the boat can run 7 miles an hour in still water. 29. A rectangle is 6 inches by 10 inches. It is to be doubled in area by equal additions to the length and the width. Find to two decimal places the increase in the dimensions. 30. Solve 29 if the area is to be made four times as great. 31. A company owns two factories that together can make 252 automobiles in 12 days. Working alone one factory requires 7 days longer than the other to make this number. Find the number of days for each factory. (Yale.) Review of Equations 395 32. If the product of three consecutive numbers is divided by each in turn, the sum of the quotients is 191. Find the numbers. 33. A man having bought an article, sells it for $ 21. He loses as many per cent as he gave in dollars for the article. How much did it cost him ? (Yale.) 34. Find the price of eggs when if two less were given for 30 ^ the price would be 2 ^ per dozen higher. (Amherst.) 35. If a ball is thrown vertically upwards with a velocity v , the distance in feet to which it will rise in t seconds is given by the formula d = v<$ — i gt 2 . (g = 32.) Solve this equation for t when v = 200, and d = 300. REVIEW OF EQUATIONS 545. Equations may be classified as to their degree into three groups : 1. Linear Equations. (Simple or first degree equations) 2. Quadratic Equations. (Second degree equations) 3. Higher Degree Equations. 546. Equations may be classified as to form into three groups: 1. Rational Integral Equations. 2. Rational Fractional Equations. 3. Irrational Equations. 547. In solving an equation we begin by simplifying as much as possible, including such steps as expanding, clearing of fractions, rationalizing, transposing, and collecting terms. All these steps aim toward some particular form. The form will depend upon the kind of equation we are solving and, in the case of quadratic equations, the method we intend to use in its solution. 396 Quadratic Equations Exceptions to the general directions just given include radical equations solved as quadratic forms without rationalizing, and fractional equations solved by substitution. Skill in selecting the best methods of solving equations, and in discovering methods of simplifying the work will be gained by experience and by a conscious effort on the. part of the student to achieve such ends. Higher degree equations, if they can be solved at all, by the methods of elementary algebra, must come under " Quadratic Forms " or under the factoring method. REVIEW QUESTIONS 548. 1. What is a simple equation? a quadratic equation? 2. What is a rational integral equation? an irrational equation ? 3. State, in full, the three different methods of solving quadratics. 4. To what form do we reduce the quadratic when we solve by " completing the square " ? by formula ? by factoring ? 5 . Can the formula be used to solve an incomplete quadratic ? 6. What do we mean by the " nature of the roots " of a quadratic ? 7. How do we determine the nature of the roots of an equation without solving the equation ? the sum of the roots ? the product ? 8. How do we form an equation with given roots ? 9. What do we mean by the extraneous roots of a radical equation ? 10. How many roots has a quadratic equation ? 11. Knowing that 3 and — 5 are the roots of x 2 -\- 2 x — 15 = 0, can you at once write the factors of x 2 + 2x— 15 ? Review Exercise 397 12. Translate into verbal language the condition that ax 2 + bx -f- c = may have equal roots ; the condition that x 2 + px = q may have real roots. 13. Do both roots of a quadratic equation necessarily satisfy the conditions of the problem from which the equation may be derived ? How can such results be checked ? REVIEW EXERCISE 549. Solve: x 2-5x 1+x 148 -5a 2 _ 2 5x+l 3-2x 3 + 13a;-10a; 2 2. -^— + ^^=25. 3. J?0 +9 _j20T 9 = a *x 2 Vx 2 4. (a>+ll)* + 3(» + ll)* = 4. u 1 x -h a x 2 5. x + a x — a a 2 — x 2 6. Vz+18+Va?-18 = 6. 7 ° ~ a a — 253 #(q — b) _ ft 'a; — 6 a; + 6 x 2 — b 2 8. a: 2 + 12 a = 2 Va? 2 + 12 a? - 4 + 67. 9. ^-1=1^. 10. Give nature, sum, and products of the roots of the following : (a) 5x 2 -7x + 2 = 0. (d) 2x 2 -6x = -m, whenra > 4|. (6) a 2 -5=4a>. (e) 5- 3a* + 7a; = 0. (c) a* + 2 = 0. (/) 3a;2-3a; + |=0. 398 Quadratic Equations 11. Form equations whose roots are : (a) 7, 3. (d) a — b, b — a. (&) h-h 00 3±V2. (<0 3,f (/) 3±«. 12. Solve7a+Va 2 -17x + 4 = 2a; 2 -27a; + 5. « Q « -+■ Vl2 a — x Va + 1 # — Vl2a — x Va — 1 14. For what value of m are the roots equal in Sxi-bx +m = 0? 15. What change is made in the roots of the complete quadratic by changing the sign of b ? 16. a; + l--^ == -^ + 5a? - 4 - 1 - X 2 x-\-l x 2 -l 17. In two years the population of a city increased from 6400 to 8100 ; the rate per cent of increase during the first year was equal to the rate per cent of increase during the second year. What was this rate ? 18. In the equation x 2 + y 2 = 1, y = ^ V3 ; find x. 19. Solve, getting the answers correct to two decimal places, 3a;2_ a ._- v /2 = 0. 20. Solvea 2 +a+3a;V3 + 4 = 0. (Yale.) One answer is 1 — V3. 21. A pedestrian having 18 miles to go to keep an appoint- ment finds that at his present rate he will be half an hour late. If he quickens his pace by half a mile an hour, he will arrive on time. At what rate is he walking ? 22. A room is two yards longer than it is wide and the floor contains 24 square yards. Find the dimensions of the room. Solve : Review Exercise 399 23 . 12^ 37-3s x - 5 T 25 - a* a; . 3a;-2 24. = I- ^-^ = = 4-. 2(z+2)^ 4 -a* 2 25. V5a?4- 1 — V3# = l. 26. V2-1 33. Find a number such that one half its square shall ex- ceed the square of one half the number by one half the number. 34. — z== = • (Apply composition and division.) Va*+1-Va*-1 6 x — -Vox . a — VaaT _ a; — a ^ (Clear of fractions and fac- 35. a + y/ax x-\-^/ax a t/01 ^ XXHL QUADRATIC EQUATIONS (Continued) SIMULTANEOUS EQUATIONS 550. The degree of an equation is determined with respect to the letters that are regarded as unknowns. For example, ax + b = is of the first degree with respect to x as unknown. (§§ 245, 246.) s = \gff- is of the second degree with respect to t as unknown. ax + by = c is of the first degree with respect to x and y. 3xy + x + y = 5 is of the second degree, regarding x and y as unknowns. 551. There is no general method for solving simultaneous quadratic systems since, in general, the elimination of one of the unknowns gives rise to an equation of higher degree that cannot be solved by methods of elementary algebra. We shall consider some special forms of quadratic systems. 552. Case I. One Equation of the First Degree and the other of the Second. (Review of Chapter XVII.) A system of equations involving two unknowns, one equa- tion linear and the other quadratic, can always be solved, since the elimination of one unknown by substitution from the first degree equation into the second degree equation gives rise to a quadratic. (See § 430.) Solve the system (x + y)(x — 2 y) = 7. (1) x-y = 3. (2) Solution. x = y + 3. (From (2).) (V + 3 + y) (y + 3 - 2 y) = 7. (Substituting in (1).) (2y + 8)(-y + 8) = 7. 400 Simultaneous Equations 401 2y2_3 y _2 = 0. y = 3±V9 + 16 =2and _l, When y = 2,^ = 2 + 3 = 5. 4 2 When |f = -|,«=-i + 8s=|. x = 5, f . V = 2, - J. Check both sets of roots. EXERCISE 553. #oZve Me following systems, grouping the answers properly at the end of the solution. Leave irrational answers in simplest radical form. Verify one set of answers in examples 1 to 6: 1. x + y = 13, 11. ab = 147, icy = 36. a : 6 = 3 : 1. 2. a? + y = 10, a; 2 + y 2 = 58. 12. M=5, x y 3. 4. ay — 5 a? =a 1, 7x — y = l. x 2 + 4:xy = 57, * + y = 7. Hint. Use - and - as un- . * y knowns. 5. 3# + 5y = 35, 13. x + y = 29, aj2 + 2 y 2 =a . y+ 8 a>— y+13. V# + Vy = l. 6. 7. 5x — y_ 7 HlN known 14. t. Use Vx and -\A/ as un- 4 ~~4 x2+ 32/ 2 = 7, a;^?/ 5 7a;2-5x'2/ = 18. o. - + - = -, V ® * 20. « 2 + a;2/ = a, ^ = 8 - if + X y= b. 9. ^-±1 + ^^ = 5, 21. aa;2 4-6a?2/=a, aj 2 +2a ;2/-32/2 = 5. /aj 0^-18. 11. (x + y) 2 =3x 2 -2, 23. 3z 2 + 3xy + 2 2/ 2 = 8, (x - 2/) 2 = 3 2/ 2 - 11. x * - ^2/ - 42/ 2 = 2. 12, (2aj+5.y)(3aj-5y)=44, 24. 6 a 2 + 5 ay - 6 2/ 2 = 0, #2_Q xy + 122/2 = 3. 2a? 2 -2/ 2 = -l. Simultaneous Equations 407 560. Case III. Symmetrical Equations. An equation is symmetrical if an interchange of the unknowns does not change the equation except in the order of its terms. Thus, x -f- 3 xy -j- y = 10 becomes, by interchanging x and y, y _j_ 3 y X + x=. 10. Therefore the equation is symmetrical. x + y = 5 becomes y -f- x = 5 and is therefore symmetrical. Is x 2 + y 2 = 5 symmetrical ? a;# + a; = 10 ? ic + 2 ?/ = 5 ? 561. A system consisting of two symmetrical 'equations can generally be solved by combining the equations in such a way as to find values of x -f- y and x — y. Solve the system, x 2 + y 2 = 17, (1 ) x+y = 5/ (2) Solution. Here we have the value of x + y ; we look for zy and thence x — y. x 2 + 2xy + y 2 = 25. (Squaring equation (2)). (3) 2 xy — 8. (Subtracting equation (1 ) from equation (3)). (4) x 2 — 2 ojy + y 2 = 9. (Subtracting equation (4) from equation (1)). B-y=±8, We now replace the original system by the two systems : A. . x -f y = 5. B. x + y — 5. ^4ns. a; = 4, 1. x-y = 3. £— y=-3. y=l, 4. Whence x = 4. Whence # = 1 . y = l. 2/ = 4. 562. 1. The student should follow some such systematic arrangement of his work as is found in § 561. 2. He should carefully study the equations to determine what steps will lead to the desired forms. No general rules can be given since the method of procedure varies with the form of the equations. 3. It will usually be found helpful to find a value of xy, as in (4) § 561, and use this value in combination with one of the preceding equations to form an equation containing some power of x + y or x — y. 4. From this equation values of x+y and x—y can be found. 408 Quadratic Equations 5. It will be well also to divide the equations, member by- member, when this is found possible. Thus, if X s -f y z = 15 and x 4- y = 3, by dividing we get x 2 — xy -f- y 2 = 5. 563. Optional Method. Symmetrical equations can be solved also by the following method: a* + y 2 = 17, (1) x H- y = 5. (2) Substitute w + -y for x and w — v for ?/. Equation (1) becomes (u + v) 2 + (w - v) 2 = 17. (From (1).) (3) 2m 2 + 2v 2 = 17. (4) (w + «) + («-«)= 5. (From (2).) 2m = 5. w = f. 2(|)2 + 2v 2 = 17. (From (4).) o 2 = f . jc = M + i? = |±f = 4orl, and y = u — ^ = ^ + 1 = 1 or 4. EXERCISE 564. Solve the following systems, and verify one set of answers in examples 1 to 5 : 1 # #2 _j_ <^2 _ 5Q ? Multiply the second equation by 2 and com- X y _ j > bine with the first equation to get values of x 2 + 2 xy + I/ 2 and a; 2 — 2 a?/ + y 2 . 2. a 2 — xy + ?/ 2 = 7, 6. x 2 + a# + y 2 = 6, SB + ?/ = 4. x 2 — xy + y 2 = — 6. 3. a 2 + #2/ + y 2 = 14, 7. x 2 + ?/ 2 + » + i/ = 146, x — y — V2. #2/ = 63. 4. x3 + 2/ 3 = 98, 8. a 3 + 2/ 3 = 28, a; 2 — a^/ -f- y 2 = 49. a; + y = 4. 5. x 2 + */ 2 = 269, 9. a?-# 3 = 26, a; — y = 3. a; — y = 2. Simultaneous Equations 409 10. (x-y) 2 -2>(x-y) = ±, x 2 + y 2 + 2 xy = 49. Solution. From the first equation x — y = 4 or — 1. (Why ?) From the second equation x + y = ± 7. (Why ?) We have, then, the four systems. A. x + y = 7, B.x + y = 7, C.x + y=-l, D.x + y = -7, x — y = 4. x — y =— 1. £ — y = 4. a; — y = — 1. Let the student complete the solution. 11. a*+0*-(a>+y)-12=O, 12. 3(aj»+2^) = 8(a? + y)-l, ajy-2(s + y)+8 = 0. ay = (x + y)+l. 13. x 2 -\- y 2 = xy = x -{- y. x 2 + 2xy + y 2 = x 2 y 2 . (From x-\-y = xy.) But z 2 + y 2 = xy. 2 zy = x 2 y 2 — xy. (Subtracting.) Let the student find the values of xy in the last equation and continue the solution. 14. x 4 + a 2 2/ 2 + ?/ 4 = 21. i5. x 2 + xy + y 2 = 91, a? 2 + a*/ H- ?/ 2 = 7. x-\- -\/xy + y = 7. Divide the first equation by the second. 16. & + y 3 = — 2 xy, x + y = -2. 21. p+pq + q = ±7, P + q = 12. 17. x* + y 3 = 280, x 2 - xy + y 2 = 28. 22. x y 2 18. x 2 + y 2 + x + y = 168, ^Jxy = 6. I + i=A. z 2 ?/ 2 36 19. x 2 + 2/ 2 + a + y = 18, 2 a# = 12. 23. r 2 + rs -f- s 2 = 21' r + i» 17. 20. ar = -jVz + 2/, y = ±£^/x + y. 24. a? 4 - t = 6 °9, a 2 + y 2 = 203. 410 Quadratic Equations GENERAL SUGGESTIONS FOR THE SOLUTION OF SIMUL- TANEOUS QUADRATIC EQUATIONS 565. In solving a quadratic system, first determine under which one of the following three cases it occurs : I. One equation of the first degree and the other of the second. II. Homogeneous second degree equations. III. Symmetrical systems. If the system does not come under one of these three cases, try to derive another system, or other systems, from the given equations that will come under one of these three cases. At all times remember that the object is to eliminate one of the unknowns. 566. Among special devices may be mentioned the following : 1. The immediate elimination of one of the unknowns by addition, subtraction, or substitution. 2. The elimination of the second degree terms. 3. Finding a quadratic form in some expression containing the unknowns and solving for the value of this expression. If this expression is of the first degree in the unknown, the given system may be replaced by two systems under Case I. 4. Dividing the equations member by member. EXERCISE 567. Solve the following systems : 1. x:y = 2:S, 5. ox - lOy + (x - 2y) 2 = 6, x 2 + y 2 = 5{x + y)— 2. xy + x+y = 7. 2. O + 2/) 2 +0»+2/)=12, 6. ^ + 5^ = 14, y 2 y 3 x 2 + i/ 2 = x -f- y + 4. 3. x 2 + y 2 = 34, x z-2y 2 + 3x = -50. x = y 2 + l. 7 . 5^+3^=8, 4. x(y-4)=U, y * y(a> + l) = 33. x 2 + y = x -f- 4. Simultaneous Equations 411 8. 4:X 2 -9xy + 5y* = 0, 7 x 2 — 3xy = 3x + 2y — l. 9. 2x 2 -3y* = 6, 3a? -2/ = 19. 10. ( x +y)(x-2y)=7, x-y = 3. 11. — - r ^- = a, 1-xy x = y. 15. 3xy-2(x-\-y)=2S f 2xy-3(x + y)=2. 16. (2x-2/) 2 -12(2»-v/) = 189, x 2 — 4 xy + 4 1/ 2 — 3 x + 6 ?/ = 54. 17. z 2 + 2/ 2 -2(x-2/) = 38, 23. 4:X 2 -9y 2 = 0, a^ + 3(a;~y)=25. 4 a; 2 + 9 y 2 = S(x + y). 12. * + y _i 1 — ay 1 a; = V3 13. » 2 + 2/ 2 + ^+y = 18, # 2 — # 2 -f x — .V = 6. 14. a? _ ^ = 40, a*/ = 21. 24. a; 2 + 2/ 2 = a^ + 189, 60(a;— 2/) = xy. 25. ^+^ = 3, 1 — ay 19. a/l+?La» g ~ y = 1 V y/ l + ajy 3 y(i + x 26. x 2 -a*/ = 3, xy-y 2 = 2. 20 _ _i_ _ = 5 ' x y ' 27. 2a;+Va#=10, a; — ^=.3. 3?/— 2yjxy = — 1. 21. Va? — 5 + \A/ + 2 = 5, x + ?/ = 16. ^ 4- 6a? + c = 0. 22. 8(z-5) 2 -3(2/-7) 2 =80, 29. x 2 + ?/ 2 + a? + 2/ = 36, 4(a-5) 2 + 5(2/- 7) 2 = 144. 2(x 2 + y 2 ) + 3 ay = 88. 412 Quadratic Equations 30. x 2 +• y 2 - xy = 7, 32. x 2 y + xy 2 = 30, a; ?/ 6 (a> — y)+a#=5. 1.1-5. 31. x 2 + »2/-« = 0, 2/ 2 + xy — 6 = 0. 33. x 2 y 2 + X y — 2=0, x+y = — l. to lor -f. 35. rc+2/+z = 2, xy = -l, xyz = — 2. 36. ic?y = 12, #z = 15, ?/2 = 20. 37. a-f-y-f z = 6, 2a-2/+z=3, a .2 + ^2_|_^ == 14 34. If x 2 — 2 y = — J and ?/ = Vl — x 2 , show that # is equal 38. xy -\-xz — 80, xz -\-yz = 108. 39. a? + y + 2! = 37, a 2 + 2/ 2 +z 2 = 481, i/ 2 = axs. 40. (z + l)(y + l) = 15, (jr + l)(s + l)«3* (« + l)(« + l)-21. ELIMINATION 568. It often happens in the study of mathematics and physics that it is necessary to eliminate one or more unknown quantities, or variable quantities, and either solve the resulting equation for the other unknown or derive an equation containing it. Some one of the methods of the present chapter will generally accomplish the desired elimi- nation. W=fs, f= ma, s = i at 2 , v = at Find W in terms of m and v ; that is, eliminate /, s, a, and t. Solution. W = ma • \at + r iJ + U' + r' after eliminating i2 + i£' + r. 10. Given v = u — gt and s = ut — \g&\ eliminate £ and solve the resulting equation for w. 11. Given (7= and C = — ; eliminate r. R + r R + nr 12. Eliminate x from the two equations ax 2 + bx + c = 0, and 2 a# + 5 = 0. 13. Given F = — m and — = i; show that F= BIL. d 2 ' m d 2 414 Quadratic Equations PROBLEMS LEADING TO SIMULTANEOUS QUADRATIC EQUATIONS 569. 1. The difference of the two sides about the right angle in a right-angled triangle is 2 inches. The hypotenuse is 10 inches long. Find the unknown sides. 2. The sum of two sides about the right angle in a right triangle is s and the hypotenuse is h. Express the values of the two sides about the right angle in terms of s and hi 3. The difference between the sides and a diagonal of a square is 3 inches. Find the side and diagonal to two decimal places. 4. The sum of the roots of a quadratic equation is 2, and their product is — 1. Find the roots and make the equation. 5. If a polygon has n sides, it has - — } - diagonals. I Two polygons have together 18 sides, while the number of diagonals of the one is to the number of diagonals of the other as 4 to 7. How many sides has each ? 6. Two polygons have together 12 sides and 19 diagonals. How many sides has each ? 7. Two adjacent square plots of unequal sides are inclosed by a continuous fence. The total area of the fields is 5200 square rods and the length of the fence is 1 mile. How large is each plot ? (Draw a figure.) 8. Besides zero there are two pairs of numbers such that their sum, their product, and the difference of their squares have the same value. Find these numbers. 9. In a proportion the sum of the means is 5 and the sum of the extremes is 7. The sum of the squares of all the terms is 50. Find the terms. (Use only 2 unknowns.) 10. Find two factors of p whose sum is s. Problems 415 11. In the figure, CD forms a right angle with AB. The sides are 9 inches, 12 inches, and 14 inches as indicated. Find the length of x and y to one decimal place. (Note that this would enable us to find the area of a triangle when we have given the three sides. ) 12. A mean proportional between two numbers equals VlO. The sum of the squares of the numbers is 29. Find the numbers. 13. The sum of the areas of two squares is 125 square inches. The sum of their four diagonals is 30V2 inches. Find the sides of the squares. 14. Find the dimensions of a rectangular room, knowing that the floor has an area of 240 square feet ; one side wall contains 180 square feet, and one end wall 108 square feet. 15. The sum of two numbers is one sixth of the difference of their squares, and the sum of the squares is 306. Find the numbers. 16. Divide 84 in two parts such that the sum of their squares is 3560. 17. If to the product of two numbers is added the greater number, we obtain 855 ; but if to the same product is added the smaller number, we obtain 828. Find the two numbers. 18. The sum of the squares of two numbers is 410. If the greater number is diminished by 4 and the smaller number is increased by 4, the sum of the squares is 394. Find the two numbers. 19. A number is formed of two figures of which the sum is 13. If 34 is added to the product of the two figures, the sum is equal to the number obtained by reversing the figures of the first number. Find the number. 416 Quadratic Equations 20. The diagonal of a rectangle is 65 inches. If 9 inches are added to the width and 3 inches subtracted from the length of the rectangle, the diagonal remains the same. Find the di- mensions of the rectangle. 21. The diagonal of a rectangle is 89 feet. If each side of the rectangle is diminished by 3 feet, the diagonal will be 85 feet. Find the length of each side. 22. The hypotenuse of a right-angled triangle is 35 feet. If the shorter side is diminished 5 feet and the longer side in- creased 2 feet, the hypotenuse will be 1 foot less. Find the two sides of the triangle. 23. Two square gardens have a total area of 2137 square yards. A rectangular lawn of which the dimensions are equal respectively to the sides of the two squares has an area of 1093 square yards less than that of the two gardens together. Find the sides of the two squares. 24. The sum of the areas of two circles is 13,273.26 square inches, and the sum of their radii is 79 inches. Find the two radii. 25. The sum of the surfaces of two spheres is 1000 square inches, and the sum of the radii is 12 inches. Find the two radii correct to two decimal places. (The surface of a sphere is AvB 2 square inches if R is the number of inches in radius.) 26. The sum of the surfaces of two spheres is 14,388.53 square inches, and the difference of the radii is 9 inches. Find the radii. 27. The sum of the volumes of two spheres is 14,778.0864 cubic inches and the sum of the radii is 20 inches. Find the radii correct to two decimal places. (The volume of a sphere equals | ttR s cubic inches if R is the number of inches in the radius.) 28. Three numbers are such that if we take the product of them two at a time the results will be 240, 160, and 96. What are the numbers ? Problems 417 29. The sum of three sides of a right-angled triangle is 208 feet. The sum of the two sides about the right angle is 30 feet longer than the hypotenuse. Find the lengths of the three sides. 30. The product of the sum and the difference of two num- bers is a; the quotient of the sum divided by the difference is b ; find the two numbers. 31. A rectangle whose dimensions are 6 inches and 10 inches is to be doubled in area by increasing the length and width by additions proportional to the present dimensions. Find the necessary addition to both dimensions. 32. Solve problem 31 if the area is to be made four times as great. 33. The dimensions of a rectangular piece of tin are in the ratio of 3 to 5. Two-inch squares are cut from the corners and the sides and ends are turned up to form a box. What were the original dimensions of the rectangle if the box holds 88 cubic inches ? XXIV. GRAPHICAL SOLUTION OF EQUATIONS 570. Graphical Solution of Equations Containing One Unknown Number. In § 369 a system of two linear simultaneous equa- tions was solved graphically. A linear equation in one un- known can be solved graphically. (Review §§ 356, 358, 359.) V 1 0/ i / ■ .V' .i X _ t 5 -i I - i I 1 . \ 1 -1 / / -9 1 / -a i / A / / —6 / B / V 571. Consider the equation 2 x — 3 = 0. For definite values of x, values of 2 x — 3 may be found and tabulated as follows : Use the pairs of values of x and 2 x — 3 as the coordinates of points and draw the line BC through these points. This line is the graph of 2 x — 3. 1 This chapter may be omitted, if desired, without interruptiDg the sequence of the work. 418 X 2z-3 - 1 -5 o — f J 1 - 1 2 1 3 3 4 5 Graphical Solution of Equations 419 This graph crosses the aj-axis at the point A whose coordi- nates are (f, 0) ; that is, at the point where the graph crosses the a>axis 2x-3=0 and x = f . The abscissa of the point of intersection of the graph of 2 x — 3 with the jr-axis is the root of 2 x — 3 = 0. 572. Consider the sec- ond degree equation X 2 _ x _ 6 = 0. We shall tabulate values of x and the correspond- ing values of x 2 — x — 6, and use these numbers as the coordinates of points. This will give us the graph of x 2 — x — 6. The intersections of this graph with the cc-axis are points whose abscissas correspond to ordi- nates 0. Since the ordinates are the values of x' 2 — x — 6, the abscissas whose ordinates are must represent the values of x for which x 2 — x — 6 = 0. In the figure these values of x are 3 and — 2. These numbers are the roots of the equation x 2 — x — 6 = 0. The result may be checked by solving this equation by one of the algebraic methods. _ 1 ±^/2E 2 1 o 7 | I / 1 i 1 / \ / \ \ •> 1 \ / \ J f \ X' X -3 -i - 1 l 1 3 4 \ / ' / \ — ? / -fl -| i / -8 / V X x 2 - x - 6 4 6 3 2 -4 1 -6 - 6 - 1 -4 - 2 -3 6 Thus, 3 and - 2. The graph just obtained is the graph of the expression x 2 — x — 6, or of the equation y — x 2 — x — 6. 420 Graphical Solution of Equations EXERCISE 573. Solve graphically as in § 572 the following equations : 1. x 2 — x — 8 = 0. 6. 2 x 2 — 3 x — 10 = 0. 2. x 2 + x-8 = 0. 7. 2 a 2 + 2 a; -10 = 0. 3. z 2 + s -i / s — o -s y / — 1 / Y\ EXERCISE 577. Solve graphically : 1. x + y = 5, 4 x — 3 y = 6. 2. x + 2y = 5, 2x-y = 0. 3. z + 2/ = 0, 3x-2/ = 2. 4. 2x — y = 5, 3x + 2y = -3. 5. x = 2ij, x + 2y = $. 6. 5 a + 4 2/ = 22, 3a; + y = 9. 7. 7x-2?/ = 3l, 4a> + 3y= — 3. 8. 6x + lly = -2$, 5y.-18*«-& 9. 6# + 22/ = - 3, 5# — 32/ = — 6. 10. 4 a + 15?/ = 7, 14 £ + 6?/ = 9. Graphical Solution of Equations 423 578. Consider the second degree system : # 2 - V = 4, x + 2y = 3. o-2-y = 4. (1) x + 2*/ = 3. (2) x y X y -4 a ±1 -3 1 1 ±2 3 ±3 5 -3 3 Locating the points tabulated and drawing the graphs we have a curve for equation (1) and a straight line for equation (2). The intersections are points whose coordinates satisfy both equations and therefore give the roots of the system. The roots are approxi- mately 86 = 2.1, -2.7. y = .4 f 2.8. If solved by the usual method, we find x = 2.1+, -2.6+. y = .45+, 2.8+. The student is not to understand that the graph- ical method of solving a system of simultaneous equations is to replace the algebraic method. The algebraic method is gen- erally much shorter than the graphical method. However, the graphical method of representing equations plays a very im- portant part in higher mathematics and in the applications of mathematics to problems of physics and engineering. It may also be noted that the algebraic methods do not 11 Y 1 (1)1 (1) 11 L _4 J ^S -3 l ZW(2) r ^^ 2 ' C ^^^ u X~ - 1 ^ / X ^t x ' J_ S^,_ x -y -r -i oi /2 F^-J V =1 I * X t U- -2 — 4 t 1 3^=* -t \ 7 ^./ *" 424 Graphical Solution of Equations always furnish the solutions of simultaneous quadratics (§ 551). The graphical method can generally be depended upon to give good approxi- mations to the real roots in such cases. Solve graphically : x 2 + y 2 = 16, x 2 — y 2 = 4. z 2 + y 2 = 16. (1) X y ±4 ±1 ± 3.87+ ±2 ± 3.46+ ±3 ± 2.64 ±4 ±5 imag. **-i 2 = 4. (2) X y imag. ± 1 imag. ±2 ±1 4- 3 ±4 ± 3.46+ ±5 ± 4.58+ The intersections give the roots approximately as follows a; = 8.1, 3.1-, -3.1, -3.1. y = 2.5, -2.5, 2.5, -2.5. X" J 7 ~F \m 4 (8)7 i 7 X 4- ^v / ^5 / 7 1 \ r \ X' _ ± X 5 — 'i H J -j J 2 - 1 C i l 2 3 4 5 _J 7 -2 r =8 -a- -4 1 Graphical Solution of Equations 425 The algebraic solution gives x = Vl\ -VlO or 3.16+, 3.16+, -3.16+ -3.16+ y=±V6, ±V6 or 2.44+, -2.44+, 2.44+, -2.44+. The curve for equation (2) is a hyperbola. The curve for a second degree equation in two unknown numbers is, in general, a circle, a parabola, an ellipse, or a hyperbola. 579. Imaginary roots cannot be found by this method. The presence of imaginary roots is indicated by a failure of the graphs to intersect. Thus, if we attempt to solve the system l Y [» (0 \ / \ / \ / \ / / / A" / - I f 1 1 / 3 '. \ 5 \ / \ / (2) -9 f - *s 4 -5 Y' x 2 — y — 4, (1), x — y — 5, (2), we shall find that the graphs have no common points. The graph of the first equation is shown in § 578. The second gives the line (2) as shown in the figure. The algebraic solution of this system gives _l± 9 2 426 Graphical Solution of Equations EXERCISE 580. Solve graphically : 1. aj« + y*=:9, (Circle.) 2. a* + y* = 9, x - ?/ = 0. & + y » 0. 3. a* + 2 y 2 _ 2 a; = 15, (Ellipse.) sb + 2 y sb 1, 4. a 2 -|- # 2 = 4, y = x — 2 V2. Solve example 4 also algebraically. 5. a 2 + 22/2_2a=15, aj8 _ 2 2/2 = _ 7. (Hyperbola.) 6. («-l)2-f-( 2/ _l)2 = 6, (Circle.) x - y = 0. 7. a 2 + a*/ + ?/2 =9, 8. « 2 + y = 7, a* + ?/ = 9. ^ + 2/2 = 11. Try to solve example 8 algebraically. XXV. THE PROGRESSIONS ARITHMETICAL PROGRESSION 581. Series. A succession of terms formed according to some definite law is a series. Thus, I, I, \ ••• and a, a 2 , a 8 ••• are series. What is the fourth term of each? 582. Arithmetical Progression. A series in which each term after the first is found by adding a constant quantity to the preceding term is an arithmetical progression (A.P.). 583. Common Difference. The constant number added is the common difference. The common difference is found by sub- tracting any term from the term immediately following it. Thus, 1, 3, 5, 7 •••, and 12, 8, 4, 0, — 4, — 8 ••• are arithmetical pro- gressions. In the first, 2 is the common difference and is added to each term to form the next ; in the second, — 4 is the common difference and is added to each term to form the next. ORAL EXERCISE 584. What is the common difference in each of the following series ? 1. 7, 11, 15, 19, — . 5. a — x, a, a -f- x, 2. 5, 8, 11, 14, .... 6. a — 3 d, a — d, a + d -. 3. i, 6J, 121 l.8|, .... 7. a, b,2b - a, « , 4. a, a -f d, a + 2 d, a -f 3 d, — . 8. 1, a, 2 a — 1, 9. Form the next two terms in each of the series in examples lto6. 427 428 The Progressions 585. Last Term. In the arithmetical progression let a represent the first term, d the common difference, n the number of terms, I the last term, and s the sum of terms. Then, if we examine the series, First term Second term Third term Fourth term nth term a, (a + d), (a + 2e*), (a + 3d), ... a +(n-l)d, we notice that in any term the coefficient of d is one less than the number of the term in the series. Hence in a series of n terms, the nth term being the last, 7=a+(n-l)d. (A) 1. Find the 7th term of the series 4, 2, 0, - 2, .... Solution. In this series a = 4, d = — 2, n = 7. .-. Z = 4 + 6-(-2) = -8. 2. Find the first 4 terms and the last term, when a = 2, d = |, w = 8. Solution. The first four terms are 2, 2f , 2f, 2$, Z = 2 + 7 • f = 4. 586. Sum of the Terms. To find the sum of a number of terms in arithmetical progression : Write the sum of the series in the usual order, (1), and in reverse order, (2), and add the two equal series. (1) S = a +(a + d) + (a+2e*) + (a + 3d)+ ••• +(J-d) +1 (2) S=l +(l-d) +(Z- 2d) +(l-3d) + .» +(g + d)+a 2 S = (a + l) + (a + I) +(a + I) +(a + l) + ... +(a + J) + (a + = n(a + 0. (Why?) Arithmetical Progression 429 Substituting the value I = a + (n — l)d from (A), we can get a formula for the sum in terms of a, n, and d. S = 5[8a + (n-l)d]. (C) 1. Find the sum of the terms of the series, 2, 5, 8, 11, — , to 12 terms. Solution. I = 2 + 11 • 3 = 35. fl = ? (a + Z) = 1? (2 + 35) = 222. A A 2. Find the sum of the series 4, 2, 0, — 2 — , to 20 terms. Solution. In the series a = 4, d = — 2, and w = 20. Hence, using formula (C) , S = 2 2 ° ■ [2 • 4 + (20 - 1) ( - 2)] = 10(8 - 38) = - 300. 3. The first term of a series is 5, the last term is 161, and the sum of the series is 3320. Find the number of terms and the common difference. Solution. Using formula (5), 3320 = - (5 + 161) = 83 n. A .-. n = 40. Using formula (A), 161 = 5 + 39 d. .\d = 4. 587. An arithmetical progression can be completely deter- mined if any two of its terms are known. The 6th and 15th terms of an A. P. are 14 and 32, respec- tively. Find the 20th term. Solution. a + 5 d = 14, and a + 14 d = 32. . •. d = 2 and a = 4. Hence the 20th term = 4 + 19 . 2 = 42. 588. Arithmetical Mean. When several quantities are in A. P. the terms between the first and last terms are the arithmetical means between them. 430 The Progressions The arithmetical mean between two numbers is equal to one half their sum. Proof. Let a and b be two numbers and A their arith- metical mean. Since a, A, b are in arithmetical progression, b - A = A - a, (Why ?) otA = ... 590. If the three numbers given and the number required are not all found in either (A) or (B) alone, these formulas may be treated as a pair of simultaneous equations after the proper substitutions have been made. Given d = 2, I = 20, S = 108 ; find a and n and the series. Solution. From (4), 20 = a + (n - 1 )2, or 22 = a + 2 n. From (JB) , 108 = - (a + 20) or 216 = an + 20 n. 2i This gives us the simultaneous system a + 2 n = 22, aw + 20 n = 216. Solving, a = 22 - 2 w. 216= (22-2w)n+20n. 2n 2 -42n + 216 =0. n 2 _ 21 n + 108 = 0. ( W _9)(w-12)=0. .-. n = 9 or 12. When n = 9, a = 4. When n = 12, a = - 2. The series is either 4, 6, 8, ••-, 20, or - 2, 0, 2, 4, •••, 20. EXERCISE 591. 1. Show that the three numbers, x — y,x,x + y form an A. P. Similarly f or x — 3 y, x — y, x + y, x + 3 y. 2. a = — 3, d = 2, n = 8 ; find Z and #. 3. a = 3, d = 3, Z = 15 ; find n and #. 4. a = 4, 7i = 12, I = 26 ; find d and A 432 The Progressions 5. d = I-, n = 3, I = 2 ; find a and & 6. a = 15,d = -i, # = 1371; find n and Z. 7. a = 4, » = 15, £ = 270 ; find d and Z. 8. d = 2, n = 15, S = 270 ; find a and Z. 9. a = 10,1 = 37, 5 = 235 ; find d and n. 10. d=-2, J = -24, # = -144; find a and n. 11. n = 13, I = 41, # = 299 ; find a and d. 12. Find 4 arithmetical means between 5 and 18. 13. Insert 6 arithmetical means between — 5 and 13. 14. The snm of three terms of an A. P. is 45 ; the sum of the squares of the terms is 773. Find the series. Note. Use x — y, x, and x + y to represent the series. 15. How many times does the clock strike in 12 hours ? 16. Find the sum of the first 20 odd numbers. 17. Find the sum of the first 20 even numbers. 18. Show that the sum of the first n natural numbers is n(n + 1) 2 19. If you save 1 ^ today, 2 ^ tomorrow, 3 ^ the next day, and so on, how many days will elapse before the total savings amount to $ 10 ? 20. The fifteenth and twenty-eighth terms of an A. P. are respectively 12 and 19. Find the first and the fiftieth terms. 21. Insert four arithmetical means between 9 and 11. 22. The sum of the first 8 terms of an A. P. is 64 and the sum of the first 18 terms is 324. Find the series. 23. The sum of the first 7 terms of an A. P. is 7 and the sum of the next 8 terms is 68. Find the series. 24. Between 6 and 10, there are 12 numbers so that the whole series of 14 numbers forms an A. P. What is the sum of the series ? Arithmetical Progression 433 25. The sum of the third and fifth terms of an A. P is 32 ? and the sum of the fourth and tenth terms is 50. Find the first term, and the sum of the first 20 terms. 26. Twenty potatoes are laid out in a straight line one yard apart. How far must a boy run to pick them up and bring them, one at a time, to a basket placed in the line and one yard from the first potato ? 27. A freely falling body falls | g feet the first second, f g feet the second second, %g feet the third second, and so on. How far will it fall in t seconds ? 28. If g = 32.16 feet, through what distance does a body fall if it reaches the ground in 6 seconds ? How far does it fall in the 6th second ? 29. If a ball is dropped from the top of Washington Monu- ment, 550 ft. high, how long does it take to reach the ground ? 30. How long does it take the ball in problem 29 to get halfway to the ground ? 31. In Italy 24-hour clocks are used. How many strokes does such a clock strike in a day ? 32. In an A. P. of ten terms the product of the first and last terms is 70 and the sum of all is 95. Find the series. 33. How many numbers of 2 figures are divisible by 3 ? (a = 12,1 = 99, d = ?) 34. Find the sum of all numbers of two figures each that are divisible by 3 ? 35. What is the sum of the first 50 multiples of 7 ? 36. The sum of n terms of the series 2, 5, 8, •••, is 950. Find n. 37. The sum of n terms of the series 87, 85, 83, •••, is the same as the sum of n terms of 3, 5, 7, •••. Find n. 434 The Progressions GEOMETRICAL PROGRESSION 592. A geometrical progression (G. P.) is a series in which each term after the first is derived by multiplying the preced- ing term by a constant multiplier called the ratio. Thus, 3, 6, 12, 24, ••-, and 36, - 6, 1, — £, •••, are geometrical pro- gressions. The ratios are respectively 2 and — \. 593. Ratio. The ratio (r) is found by dividing any term by the term immediately preceding it. ORAL EXERCISE 594. What is the ratio in each of the following series? 1. 2, 6, 18, 54, .... 5. 1, V2, 2, .... 2- 12,6,3,|,.... 6 . «,6, ^, .., 3. 5,-10,20,-40. i * i . _L, J., _L, ". 4. a, ar, ar 2 , ar 3 . 8. 1, i, — 1, •••. 9. Form the next two terms in each of the series in ex- amples 1 to 8. 595. Last Term. If a is the first term, I the last term, r the ratio, and n the number of terms, we have the following from the definitions : 1st term 2d term 8d term 4th term 5th term nth. term a ar ar 2 ar 3 ar 4 ••♦ ar n_1 By examining this series we notice that the exponent of r is always one less than the number of the term in the series. Hence, in a series of n terms, the nth term being the last, / = ar"-*. (A) Thus, the 8th term of 3, f , f, •••, is 3 • Q) 7 = T f*, and the last term of 1, 5, 25 to 10 terms is I = 1 • 5 9 = 5 9 . Geometrical Progression 435 596. Sum of the Terms. To find the sum of the terms in a geometrical progression : Write the sum of the series, S = a + ar + ar 2 + ar 3 + ••• + ar"" 1 (1) (1) x r, rS = ar + ar 2 + ar 8 + • • • + ar"- 1 + ar" (2) (l)-(2) S-rti=a -ar" ,.S = ^=^or^^. (J) 1 — r r — 1 Since or" -1 = Z or ar n = rl, the formula may be written a formula that is sometimes useful. Find the sum of 6, 3, 1-J, •••, to 10 terms. Solution, a = 6, r = |, n = 10. .-.from (2?), we have £ = G ~ 6 (i) 10 = g^ff = lif 5f. * y 1 - 1 256 * 5 * 597. Geometrical Mean. If several quantities are in G. P., the terms between the first and last terms are the geometrical means between them. The geometrical mean between two numbers is the square root of their product. Proof. Let G be a geometrical mean between a and b. — = — , for each fraction equals the ratio of the series. a G .-. G 2 = ab and Q = ± Vo&. Find a geometrical mean between V8 and V2. Solution. G = ±v V8 • V2 =± 2. The student should notice that the geometrical mean and the mean proportional are the same. See § 326. 436 The Progressions We may also insert several geometrical means between two given numbers. Insert 4 geometrical means between 16 and ^p. Solution. Here a = 16, n = 6, (Why ?) I = *$* From (A), ^ = 16^. r* = W- r = f. Let the student complete the solution. 598. Application of the Formulas. The formulas in geometri- cal progression to be remembered are : l = ar n ~K (A) s = a^a 0T a-ar^ t r— 1 1 — r S = ^or^. (C) r- 1 1 - r In (4), we have a, w, r, Z. In (B), we have a, n, r, JS. In (C), we have a, r, Z ? S. 599. The suggestions of § 589 hold here except that we may not be able to solve for n or r. In most cases the use of n as an unknown introduces equations of a type wholly new to the pupil, that is, with the unknown number an exponent. When r is unknown it may become necessary to extract roots higher than the second or third. Both these problems can be solved by inspection in some simple cases ; for example, 2 n = 8, .-. n = 3 ; and r 5 = 243, .-. r = 3. Logarithms may also be used in such solutions. 1. Given a = 1, I = 2, n = 4 ; find r. r, a, Z, and n are all in formula (A). Hence we may write 2 = 1 • r 3 . r3 = 2. The series is 1, y/2, y/4, 2. Infinite Geometrical Series — Repeating Decimals 437 2. From the data in example 1 find S. Both formulas for S involve r. Using the result obtained in example 1, we may write from (c), » = 2v / 2-l W-l ■ The result may be left in this form. EXERCISE 600. 1. Find the 6th term of 1, 2, 4, 8, .... 2. Find the sum of 1 + 2 + 4 ••• to 6 terms. 3. Insert 3 geometrical means between 5 and 8. 4. Find the difference between the arithmetical mean and geometrical mean of 1 and 2. 5. The fourth term of a G. P. is 54 and the fifth term is 486. Find a and r. 6. Find the sum of the first five terms, when a = 1, r = f . 7. Find a fraction whose value is l + aj + a£-fa? + .-.a; 15 . 8. Find the sum of the first 5 terms of 1, -J, J, •••. 9. Find the geometrical mean between 14 and 686. Be- tween 38 and 123 to two decimal places. 10. If Z= 128, r = 2, n = 7, find a and S. 11. If a = 9, I = 2304, r = 2, find S and n. 12. If a = 2, Z = 1458, S = 2186, find r and w. INFINITE GEOMETRICAL SERIES — REPEATING DECIMALS 601. The student will recognize the identity £ = .3333... This means that the repeating decimal .333 ••• approaches in value the fraction £. 438 The Progressions It is evident also that the repeating decimal equals .3 -h .03 + .003 + -. This is a geometrical series with a = .3, r = .1 and n indef- initely large. Using formula (B), we have q _ a — ar n _ a ar n 1 — r 1 — r 1 — r .3 .3 x .l n l-.l l-.l _1 .3 x .1" 3 .9 O 1 The second term, — x.l n =-x.l n , becomes smaller as n becomes larger. Thus, when w = 6, -x .l n = - of .000001. When n becomes infinitely large the term becomes so small that it may be neglected, and we have the sum of .3 + .03 + .003 + ••• indefinitely = -. o 602. Formula for an Infinite Geometrical Series. The result found in the last article may be generalized in the following statement : When r is less than unity and the number of terms is infinite, ■n & a — ar n a ar n Proof. S = 1 — r 1 — r 1 — r When r < 1 and n is infinitely large, r n is smaller than any assignable number and therefore the term may be neglected. This leaves S= a 1-r Infinite Geometrical Series —Repeating Decimals 439 1. Find the value of l-}-| -f \ ♦•• to an infinite number of terms. i a = 1, r = £, £ 1-1 2. Find the value of 3.2727 .... Solution. Note that 3 is not part of the infinite geometrical series that follows it. First find the value of .2727 • •• = .27 + .0027 -f •••• Here a = 27, r = .01. B _ .27 _.27_3 | 1 - .01 .99 11 ' .-. 3.2727.-. =3^. EXERCISE 603. Find I and S : 1. When a = 2, r = 2, n = 7. 2. When a = 5, r = 4, n = 9. 3. When a = 6, r = f , n = 6. 4. a = 40, r = f , n = oo ; find S. (ao is the symbol for an infinitely large number.) 5. a = 9, r = |, ?i = oo ; find £. 6. If the first term of a geometrical series is a, and the second term is b, what is the ratio ? 7. What is the sum of the first four terms of the series in example 6 ? 8. If a > b and n is infinite, show that the value of S in example 6 is a — b 9. Find the sum of the infinite series -7 + -, r— m + 1 (m -f l) 2 when m > 0. (m + 1) 3 10. What is the significance of making m > in example 9 ? Find the series and the answer in 9 when m = 2. 11. What common fraction reduces to the repeating decimal .777-.? 12. Find the fractional form for 3.25757 •••. 440 The Progressions Some of the problems that follow will require the use of the formulas of A. P., and some will be in G. P. 13. How many numbers of two figures each are exactly divisible by 7 ? 14. How many numbers of three figures each are multiples of 7? 15. How many numbers under 1000 are powers of 2 ? 16. What is the sum of all the three figure numbers that are multiples of 5 ? 17. In an A. P., S = 20 + 13 V2, n = 10, I = 2.6 V2 ; find a and d. 18. What kind of series is - 2, V2 - 1, 2 V2, ... ? Write the next two terms. 19. Given a + V2, 2a + 2, 3a + 2V2, 4a+4-... Show that the sum of this series to 10 terms is 55 a + 31(2 -f V2). 20. Indicate the sum of n terms of the series in example 19. 21. The first term of an infinite geometrical series is 3 and the second term is 2. Find the sum. 22. Find the 10th and 15th terms of 2LzJt, a-2 a-S ^ a a a (Williams College.) 23. Find the sum of 10 terms of 6, — 4, f ... and the sum of 12 terms of - 5, - 1, 3 .... (Williams College.) 24. Determine whether 3§, 4|, 6f ••• are in A. P. or G. P. and find the sum of the first 6 terms by the general formula. 25. Find the geometrical mean between 6 -f V2 and 6 — V2. Find the arithmetical mean between the same numbers. 26. The side of an equilateral triangle is 10 inches. The midpoints of its sides are joined, forming another equilateral triangle, and this process is repeated indefinitely. Find the sum of all the lines. Infinite Geometrical Series — Repeating Decimals 441 27. Lines are drawn joining the middle points of the sides of a square, thus forming a second square, and the middle points of the sides of this square are joined. If this process is repeated indefinitely, find the sum of all the lines, if a side of the origi- nal square is 6 inches. 28. What number added to each of the numbers 1, 8, 22 will make a G. P. ? 29. What distance will an elastic ball travel before coming to rest if it falls 20 feet and rebounds each time f of the dis- tance of its last fall indefinitely, that is, until it comes to rest ? 30. The difference between two numbers is 48. Their arithmetical mean exceeds their geometrical mean by 18. Find the numbers. (Yale.) 31. Find the sum of n terms of 9 - (5-2^) 6 . 19 . (l + V^3)3. 10. (tf-y) 1 . 20. (3+V^5) 7 . 11. (f-fa) 7 - 21. (a -\-biy- (a -bi)*. 12. (±-3yy. 22. (1 + Va) 7 - (1 - Vz) 7 . 23. What are the signs of the terms in the expansion of (a -b) 7 ? of (-a -{-b) 7 ? of (-a +6)6? of {-a -by? of (_ a _6)6? 24. Expand (2-3i) 5 . 25. Find the first 4 terms and the last term of (a^ — 6*) 20 . 26. Find the first fonr terms and the last term of (a + b) m . 27. How many terms are there in the expansion of (a -f b) b ? of (a + b) 6 ? of (a + b) n ? 28. What are the exponents of a and b in the fourth term of (a + 6) 5 ? in the fifth term of (a + b) b ? in the fourth term of (a + b) 6 ? in the fifth term of (a + 6) 6 ? 29. What is the sum of the exponents of a and b in each term of the expansion of (a + b) 5 ? of (a + &) 6 ? of (a + 6) n ? 607. Binomial Coefficients and Exponents. The coefficients in the expansion of a -f 6 to any power, and the exponents of a and b in this expansion, are called respectively the binomial The Binomial Formula 445 coefficients and the binomial exponents of the expansion. These terms are used to distinguish them from the reduced results in cases when a, for example, is represented by 2 x* and b by a similar expression. The expansions should always be made first in binomial coefficients and exponents. (Why ?) An interesting relation among the binomial coefficients of successive powers of a binomial is shown in the following scheme known as Pascal's Triangle : The numbers in the first line are the coefficients of (a 4- b) ; in the second line of (a -f b) 2 ; in the third line of (a 4- b) 3 , etc. Any number in this scheme equals the number directly over it plus the number at the left of the one over it. The coeffi- cients of the 5th power of a + b are found from the coefficients of the fourth power as follows : 1+4 = 5; write 5 under 4 ; 4 + 6 = 10 ; write 10 under 6 ; etc. Let the student write the coefficients for n — 1 and n — 8. w = l 1 1 n = 2 1 2 1 w = 3 1 3 3 1 n =4 1 4 6 4 1 n = 5 1 5 10 10 5 1 w = 6 1 6 15 20 15 6 M 608. Any Required Term. Writing the binomial formula and numbering the terms, we may make a formula for any term. 1st 2d 3d 4th (a+6)»=a»+»a»-'6 + 2^1 a -W + ^ W 7^ ( "~ 2) a-W+.~ l. • Z 1 • Z • o 6th »(n-l)(»-2)(n-3)(n-4) + 1-2.3.4.5 a ° + 10th n(n-l)(n-2)-.(n-8) aB _ 969 ... I.2.3-9 T * A simple way to find any required term in the expansion of (a + b) n is to start with the exponent of 6. What is the exponent of b in the 2d term? in the 3d term? in the 4th term? How does it compare with the 446 The Binomial Formula number of the term ? In the rth term it is r — 1. In a similar way the exponent of a is n minus the exponent of b; that is, n — (r — 1) or n — r + 1. By further comparison of the coeffi- cient of any term with the number of the term it will be seen that the rth term is 1.2.3..(r-l) r ' 1. Find the 7th term of (1 a 2 — 2 b%) 12 . First write the binomial in the form [(Ja 2 ) + (— 2 6*)] 12 . Applying the formula for the rth term when r = 7 and n = 12, 3 4 we have ^'^'/^l^ a a 2 ) 6 (- 2 &*)6 = 924 oK&». 2. Find the middle term of (a? - Vy) 14 . How many terms are there in the expansion ? What is the number of the middle term ? 2 3 l-g^V/fl-V ^ 7 (-VF) 7 =-3432 *yVy. EXERCISE 609. ^Vwc? o^/y £/ie terms asked for : 1. The third term of (a -b) 9 . 2. The third term of (- a + z) 10 . 3. The third term of (a + Z>) 100 . 4. The sixth term of (x 2 — or 1 ) 8 . 5. The seventh term of (2 ar* -y - *) 6 . 6. The middle term of (a; + -\ * 7. The middle term of (3 x - $)». 8. The sixth term of (1 + x)\ The Binomial Formula 447 9. The two middle terms of f x ) . \ x ) 10. The term containing a,* 5 in (a — x) 7 . 11. The term containing a 5 in (a — x) 9 . 12. What term of what power of a -f b contains a?b n ? a?b 11 ? 13. Write the first four terms, and the last term of (2 a 2 — b%) 8 . 14. Find, in simplest radical form, the value of (V2 + V3) 4 . Write the first three terms, and the last term in the following : 15. (2 a* -by. 19. f-*-<2iA\ 16 ' (I -3 )* 20. (-*a-§by. 17. (1- fz) 6 . 21. (3-2a>)». 18. (— |a;-f fy) 5 . • 22. (-3 + 2z) 8 . 23. Find the 7th and 8th terms of (a + 6) 10 . 24. Find the 4th term of (a + 6) 11 . The 4th from the last. 25. Write the next to the last term of (3 a* — b*) w . 26. What is the exponent of x in the first term of (x -f ar 1 ) 12 ? in the second term ? in the third term ? Find the term that does not contain x. 27. Is there a term in (x -f a? -1 ) 11 that does not contain x ? 28. Find the last three terms of (V2 — b?f. XXVII- VARIATION 610. In numbers that are related to each other through mathematical equations, some of the numbers may be changing in value, while others may have fixed values. If a train travels at a uniform rate of r miles per hour, we may express the distance it has traveled after the lapse of any time by the equation, d = rt. In this equation d and t vary in value from moment to moment, but r is a constant, for by the conditions, the rate is uniform. 611. Variable and Constant. A number that is changing in value is a variable; a number whose value does not change is a constant. The formulas of algebra, geometry, physics, and their prac- tical applications, involve, in general, variables and constants. In the illustration just given, d = rt, d and t are variables and r is a constant. In A = 7ri?' 2 , A and B are variables and ir is a constant. 612. Direct Variation. If two variable numbers are so related to each other that through all their changes in value their ratio remains unchanged, one of these numbers varies directly as the other, or simply varies as the other. 613. Constant of Variation. The constant value of the ratio of the variable numbers in direct variation is the constant of variation. We may write d = rt, when r is constant and d and t are variables, in the form - =r. Then by definition, we have "distance varies as the t time." The student must remember that d, t, and r are abstract numbers. They represent the numerical measures of concrete magnitudes ; that is, d equals the number of miles traveled in t hours. 448 Variation 449 614. Notation. The symbol for variation is oc. a oc b is read " a varies as b" It is customary to use a letter with different subscripts to represent different values that a variable number has at dif- ferent periods of its variation. Thus, di, d 2 , d s •■• (read d-sub one, etc.) are used to represent the dis- tances traveled in the times ti, t*., h •••, respectively. 615. In agreement with this notation and the definition of variation, we have, for uniform motion, * = ,, d2 = ds = t( , h to t Z where r is the constant of variation, in this case the uniform rate of motion. It is evident that the constant of variation can be found in any particular case, if we know a set of corresponding values of the two variables. Thus, if d\ = 140 miles, and t\ — 4 hours, r = ^^ = 35, the number of miles per hour. 616. If a and b are two variable numbers, and accb, then a x : a 2 = b x : b 2 , where cti and b lt a*, and b 2 are sets of corre- sponding values of the variables. Proof. We have given accb. .-. ^-=k, and ^1— k, where k is the constant of variation. &1 &2 a l _a 2 b x b 2 . fit— 5s. 4 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 N 1 2 3 4 5 6 7 8 9 464 Logarithms Find log 3847. Mantissa of log 3850 = .5855. Mantissa of log 3840 = .5843 . 10 0.0012. Mantissa of log 3847 = .5843.+ ^ of 0.0012 = .5851. The difference between 3840 and 3850 is 10 ; the difference between the mantissas of their logarithms (.5855 — .5843) is 0.0012. Assuming that each increase of 1 unit between 3840 and 3850 produces an increase of 1 tenth of the difference in the mantissas, the addition for 3847 will be 7 tenths of 0.0012 or 0.00084. .5843 + 0.00084 = .5851. Therefore, the mantissa of log 3847 = .5851. EXERCISE 640. Find the logarithms of: 1. 1845. 2. 6.897. 3. 0.04253. 641. To find the number corresponding to a given logarithm : The number corresponding to a logarithm is its antilogarithm. The characteristic determines the position of the decimal point. (1) If the mantissa is found in the tables, the number is found at once. Find antilog 3.5877. The mantissa is found at the intersection of row 38 and column 7. \ antilog 3.5877 = 3870. (2) If the exact mantissa is not found in the tables, the first three figures of the corresponding number can be found and to them can be annexed figures found by interpolation. Find antilog 3.5882. log 3880 = 3.5888 log required number = 3.5882 log 3870 = 3.5877 log 3870 = 3.5877 10 0.0011 log req.no. - log 3870 = 0.0005 3870 + (A of 10) = 3874.54+. Use of Tables 465 The two mantissas in the table nearest to the given mantissa are .5888 and .5877, differing by 0.0011. The corresponding numbers, since the characteristic is 3, are 3880 and 3870, differing by 10. The difference between the smaller mantissa 5877 and the required mantissa 5882 is 0.0005. Since an increase of 11 ten thousandths in mantissas corresponds to an increase of 10 in the numbers, an increase of 5 ten thousandths in mantissas may be assumed to correspond to an increase of T 5 T of 10 in the numbers. Therefore the number is 3870 + (^ of 10)= 3874.54+. The last two figures are uncertain. EXERCISE 642. Find the antilogarithms of: 1. 2.9445. 3. 1.6527. 5. 1.9994. 2. 2.4065. 4. 3.7779. 6. 0.7320. 643. The cologarithm of a number is the logarithm of its reciprocal. The cologarithm of 100 equals the logarithm of rfo>> that is > ~ 2 - Since log 1 — 0, .-. log- = log 1 — log n = — log n, n therefore colog n = — log n. As the cologarithm of a number equals the logarithm with its sign changed, adding the cologarithm will give the same result as subtracting the logarithm. The former is sometimes more convenient. To avoid negative results it is often more convenient to add and subtract 10. 1. Find colog 47.3. In subtracting 1.6749 or any log 1 = 10.0000 — 10 other logarithm from 10, the result log 47.3 = 1.6749 may be obtained mentally by sub- colog 47.3 = 8.3251 — 10 tracting the right-hand figure from 10 and all the others from 9. 2. Find the value of 4 5371 x 29 466 Logarithms 4.PC9 v 93 l0g 5371x29 = l0g 452 + l0g 23 ~ l0g 5371 ~ l0g 29 = log 452 + log 23 -J- colog 5371 + colog 29. log 452 = 2.6551. log 23 = 1.3617. colog 5371 = 6.2699 - 10. colog 29 = 8.5376 - 10 . Adding 8.8243 - 10. antilog 8.8243 - 10 = 0.066728+ Theref0le 5^1 = °- 066728+: 3. Find 50*. log 50? = flog 50. log 50 = 1.6990. | log 50 = | of 1.6990 = 1.2742. antilog 1.2742 = 18.80. .-. 50* = 18.80. 644. Compound Interest. Problems in compound interest that involve long computations can readily be solved by means of logarithms. To find the amount (4) at the end of n years of a given sum of money (P) invested at compound interest at a given rate (r) : The amount of P dollars at compound interest, at the end of the first year is, A — P + rP = P(l + r). At the end of the second year, A = P(l + r)+ rP(l + r) = P(l + rf. At the end of the third year, A = i?(l + r)2 + rP(l + r) 2 = P(l + r) 3 . At the end of the nth year, A = P(l + r)»-i + rP(l + r)»~i = P(l + r)». Use of Tables 467 What will be the amount of $ 1500 for 12 years at 4 %, the interest being compounded annually ? Here A = 1500(1 + .04)" log A = log 1500 4- 12 log 1.04 = 3.1761 + 12 x .0170 = 3.3801. .-. A = $ 2405.55. EXERCISE 645. 1. Find from the tables the logarithm of each of the following numbers: (a) 74; (6) 129; (c) 2004; (d) 16.21; (e)9.547; (/).018; (g) .21 j (ft) ft; (i) ft; (j) 7|; »&■ 2. Find the logarithm of each of the following numbers : (a) 7 5 ; (b) 212 14 ; (c) 3.171 4 ; (d) 31.2; (e) 918.4; (/) .00084; to) 42.5 3 ; (ft) .1871 3 ; (i) .00427. 3. Find from the tables the numbers' corresponding to the following logarithms: (a) .7412; (b) 2.9983; (c) .9060; (d) .7033 ; (e) 4.9883 ; (/) 1.0881 ; (gr) 3.6538 ; (ft) 3.5051. 4. Perform the following operations by means of loga- rithms : (a) 256 x 311 ; (6) 451 X 215 ; (c) 7643 -r- 213 ; (d) 972 + 41 ; (e) 158 x ^39 ; (f) 7 4 x 4 11 ; (g) 615 x_53 ; (ft) 61 3 -h 17 4 ; (I) 19 3 x 8 10 ; (J) 17 V29 ; (ft) 41 ' 1 ^ 613 i (0 36« x (A)*- 5. At birth a child has $ 500 placed in the bank for him, to accumulate at 4 % compound interest till he is 21. What amount will he receive when he is 21 ? 6. The first Folio of Shakespeare, regarded as the most valuable book printed in the English language, was published in 1623. The original cost was £1 or approximately $5. The last copy offered for sale in 1912 brought $ 9000. One 468 Logarithms would naturally think that the purchaser of this first Folio in 1623 made a fine investment. What would an original invest- ment of $ 5 amount to in 1912 at 6 % compound interest ? 7. If at the beginning of the year 1, one cent had been invested at 4 % compound interest, what would the amount be in 1915 ? What would be the radius of a sphere of gold that would represent the value of the investment in 1916, if a cubic foot of gold is worth $ 362,900 ? 8. If log 2= .3010 find the value of x in the equation 2 X = 10. 9. Compute the value of 3 2 by means of logarithms. (Harvard.) 10. About 300 years ago the Indians sold Manhattan Island to Peter Minuit for $ 24. Suppose this money had been put out at compound interest at 6%, how much would it have amounted to at the present time ? 11. According to the will of Benjamin Franklin, the cities of Boston and Philadelphia each received £ 1000 in July 1791 to be invested at 5 % compound interest for 100 years. In July 1891 the total amount of the fund in Boston was $ 391,168.68 and in Philadelphia $ 100,000. How much should have been realized by the terms of the will ? (£ 1000= $ 5000.) 12. A chain of letters is started for the purpose of aiding an old railroad man who is ill. Number 1 sends a letter to each of 5 friends, each of them in turn sends a letter to 5 friends, and so on. If the chain ends with letter number 50 and each person who receives a letter sends 10 cents, how much does the man receive ? XXIX. GENERAL REVIEW 646. 1. If a = 3, b = 2, c = 1, find the value of each of the following expressions : (1) 2 a 2 - b 2 ; 2(a 2 _ 52) ; (2 a* _ &*)2. 2(a 2 - 6 2 ) 2 . (2) a&c-(a + 6 + c). (3) (a 2 + b*)(a + 6)(a - 6). (4) [a*+(b-c)a-bc](b-c). (5) V(a + 6 + c) a&c. 2. Verify the following identities : (2)^ = ^ + 9) 2 -^-?) 2 . (3) 1 + 2 + 3 + 4+.. > + n = n(?l 2 + ^ » (4) 1 + 3 + 5+ ... +(2n-l)=n 2 . 3. Solve the following problems by translating the verbal language of the problem into an equation with one unknown : (1) In five years a boy will be double the age he was five years ago. How old is he ? (2) I have as many brothers as sisters said a boy. And I, said one of his sisters, have twice as many brothers as sisters. How many brothers and sisters were there ? (3) Can there be three consecutive integers such that their sum is three times the smallest ? (4) The sum of three consecutive numbers is three times the middle number. What are the three numbers ? Does this problem lead to an equation or to an identity ? 4. Express in algebraic language the following theorems : (1) Thcproduct of two numbers is equal to the difference of the squares of their half sum and their half difference. 469 470 General Review (2) Every integer that is a perfect square diminished by unity is equal to the product of the number that is one less than its square root by the number that is one more than its square root. (3) The difference between the squares of two consecutive integers is an odd number, obtained by increasing by unity twice the smaller of the two numbers. 5. The lengths of the sides of a triangle are a = 5 inches, 6=4 inches, c = 3 inches. Indicate the semi-perimeter by s = , and find the area, A, of the triangle, if A = Vs(s — a)(s — b)(s — c). 6. If 2 s represents the perimeter of .a triangle and a, 6, c its sides, verify the following : (1) - a + 6 + c = 2(s - a). (4) a = 2 s -(b + c). (2) a -b + c = 2(s-b). (5) b = 2s-(a + c). (3) a + b - c = 2(8 - c). (6) c = 2s -(a + b). 7. Having given the trinomials r + 8 + t, r + s — t, r — s + t, — r -f s + t, from the sum of the first three subtract the sum of the last three increased by the sum of the second and third. 8. If A=(p + q) + (r + s), B = (p + q)-(r + 8), C=(p-q) + (r — s), D=(p-q)-(r-s), find the value of A + B+C + D and of A x D by type multiplication. 9. Prove that [m — (p + q)+ r] — \m — [(p + q)— r] \ -f \m — (p + r)+q\ = m — p + q — r. 10. Apply the general formulas to the following : (1) (tf-4--Jj9; }m-|w. 18. Square each of the following : a 3 — a 2 -|-a — 1; x — y + z + 1; ax 2 -\- bx 4 c. 19. Verify the following identities : (1) (a 2 + 6 2 ) 2 = (a 2 - 6 2 ) 2 + (2 ab) 2 . (2) (a 2 4- b 2 + c 2 ) 2 = (a 2 + b 2 - c 2 ) 2 + (2 ac) 2 + (2 6c) 2 . (3) [(ti 4- 2) 2 - (n + l) 2 ] - [(n + l) 2 - n»] = 2. (4) Show that the identity (n 4- l) 2 — » 2 = 2?* 4- 1 expresses that the difference between the squares of two consecutive integers is always an odd number. 20. Evaluate each of the following expressions : (1) 2 x 5 4- 12 -f- 4- 7 +6-tAl. 6 (2) 6 x 7 - 3 2 x 5 4^8 X 5 - 7. 21. Extract the square roots of the following polynomials : (1) 3a 2 -2a + l-2a 3 4-« 4 . (2) 16a 8 f 9y s -S0x 2 f + 4,9xY — i0x 6 y 2 . 472 General Review 22. Factor into prime factors : (I) 7 a 4 + 7 a 2 b 2 - 14 a 3 b. (2) m 2 (a - b) + n 2 (6 _ a ). (3) 7pga; 2 - 42_pga; + 63 pg - 7pra 2 + 42prx - 63pr. (4) a;y - x + y — 1. (5) xa' + a#" + x 2 + «V. (6) a,* 3 4- ?/ 3 — a; 2 ?/ — a,^ 2 . (7) a 2 b + 6 2 c + c 2 a + a 2 c + 6 2 a + c 2 b 4- 3 a&c. (8) 7 a; 2 -28 a 4 . (9) — 2uv—u 2 — v 2 —2uw — 2vw — w 2 . (10) x*(x 2 -y 2 )-f (x 2 - y 2 ) -xy(x- y) 2 (x + y). (II) (a 2 + b 2 ) 2 - (a 2 - 6 2 ) 2 . (12) x(x - 1) - (x - l) 2 + & - 1. (13) a 3 — a? 4- a 2 a; — aa; 2 — a 4- a\ (14) ^ _ aty. (15) ^ + 10 a^ + 21. (16) 5 a 2 a; 4 - 5 a 4 a; 2 - 5 a 2 6 2 a; 2 4- 5 a A b 2 . (17) a 4 — 4 nx 3 + 6 n 2 a; 2 -4A + n 4 . (18) 3a 3 - 16a; 2 - 3a; 4- 16. (22) x 2 + 2ax - b 2 + a 2 . (19) a; 2 - 7 a; + 10. (23) x* - 3 a 2 a; 2 + a 4 . (20) 3 a; 2 + 12 x + 9. (24) a* - (a 3 + & 3 ).^ + a 3 6 3 . (21) 9 a; 2 - 12 x - 5. (25) ax 2 + (a + b)xy + by\ (26) 4(a6 4- cd) 2 - (a 2 + 6 2 - c 2 - d 2 ) 2 . 23. Solve the following equations by factoring : (1) x 2 -(a + b)x + ab = 0. (6) a? - 2a; 2 - 4a;4-8 = 0. (2) p 2 + 3i) + 2 = 0. (7) 3a? + 7a; 2 = 3a; 4- 7. (3) t* - t 2 = 9 1 - 9. (8) v 3 + v 2 - v - 1 = 0. (4) z 2 + 2 - 30 = 0. . (9) s 4 4- s 2 - 12 = 0. (5) x 2 4- £a> - £ = 0. (10) & 3 4- k 2 = 0. (11) Find a number such that if 3 and 5 are subtracted from it in turn, the product of the two remainders is 120. (12) Find two numbers such that their difference is 2 and the sum of their squares is 130. 24. Find the H. C. F. of a; 2 - 3 a; 4- 2, a; 2 - 2 a; 4- 1, x 2 + x-2. 25. Find the H. C. F. of x 2 + 2a;4- 1, x 4 - 10a; 2 4- 9, x* 4- 2a? 2 -5x -6. General Review 473 26. Find the L. C. M. of x 2 +(a + b)x + ab and x 2 + (a — b)x — ab. 27. Find tfie L. C. M. of p 3 + q*, p 2 - q 2 , p 2 + 2pq + q\ 28. Simplify the following fractions : ( x + a)2_(p + c y 30 a; 2 - 18a;- 12 1 ; (x + b) 2 -(a + c) 2 ' U 16a; 2 + 4a; -20 ' a;2_6a;H-5 ^ 2a 3 - 7a; 2 + 7x - 2 { } x 2 -11a; + 10' ^ ' 3a? - 10a; 2 + 9a; -2* 29. Reduce each of the following groups of fractions to groups having a common denominator : (I)-,-,-- (2) m p a b 7 c x ' a + 6' a-b' a?-b r (S) P + 1 P- 1 l>-7 V ; p 2 -8p + 7' p 2 -6p-7' p 2 -l o^ Q v 4.1, 4. a; 2 + 8 a; + 15 x — 1 4 30. Show that a; 2 + 7 a; + 10 a; + 2 a; + 2 31. Show that =- — 1 ; ?\ = -- -.1 a(a — l) a a n + 1 n 1 32. Show that 2n + 3 2» + l (2n + l)(2n + 3) 33. Show that ^ ^- = ^ x -%. From this relation b a+b b a+b find two numbers such that their product is equal to' their difference. 34. Simplify the following : 2c? K^J c 2 - -±d 2 N a-36' (2) e + iX' '-i)( i+ a; 2 \ (3) +_»4.lYl-S — > 474 General Review <*» (!-!)•• , 6) g» + 3a? + 2 x (5) (2+tf x 2 + 6 a; -f 9 ».+ 3 x (7) — +— • V ; 6a 2 3a x 2 + 4x + 3 # 2 -f 4 a; + 4 (10) a+^L. 6 + (8) (9) x 2 — y 2 . »+;jf a 2 + 2a;zH-z 2 cc + 2 (fyP — X* ^ ga;2-}-a*» a 3 — a? 3 ' a 2 +aa;-|-a; 2 ' (11) (12) z + ac 6 + c 35. If 2/ 1 + z 2 and z = 1 + x , express y in terms of x. 36. If x -f- - = 8, show that x 2 + — = s 2 — 2 and a? + — a? = s3-3s. 37. Show that «j a.* -2/ 8 » 4 + 2/7 « 2 + ?/ 2 J » + 2/J » — # 1. 38. How much water must be added to 80 pounds of a 5 per cent salt solution to obtain a 4 per cent solution ? (Yale.) 39. Simplify x + y- x + y- xy x + y. x 3 — y 3 x l — y l (Cornell.) 40. What is the price of eggs when 2 less for 24 cents raises the price 2 cents a dozen ? (Yale.) 41. What values of x will make the product (x — a) (x — b) (x — c) equal to zero ? General Review 475 42. Factor ar 3 -f- 10 x 2 -f 21 x and indicate the values of x that will make the expression zero. 43. Simplify the expression : ftx(x + l)(x + 2)+ aj(* - l)(a? - 2)]+ f(* - l)x(x + 1). 44. Divide (a 3 - l)a 3 - (.t 3 + x 2 - 2)a 2 +(4 x 2 + 3 ar + 2)a -3 (a + 1) by (x - l)a 2 -(x - l)a + 3. 45. Show that J (a 2 + ?/ 2 ) + z 2 — J. a?y + xz — yz becomes (y — z) 2 or {z — y) 2 when —x = y. 46. By what transformation can a(x — b) be put into the form (a -f b)x — (a + x)b ? 47. Solve the following equations : (1) 2(x - 1)= 6. (3) 3(a - 5)+ 8 = 17. (2) 13(12 - z) = 14. (4) 5 a + (7 -2x)= 11. (5) 8(37 -5 a) =4(3 a- 17). (6) 28 + 2?/ - 16?/ -62/ -12 + 2?/ = 0. (7) 9x + 22 - 2x = 193 - 22 x - 84. (8) 5x -.3 a = 4.5 a? + 2. (9) .9 05 - 1.5 a = a- 3.5. (10) .25 * + .943 = 1.9 a? - 6.812. (11) .15 x + 1.575 - .875 x = .0625 x. (12) 1.111 -.Hilar = .3333. (13) .5x + 2-*x=Ax-ll. 48. Solve the following equations : (1) x + 5x-b =2a. (2) 3a + 2z-4b = 5z - b. (3) k(k + 3 aca + 3) = kx + 3 a&A; - A; 2 - ac&x. (4) (x-2a) 2 + (x + 2b) 2 =2(x-2c) 2 . (5)x-™=p. B±* (8) x p p q in (7) 2.+,. » +',. (9) 5x 86 476 General Review 49. Form a proportion with the numbers 75, 18, 27, 50. 50. Knowing three terms of a proportion, how can the fourth be found ? 51. Solve each of the following proportions for x: (1) ^ = 57. (3) 3.15: z = 6.75:20. K) 10 x' (4) a:3f = 4i:lli (2 ) P 2 -g 2 . (P + g) 2 = a--6 , 6 x 3^.12a = 14c w a + 6 a 2 -6 2 p + g" ' W 56 ' 7c 15 6* 52. Form as many proportions as possible from each of the following equations : (1) xy = vt. (3) (a + 6)2 = m 2 - n 2 . (2) m 2 = rs. (4) x 2 = a 2 — b 2 . 53. Find a fourth proportional to each of the following sets of numbers : (1) 27, 90, 45. (2) p, q, r. (3) I, 1, 1. a o c 54. Find a mean proportional between each of the following : qx a?/ ab ,«\ 2(a 2 — aft) — 10 a U 6 ' ?/ ' 35 6 ' 7(a6 - 6 2 ) ' , 55. Find a third proportional to each of the following : (1) 8,9. (2) 36a 2 & 2 4(a 2 -q&). W W (a 2 - 6 2 ) 2 ' b(a + &) 2 56. If - = - , prove each of the following relations : o a K } b d { ] a-b c-d (9\ a ± b _ c ± d ,,* a-\-c_a 1 ' a c U b + d~b' 57. Find the values of x and y in each of the following : (1) ? = ^i,whenx + y = 9. (2) ® = |, when x + y = 15. (3) x:y = 3.5 : 4, when x — y = 2.5. General Review 477 58. Combine each of the following proportions so as to eliminate x and leave the new proportion in its simplest form : ^ } d~x' g~~f K ^ a-b~ (c + d)* ro\ l — n ® — -. a _3 c . 2c m x' p r 14 6 lb a 59. The ratio of the sun's diameter to the earth's is 542 : 5 ; of the earth's to the moon's, 11 : 3. Find the ratio of the di- ameter of the sun to that of the moon. 60. The age of a father to that of his son is as 7 to 4. What is the age of each if the father is 24 years older than his son ? 61. If in the composition of powder, the ratio of niter to carbon is 31 : 9 and of carbon to sulphur is 9 : 10, how much of each must be used to make 1200 pounds of powder ? 62. Solve by two methods : 2 x -+- y = 11, 3 x — y = 4. 63. Solve the following systems of equations : (1) 15a-7y = 9, (6) 2f */ - f a; = 90, 9y-7x = 13. 2| = 4y. . 4 2 3' (4) 5 + 4a> = 16y, (8) .25a + 3y = 10, 5 x + 28 y = 19. 4.5 x - 4 y = 6. (5) | a; + 1 2/ = 34, (9) 25.9 v - 60.1 u = 1, 7 a + ^ y = 12. 24.1 v - 55.9 w = 1. (10) ,2y + .25x = 2(y-x), .8 a- 3.7 2/ = -15.3. 64. The formula for converting a temperature of i^ 7 degrees Fahrenheit into its equivalent temperature C degrees centi- grade is <7=|-(F— 32). Express F in terms of (7, and com- pute F when C = 30° ; when C = 28°. 478 General Review 65. Find two numbers such that their sum and difference are in the ratio 5 : 1 and their sum to their product in the ratio 5 : 8. 66. A servant is given $ 2 to buy 10 pounds of sugar and 4 pounds of cheese and should have 60 cents left. She makes a mistake and buys 10 pounds of cheese and 4 pounds of sugar and lacks 24 cents. What is the price of cheese and of sugar ? 67. Simplify the following : (1) a 2m ~ n a m+n . m (tf)X^j- (2) (a + 6)-(a + b): (3) (p + q y->(p + gf. (11) ifl x ^JC a n n X xP (4) (a* + &*)(a*-&"). J (5) (m»-n")». (12) ^J* (7) _^_ + _# !L (14) [(aa;) 3 ^ 4 *] 51 '. m* + * m^ 1 m p * (15) (^p _ l) + ( x p - 1). (8) ( - a) 3 ( - a) 5 - (16) (x + yy-» : (x + y) 6— a 7 — c \ V5 — m\ 6 n 7 >. ^ w+ra ^ m43n 68. Perform the following operations and write the results so that each term shall have the integral form affected by negative exponents where necessary : (2) f 2x2 x 1 a Sa,2 \ --f 2a Sa ' ^ ' \a 2 a x x 2 J \ x x 2 69. Write the following expressions without using either the negative exponent, or the exponent zero, and simplify the results : (1)*. (2)1- (3) (S*E 70. Verify the identity a 2 (5-i _ c -i) + fe 2 ( c -i _ a -i) 4. C 2( a -i _ 5-1) _ a + fr + c a(6~ 2 - c" 2 ) + 6(c" 2 - a~ 2 ) + c(a~ 2 - b~ 2 ) a" 1 + b~ l + c" 1 ' 71. Write the following expressions without using the radical sign or negative exponents : (1) -Va\ (2) (V^) 6 . (3) -V&. (4) Vp^. (5) V{a + b) m+ *. (6) ^a 3 -3a 2 6 + 3a& 2 -& 3 . (7) V FT. (10) ^/^ "£ < 8 > ^ (11) *L ^ 72. Write the following expressions using only radical signs and positive integral exponents : (1) a*. i (3) A (5) dl (7) e* (2) m*. (4) y m . (6) *»« (8) e *. 73. Simplify the following radical expressions : .(1) 2Vl08a 4 6 7 . (5) ^/ a 2m+n b 2mn c m+2n . (2) V 7(14a-21&). (6) \a?_2_a + 1 (3) V(n»-w)(w + l). ^ c2 c (4) V(a 2 H-6 2 ) 2 -(2a6) 2 . (7) V98"^7VSf. (8) V21+Vl3+V7+V4. (9) VoVaVS. ( 10 ) \ (&+c)< 480 General Review (11) 4V32-5V50 + 3Vl8. v (12) ( Va + 3 V6) - (15 VS-2V5) + (4 V6 + 7 Va). /-.ox Vc-hVa Vc — Va ,.,., /- /- V» (1^) -—= — — -• (14) Vpxvgx-^- Va? — Va V» -f Va Vg (15) ( Vp + g - Vj> - g)(Vp + g + Vj> - g). (16) Vw«-n-4-Vw + l. (17) (x + y) + iV&^jp M 74. Simplify the following imaginary expressions : (1) V^50+V^5+V^18+V^I-V^4 + 2V^2. (2) V-3V8V-6. VIP (3) (s+V=6)(a>-V=6). V^6 (4) Vl + V3lVl-V^l. a^ < _ (a ._ y) s (5) Va; — 2/V.y — a?, a; — y * (x + yf (8) (-l+V^3) 3 +(-l-V^3) 3 . (9) ?? (10) LzJ + l±i. ; 7 + 2V^5 ^ ' l + » l-< 75. Solve the following equations : (1) V5a^ = 20. (5) VlO?/-4 = V77TTl. (2) Va7+9 = 5 VaTT3. (6) 3Vl6a; + 9 = 12V4al- 9. (3) 5 -V32/ = 4. (7) j + xl= 3 . (4) V2v + 8=V5i; + 2. (» _ 3)* (8) * U ±— m U x + V4-x 2 a;-V4-a; 2 7 (9} Vo a; — 4 + V5 — a; _ V4a; + 1 # V5 x — 4 — V5 — a; V4 a; — 1 76. Solve the following quadratic equations by completing the square : (1) z 2 -8a: = -7. (2) aj 2 + 12 = 7». General Review 481 (3) x(x - 1) = 380. (5) 250 + 2 sfi = 3 x> - W x. (4) t 2 + 10 £ - 56 = 0. (6) 6 a; 2 - | + 4 a? = x + 5 a& (7) (3 a? - 2) 2 = 8(aj + l) 2 - 100. (8) (a; -5)2= 4. (9) 3a; 2 + 5x- 42 = 0. (10) 2a; 2 -8 = 3a; + 12. (11) 5a;(a;-2)+2=-4-a;(4a;-5). (12) 3(z 2 + 2) 2 - 54 = 3 z* + z(5 z - 7). (13) 5(1 + u) 2 + 4 u = (3 - uf + - 4 ^. (14) a7Tl + 2^1) = ^ (16)3m 2 -6m=-f. (15) v _3 + ^±| = 2v-7. (17) 5a; 2 -8a; + 3 = 0. (18) (px- 2){x + l) = (a? - f)5 - 5. (19) (1.2 - a;) 2 + (a; + -8) 2 = 2(6 * - .2) 2 . , 20 \ 8-2 _ 8-3 s-4 _ 7 ^ ' s _i s + 3 s _i 4* 77. Solve the following equations as quadratics : (1) ^ + 4^ = 96. (3) ax 11 + bx 9 + ex 7 = 0. (2) a/**- 16 a>* = 512. (4) 2ar> - 6 - a?i = 0. (5) IV* 2 + \x + 81 = J(63 - 2 a; 2 - x). (6) a: 2 - 5 a; + 2Va; 2 -5a; + 3 = 12. Hint. Add 3 to both members and treat x 2 — 5 x + 3 as the unknown. (7) 2 a; 2 - 4 a; + 3 Va; 2 - 2 x + 6 = 15. (8) a; 2 + 2Va; 2 + 6a; = 24 - 6a;. (9) 3a; 2 - 4a; + V3a; 2 - 4a; - 6 = 18. (10) 8+9V(3a;-l)(a;-2)=3a; 2 -7a;. (11) 3a; 2 -7 + 3V3a; 2 -16a; + 21 =16 a;. (12) V4 a; 2 - 7 a; - 15 - Va; 2 - 3 a; = Va; 2 - 9. Hint. The factor y/x — 3 can be removed from each expression. (13) V2a? 2 -9a;-r-4 + 3V2a;-l=V2a; 2 + 21a;-ll. 482 General Review (14) gfi + x +- + ~ =s4 x x 2 Hint. {& + % + 1\ + x + ^ = 6, or (a + ^) 2 + (x +1]- 6 = 0. (15) X 2 +x + -+* = X X 2 Hint. (16) tf + s + 1+1 = 6 f. (17) 12o 4 -56a; 3 + 89a; 2 -56a; + 12 = 0. Dividing by x*, 12 (x 2 + -M - 56 ( x + -\ + 89 = 0. Put x + - ^2 _i 1 _ -,2 _ 9 = z, then x 2 + — = z 2 — 2 (18) a^ + ar 5 - 4a; 2 + a + 1 = 0. (19) a 4 + l-3(ar 5 + a-)=2a; 2 . (20) Vx 2 + x + s/x* — x 2 ' 78. Solve the following systems of quadratic equations : (1) x 2 + I/ 2 = io, (8) x 2 y + a*/ 2 = 120, x — y = 2. a? + f = 152. (2)* + 2/ = 23, (9)a 2 -2/ 2 =4, t x + xy = 144. a? + 2/=|. (3) a-y=20, (10 ) l+l_ + i = 47, a* _ xy = ioo. * xy y (4) ar2_4 2/ 2 ==9) 1 + 1=12. (5) 2 1? + 3 « * 20, Suv- u 2 = SS. (11) x + y 4- x 2 + ,v 2 = If, 2/ — a; + ?/ 2 — x 2 = — 1. (12) ar> - f = 279, (6) a + 2/ = a, X 2 + a . y + 2/2 = 93. a? 2 + y 2 = 6^2/. 1 (13) J- 1 1 t/2 + 2/2 — 74. m ( 13 ) 1-1=1304, (7) y - z = 1, v y t» w 5 2/* = 20, 1_1 = 8 General Review 483 (14) aj» + y»=.152, (21) 2 * - 5 y = - 18, x 2 — xy -f y 2 = 19. 3 a# = 264. (15) 2a? 2 -3x# + 2/ 2 = 24, (22) x 2 - y 2 = 120, 3* 2 _ 5xy + 2y 2 = = 33. x + ?/ = 20. (16) *2/ + x = 104, (23) a; 2 + y2 = 250, xy -y = 84. a; - 2/ = 22. (17) 5* +2y =29, (24) * 2 + y 2 _ a- + 2/ = 32, 6 xy = - 105. 2 ^2/ = 30. (18) 3^-2/2 = 83, (25) afy* + 3*2/ = 18, a; 4. ?/ = 15. = 8, ic + 2/ = 5. (19) x + y + 2Vx+y (26) 3*2-2/ 2 = 23, x 2 + xy = 8. 2*2 _ xy = 12. (20) ^ - 80, (27) ^+»y + ^= 931, *=5. x 2 + xy +y 2 = 19. .V 79. If r x and r 2 are the roots of the quadratic equation x 2 + px + q = 0, show that r x -f- r 2 = — p and r^ = +3a)«. (4) ( V^ + xf. 100. Find the 6th term in the expansion of (3 + 2 # 2 ) 9 and the 7th term in the expansion of (J a — x) 17 . 101. Expand (2x* — y 3 ) s . (University of Michigan.) 102. Raise 98 to the 5th power by the Binomial Theorem. (Write 98 = 100 - 2.) (Yale.) 103. Find the first three terms of (1 + 2 a;) 8 by the Binomial Theorem. (Sheffield Scientific School.) 104. Find the coefficient of a? and x 4 in the expansion of (1+2 x) s , using the Binomial Theorem. (Sheffield Scientific School.) 105. Expand the expression (x* — x~*y and write the result in a form free from negative exponents. (Harvard.) 106. Define logarithm ; base. In the equation a* = N, what is x ? what is a ? 107. What are the logarithms of 3, 81, 243, 729 in a system of which the base is 3 ? 108. What are the logarithms of J, ^ 7 , -fa, ^ in a system of which the base is 3 ? General Review 487 109. Given log 10 3 = .477 ; find log 3 10, log 3 .1, log 3 .01. (Sheffield Scientific School.) 110. Compute the value of x from the equation = (39.71) 3 Vl3:i6 X (46.71) 4 using logarithms. (Sheffield Scientific School.) 111. Perform the following operations, using logarithms : (1) 9 5 x3 4 . (3) ^76245. (5) 15 4 -23 5 . (2) ^4158. (4) ^2. (6) **g». 112. Solve the following equations : (1) 2* = 1024. (3) 12- = 20,737. (2) 10,000- = 10. (4) 31** = 4. 113. Write the roots of (x 2 -\-2 x)(x 2 -2 x-3)(x 2 -x + l) = 0. (Sheffield Scientific School.) 114. Solve the equation x 2 — 1.6 x — .23 = 0, obtaining the values of the roots correct to 3 significant figures. (Harvard.) 115. The distance s that a body falls from rest in t seconds is given by the formula s = 16 t 2 . A man drops a stone into a well and hears the splash after 3 seconds. If the velocity of sound in air is 1086 feet a second, what is the depth of the well ? (Yale.) 116. A man spent $539 for sheep. He kept 14 of the flock that he bought and sold the remainder at an advance of $2 per head, gaming $28 by the transaction. How many sheep did he buy and what was the cost of each ? (Yale.) 117. Solve by factoring a? 3 + 30 x = 11 x 2 . (Colorado School of Mines.) 118. Solve the equation .03 x 2 - 2.23 x + 1.1075 = 0. (Colorado School of Mines.) 488 General Review 119. How many pairs of numbers will satisfy simultaneously the two equations 3 x + 2 y = 7 and x + y = 3 ? Show by means of a graph that your answer is correct. What is meant by eliminating x in the above equations by substitution? by subtraction? (Colorado School of Mines.) 120. An automobile went 80 miles and back in 9 hours. The rate of speed returning was 4 miles per hour faster than the rate going. Find the rate each way. (Cornell.) 121. A goldsmith has two alloys of gold, the first being f pure gold, the second -^ pure gold. How much of each must he take to produce 100 ounces of an alloy which shall be -§ pure gold ? (Harvard.) 122. A man walked to a railway station at the rate of 4 miles an hour and traveled by train at the rate of 30 miles an hour, reaching his destination in 20 hours. If he had walked 3 miles an hour and ridden 35 miles an hour, he would have made the journey in 18 hours. What was the total distance traveled ? (Mass. Institute of Technology.) 123. A page is to have a margin of 1 inch, and is to contain 35 square inches of printing. How large must the page be, if the length is to exceed the width by 2 inches ? (Mount Holyoke College.) 124. Factor the following expressions : (a) a* -6* (b) xY* 2 — ® 2 z — y 2 % 4- 1. (c) 16(0? + y) A - (2 x - y)\ (Mount Holyoke College.) 125. If four quantities are in proportion and the second is a mean proportional between the third and the fourth, prove that the third will be a mean proportional between the first and the second. (Princeton.) 126. Solve x 2 + y 2 — xy = 7. x + y = 4. (Smith College.) General Review 489 127. The diagonal of a rectangle is 13 feet long. If each side were longer by 2 feet, the area would be increased by 38 square feet. Find the lengths of the sides. (Smith College.) 128. A field could be made into a square by diminishing the length by 10 feet and increasing the breadth by 5 feet, but the area would then be diminished by 210 square feet. Find the length and the breadth of the field. (Vassar College.) 129. Simplify — — ~*~ , and compute the value of the V2-V12 fraction to two decimal places. (Yale.) 130. In going 7500 yards a front wheel of a wagon makes 1000 more revolutions than a rear one. If the wheels were each 1 yard greater in circumference, a front wheel would make 625 more revolutions than a rear one. Find the circumference of each. (Yale.) 131. In the expansion of (2 x — 3 a; -1 ) 8 , find the term that does not contain x. (Princeton.) COLLEGE ENTRANCE EXAMINATIONS UNIVERSITY OF CALIFORNIA Elementary Algebra 1. At a football game there were sixteen thousand persons. The num- ber of women was six times the number of children and the number of men was three thousand less than twice the number of women. How many men, women, and children were there ? 2. Solve : (a) 2x + 3 = 0. (&) 3n + 2(n + 4)=4n + 14. 3. Factor : (a) a 4 - 7 z 2 + 12. (6) &s_27. (c) (3s- l) 2 -(x 2 + 4y 2 -4xy). 4. The difference between two numbers is 14 and their product is 176. Find the numbers. 5. Solve for x : (a) ? = — 2L. • v J b c-x (6) x(x 2 - 4) (z 2 - 9) = 0. 6. («) Describe the method of locating a point on squared paper. (6) Construct a graph of (a) 2 x = 5 y + 10, (&) x = 5. 7. Solve graphically : j* + 2 y= 4 ' 2x + y=-l. CORNELL UNIVERSITY Elementary Algebra 1. Multiply 1 + 2 x — x 2 — \ x s by itself, and then find the value of the result if 1 - 2 x = 3. 2. What is the value of x* + y 3 if x + y = 4, and 2x 2 -f2 y 2 = 17 ? 490 College Entrance Examinations 491 3. (a) Add -, , x ~ — , and — — , and express the result v J x 1 -x (x + l) 2 x + 1 as a fraction in its lowest terms. Vx — 4Vx— 2 (6) Rationalize the denominator of ,- 2Vx + 3Vx-2 4. Find a root of x 2 — x — 1 =0, and verify correctness of the result. 5. Solve : 2x + 4y + 5z = 19, -3x + 5y + 7z = 8, Sx-Sy + 50 = 23. 6. A takes three hours longer than B to walk 30 miles; but if A doubles his pace he takes two hours less than B. Find the rate at which A and B each walk. 7. Find the time between three and four o'clock when the minute and hour hands are opposite each other. Intermediate Algebra 1. Solve for x and check results : x 2 - 1 x-1 x+1 2. Solve and check : x + y + 2 y/x + y — 1 = 25, x — y + 3Vx— y + 1 = 9. 3. For what values of m will the roots of 2 m 2 + x 2 — 2 mx + 4 x — 6 m + 4 = be real and distinct ? 4. Evaluate (81)^+ ( - 27)^ 3(9)"* + (27)" 1 5. Find the greatest common divisor of x 4 + x 3 — x 2 — x and x 4 + 4x 3 + 3 x 2 — 4 x — 4. Also find the least common multiple. 6. What is the sum of 1 + 3 + 5 + ••• + (2 n — 1), n being a positive integer ? What is the least odd integer such that the sum of all the posi- tive odd integers up to and including it will exceed 45,370 ? 7. From a thread whose length is equal to the perimeter of a square, 1 yd. is cut off. The remainder equals the perimeter of a square whose area is | that of the first. What was the original length of the thread ? 492 College Entrance Examinations PRINCETON UNIVERSITY Algebra A I 1. Simplify — ■*-(«- 8). 2 + *i±_3 2. Simplify {x~%{xy- 2 )~^(x-^)Y ', 3 V} + V40 + V| - v'J, 3. Factor x 2 - 4 ax - 4 b 2 + 8 a& ; a 2 + cd - a& — bd + ac + ad ; (x + l)(6x 2 -x)-15(x + 1). 4. Find the H. C. F. of x 4 - 2 « 3 - Sx 2 - 2 x - 4 and x 4 - x 3 - 7 x 2 - 2x + 4. 5. Solve Z + 2/+ (si, 2x + ?/ + 3£=4, 3x + ?/ + 7£ = 13. 6. A gave B as much money as B had ; then B gave A as much money as A had left ; finally A gave B half as much as B then had left. A ends with $ 4 and B with $ 36 ; how much had each originally ? 1. Solve (a) x-^ = b Algebra A II a _ b b a (&) Vx+ 1 5 x + Vx + 1 n 2. Solve the following equations, pairing the corresponding values of x and y and testing one solution in each case : (a) *+2> = 4 ' (b) 2(x 2 + 2/ 2 )+x + t/= 11, x j/_5. xy = l. y x 2 3. Show that the series whose terms are the reciprocals of the terms of a G. P. is a G. P. How many terms of the progression f, |f, -^ ••• must be taken to make the sum 36£ ? 4. A earned $6, and B, who worked 4 days more than A, earned $ 14. Had their wages per day been interchanged they would together have earned $ 19. How many days did each work ? College Entrance Examinations 493 YALE UNIVERSITY— SHEFFIELD SCIENTIFIC SCHOOL Elementary Algebra (Omit one question in 1-3 and one in 7-9) 1. Solve 3 _ 1 3 5 x + 2 4z-6 2s 2 + 3-6 2. Solve and verify V5 - 2 x + Vl5 — 3 x = V26 — 6 a. 3. Draw the graphs of the equations : y = x-l, y = x 2 — 4 x + 5. Solve them simultaneously and explain the relation between the graphs and the solutions. 4. Simplify (a) ^-±-^ 2(1 ^—\ * * K J x* + 4x + 3 V 1 + aJ 27 + x s ^ 2 n(m — n) ,-. s m + n (6) i^ + n' ! mn + w 2 6. Simplify (a) -JI+-JS"- (6) - J m ~* n + m-^' 8 lb mhm-n)» 6. Write the first three and last three terms of the expansion of (a- 1 + 2 a^) 8 and simplify the result. 7. A and B started in business at the same time. The first year A lost .$> 5000, but during the second year gained 25 % on the amount left at the end of the first year. B started with f as much money as A and gained 20% the first year, hut lost $2050 the second year. He then had the same amount as A. How much had each at first ? 8. A 13-foot ladder leaning against a building lacks 3 feet of reaching a window, while a 17-foot ladder with its base placed 3 feet farther from the wall just reaches it. How high is the window from the ground and how far was the bottom of the first ladder from the wall ? 9. Divide S 700 between A, B, C, and D, so that their shares may be in geometrical progression and the sum of A's and B's shares equal to $ 262. APPENDIX REMAINDER THEOREM, FACTOR THEOREM, AND SYNTHETIC DIVISION 647. Remainder Theorem. 1. Divider 2 — 5x+8byx — a. 2. Divide x 2 — 5x+8 by x — 2. x*-5x + 8 £ — a X 2 -5z + 8 z 2 -2sc -3* +8 -3z + 6 z-2 x 2 — ax x + (a - -5) x-3 (a - 5)z + 8 (a - 5)x - a 2 + 5a a 2 - 5 a + 8 = Rem. 2 = 2 2 — 5-2 + 8 = Rem. Note that the remainder in the first division is the same as the dividend except that a has replaced x. The remainder when dividing by x — 2 is the result of substituting 2 for x in the dividend. 648. If a rational and integral expression in x is divided by x — a until the remainder does not contain x, the remainder is the expression obtained by substituting a for x in the dividend. Proof. Call the dividend D x , and let J) a be the result of putting a for x in the dividend. Call the quotient Q x , and let Q a be the result of sub- stituting a for x in the quotient and let the remainder be B. D x =(x — a) Q x + B. (Dividend = divisor x quotient -f remainder.) This equation is an identity ; that is, it is true for all values of x. Sub- stitute a for x in both members. Then D a = (a- a) Q a + B. .-. D a = 0- Q a + B or D a = B. q.e.d. The student should note that R is not changed when a is substituted for cc, since B does not contain x. 404 Appendix 495 Examples 1. Find the remainder when sc 3 — x + 7 is divided by x — 3. Solution. Substitute 3 for x. 33 _ 3 + 7 — 31, the remainder. 2. Find the remainder when x 3 — 5 x 2 + 2 is divided by # 4- 2. Solution. x + 2 = x -( — 2). Substitute — 2 for x. (_ 2) 3 — 5(- 2) 2 + 2 =— 8 — 20 + 2 =— 26, the remainder. EXERCISE 649. Find the remainder when each expression is divided by the binomial opposite it: 8. x 3 — 2 a + 3, x + 1. 9. x* + l, a> + l. 10. ic 5 + 1, x + 1. 11. # 7 + l, aj + 1- 5. 3^-17x-30, z-3. 12. a 3 -l, a?— 1. 6. y 3 — 5 ?/ + 4, ?/ - 3. 13. a; 3 + # 2 + x + 1, a + 2. 7. a 8 — 2s 8 + 3, z-1. 14. a? + a; 2 + a> + l, a + l. 650. Factor Theorem. The student will observe that in some of the examples in § 649 the remainder was 0, and, there- fore, the divisor was a factor of the dividend. 651. If a rational integral expression containing x vanishes (becomes equal to 0) when a is put for x, then x — a is a factor of the expression. Proof. If the result of substituting a for x is 0, the remainder when the expression is divided by x — a is 0, and the divisor is a factor of the dividend. (§648.) Examples 1. Show that x — 1 is a factor of x? — 2 x + 1. Solution. Substituting 1 for x gives 18_ 2. 1 + 1 = 0; .'. x — 1 is a factor. By dividing x 3 — 2 x + 1 by x — 1 the other factor is found to be x* + x-l. .-. x*-2x+ 1 = (x-l)(se 2 + x- 1). 1. a? -3a; + 8, aj-1. 2. a**_2ic + 3, a -2. 3. 2 « 2 - 5 a + 4, ic — 5. 4. #-2 x 4-1, x-1. 496 Appendix 2. Factor a? - 5 x - 12. The number to substitute for x must be found by trial. Try different factors of 12. 3 a_ 5 . 3 _ 12 = o. .\x — 3 is a factor. z*-5x- 12 = (x-3)(x 2 + 3x + 4). EXERCISE 652. Factor by the factor theorem : 1. x* -7x + 6. 4. a; 2 -5a; -6. 7. 3a? + a; 2 -28. 2. a? — 8. 5. ar>-2a; 2 -9. 8. 2a? + a; 2 — a;- 2. 3. a; 2 -5a; + 6. 6. a? - 2a; - 56. 9. x* + 2x + l. Solution. All terms are positive, so no positive number need be tried. (Why?) ( _ 1)4 + a( _ 1)+1 = ft . \ x— ( — 1) or x -f 1 is a factor. a 4 + 2x + l=(x+l)( ? ). 10. x 3 + 3a;-f-14. 13. 2a? + 3a!+5. 11. a?-6a; 2 + lla;- 6. 14. 5 x i - 21 a; - 38. 12. a? -6ar> + 13a; -10. 15. 7 a? + 9 a; + 16. 653. Synthetic Division. Division of polynomials containing x by divisors of the form x — a can be performed very ex- peditiously by the method known as synthetic division. x s _ 6 x 2 + 11 x + 2 x-2 - 4 x 2 + ]>£ z^Ar&+ 8x x 2 - 4 x + 3 3x+# 3^-6 8 The coefficients of the first term and of each partial remainder, except the last, are the coefficients of the terms of the quotient. The terms crossed off could as well as not be entirely omitted from the work. The Appendix 497 se's could all be omitted, since the orderly arrangement of the work would enable us to replace each x with the proper exponent. The subtractions that we made in the original division, — 2x 2 from -6a; 2 ; Sx from 11 x; — 6 from 2, can be changed to additions by using 2 instead of — 2 as a multiplier. The work is thus reduced to the following : 1-6 + 11 + 2 2- 8 + 6 1 — 4 + 3 1 + 8, remainder. 1. Write the coefficients of the dividend in order with their signs and write the second term of the divisor with its sign changed to the right. 2. Bring down the coefficient of the highest degree term, 1, as the coefficient of the first term of the quotient. 3. Multiply this number by 2 and write the product, 2, under the second term and add. This gives — 4, the coefficient of the second term of the quotient. 4. Multiply — 4 by 2 and write the product under 11 and add. 6. Repeat this process to the end of the polynomial. 6. The first three numbers on the last line are the coefficients of the quotient x 2 — 4 x + 3, and the last number is the remainder. If any power of x from the highest down to the absolute term is miss- ing, a zero coefficient must be supplied in its place in the detached coeffi- cients. Examples 1. Divide £c 4 + aJ 2 +-8byz-l. Division : 1+0+1+0 + 8 1+1+2+2 1+1 + 2 + 2 1 + 10 , remainder. The quotient isx z + x 2 + 2x + 2 and the remainder is 10. 2. Divide x 3 + 6 x 2 -16 by x + 2. Division : 1+6 + 0— 16 1-2 -2-8+161 1 + 4-8 10 The quotient is x 2 + 4 x - 8 and the division is exact 498 Appendix EXERCISE 654. Divide the following by synthetic division : 1. (a;2_9x + 20)--O-l). 7. (3x?-5x + 2) + (x-2). 2. (;xZ-9x + 20) + (x-2). 8. (x 3 4- 2x + l)-s-(a> + 1). 3. (a* -9 a? + 20) -*-(»- 4). 9. (^ + 1)^ + 1). 4. (x 2 -9x + 20) + (x-5). 10. (x 3 -1)h-(«-1). 5. (« 3 -9ic + 20)--(x-2). 11. (a*-l) + (x + l). 6. (2#-9« + 3)-i-(aj-3). 12. Factor a 4 - 5a 3 + 3 x 2 + 15 a - 18. Solution. Substituting 2 for x makes the expression vanish. There- fore x — 2 is a factor. Divide by X — 2. 1_5 + 3 + 15_ 1812 2-6- 6 + 18| 1_3_3+ 9 |_o The quotient x 3 — 3x 2 — 3x + 9 vanishes for x = 3. Divide by x — 3. 1 -3-3 + 913 3-4-0 — p| l + 0-3 [0 The quotient is x 2 — 3. .-. x 4 - 5x3 + 3x2 + 16x _ 18 = ( X _ 2)(« - 3) (x 2 - 3). Factor the following : 13. o# — 10.S + & 17. a? + a? 2 + a; — 3. 14. 2a 3 +-5# 2 -4. 18. a 3 +-a + 2. 15. ^-# + 24. 19. a 3 - 3z -322. 16. a?— 13 a* + 49 a> — 45. 20. a 3 + 5a; + 150. INDEX [The numbers refer to pages.] Abscissa, 245. Absolute terra, 290. Absolute value, 20. Addition, 32-49. of fractions, 179. of imaginary numbers, 369. of like monomials, 35. of polynomials, 42. of radicals, 341. of signed numbers, 18, 20. of unlike monomials, 40. Addition and subtraction, elimination by, 255. Algebraic expression, 32, 93. Algebraic fraction, 1(57. Algebraic improper fraction, 174. Algebraic number scale, 17. Algebraic numbers, 18. Algebraic signs in fractions, 171. Algebraic sum, 22. Alternation, proportion by, 226. Antecedent, 218, 219. Antilogarithms, 464. Appendix, 494-498. Area of circle, rectangle, triangle, 1. Arithmetical mean , 429. Arithmetical number, square root of, 283. Arithmetical progression, 427. Arranging terras of a polynomial, 39. Ascending powers, 39. Axis, 244. Base, 33. Binomial, 32. Binomial coefficients and exponents, 444. Binomial formula, 442-447. Binomial quadratic surd, 356. Braces, 58. Brackets, 58. Briggs's logarithms, 458. Cancellation, 168. Characteristic, 458, 459. Check, 8, 44. Circumference of Circle, 1, Clearing of fractions, 197. Coefficient, 33, 444. Collecting terms, 41. College Entrance Examinations, 490. Cologarithra, 465. Common denominator, 177. Common difference, 427. Common factor, 162, 163. Common logarithm, 458. Common multiple, 165. Complete quadratic, 290, 294, 374. Completing the square, 294, 296. Complex fraction, 194. Complex number, 368. Composition, proportion by, 227. Composition and division, proportion by, 230. Compound interest, 466. Conditional equation, 104. Conjugate quadratic surd, 350. Consequent, 218, 219. Constant of variation, 448. Coordinates, 244, 245. Cube, 5. Cube root, 5. Decimal, repeating, 437. Deduction, symbols of, 5. Degree, 162. Degree of an equation, 400. Denominator, 167. lowest common, 177. rationalizing, 352. Descending powers, 39. Difference, in subtraction, 22. common, 427. Direct variation, 448. Discriminant, 386. Dissimilar terms, 35. 499 500 Index Dividend, 92, 167. Division, 92-103. defined, 27, 92. of fractions, 190. of imaginary numbers, 369. of monomials, 94. of polynomials, 96, 99. of radicals, 344, 350. of signed numbers, 27. proportion by, 228. synthetic, 496. Divisor, 92, 167. Elimination, by addition and subtrac tion, 255, 412. by substitution, 259, 412. defined, 252. Equality, symbol of, 2. Equation, complete quadratic, 290. conditional, 104. defined, 7, 104. degree of, 400. formed with given roots, 387. identical, 104. incomplete quadratic, 290. indeterminate, 250. integral, 107. linear, 107. members of, 7. principles used in solving, 8. radical, 362. simple, 104, 107. type form ax= b, 108. Equations, containing fractions, 197- 217. determinate system, 250. graphical solutions, 253-254, 418-426. homogeneous, 402. independent, 251. linear simultaneous, 250, 251. literal, 201. quadratic form, 388. quadratic simultaneous, 305. symmetrical, 407. Evolution of radicals, 355. Exponent, defined, 5, 33, 322, 454. fractional, 316. negative, 318. zero, 318. Exponents, 315-330. / Exponents, definitions of, 322. laws of, 67, 93, 315, 322. Expression, integral algebraic, 93. mixed, 174. radical, 332. Extraneous root, 363. Extremes, 219. Factor, common, 162. defined, 5, 25, 132. highest common, 163. rationalizing, 349. Factor theorem, 495. Factoring, 132-160. cases of, 133-150. solution of equations by, 156. square root by, 280. summary of, 152. Formula, binomial, 442. defined, 1. for solving quadratic equations, 381. Formulas, rules and, 123, 124. Fourth proportional, 220. Fraction, algebraic, 167. improper, 174. lowest terms of, 168. reduction of, 168. terms of, 167. Fractional equations, 197, 214. Fractional exponents, 316. Fractions, 167-196. addition of, 179. clearing of, 197. complex, 194. division of, 190. multiplication of, 186. reduction of, 168. subtraction of, 179. Function, defined, 245. graph of, 246. General review, 469-489. Geometrical mean, 435. Geometrical progression, 434. Graph of function, 246. Graphical solution of equations, 253, 254, 418-426. Graphs, 237-249, 253, 254, 418-426. Index 501 Highest common factor, 161-165. defined, 163. Homogeneous equations, 402. Identity, 104. Imaginary numbers, 367-372. addition and subtraction of, 369. denned, 331,367. division of, 369. multiplication of, 369. operations with, 368. reduction of, 369. unit, powers of, 368. Improper fraction, 174. Incomplete quadratic, 290, 291. Independent equations, 251. Index of radicals, 332. Infinite geometrical series, 437-438. Inserting parentheses, 62. Integral algebraic expression, 93. Integral equation, 107. Integral polynomial, 161, 162. Integral term, 161. Interest, compound, 466. Introduction, 1-14. Inverse variation, 451. Inversion, proportion by, 226. Involution of radicals, 355. Irrational equation, 362. Irrational numbers, 331. Joint variation, 452. Last term, 428, 434. Laws of exponents, 67, 93, 315-322. Like terms, or monomials, addition of, 35. defined, 35. subtraction of, 50. Linear equations, 107, 250-278. Literal equations, 201. Logarithms, 454-468. common or Briggs, 458. laws of, 456-457. table of, 461-463. Lowest common denominator, 177. Lowest common multiple, 165, 166. Mantissa, 458, 459. Mean, arithmetical, 429. Mean, geometrical, 435. Mean proportional, 220. Means, 219. Members of equations, 7. Minuend, 22, 23. Mixed expression, 174. Monomial, addition of, 35, 40. defined, 32. division of, 94. multiplication of, 69. Multiple, lowest common, 165. Multiplication, 67-91. of fractions, 186. of imaginary numbers, 369. of polynomials, 71, 74. of radicals, 344. of signed numbers, 25. type forms of, 78-87. Nature of roots, 385. Negative exponents, 318. Negative numbers, 15. v Number scale, 17, 18. Numbers, algebraic, 18. complex, 368. literal, 1. positive and negative, 15, 16. prime, 132. rational and irrational, 331. real and imaginary, 331, 367° signed, 18. symbols representing, 1. unknown, 7. Numerator, 167. Numerical coefficient, 33. Operation, order of, 28. symbols representing, 2. Order of radicals, 332. Ordinate, 245. P-form of quadratic, 295. Parenthesis, explained, 57. inserting in, 62. removal of, 59. Polynomials, addition of, 42. defined, 32. degree of, 162. division of, 96, 99. multiplication of, 71, 74. 502 Index Polynomials, rational and integral, 161. square root of, 280. subtraction of, 54. Positive and negative numbers, 15-31. Power, defined, 33, 279. of imaginary unit, 368. Powers, ascending, 39. descending, 39. of monomials, 355. Prime factor, 132. Prime number, 132. Principal root, 332. Principles used in solving equations, 8. Problems, hints on the solution of, 116. Product, 5, 25, 132. Progression, arithmetical, 427-433. geometrical, 434- 41 1. Proportion, terms of, 218, 219. Proportional, fourth, mean, third, 220. Proportions, properties of, 222. Quadrants, 244. Quadratic, complete, 290, 294, 374. equations in the form of, 388. incomplete, 290, 291. p-form, 295. relation between roots and coeffi- cients, 384. solution by completing the square, 296, 374. solution by factoring, 156, 377. solution by formula, 380. surd, 332, 350. theory of, 384. Quadratic equations, 290-304, 373- 399. formula for solving, 381. simultaneous, 305-314, 400-417. theory of, 384. Quadratic trinomial, 146. Quotient, 27, 92. Radical equations, 362-365. Radical expressions, 332. Radical index, 332. Radical sign, 133. Radicals, 331-361. addition of, 341. division of, 344. Radicals, involution and evolution of, 355. multiplication of, 344. order of, 332. reduction of, 333. similar, 341. simplest form of, 337. subtraction of, 341. Radicand, 332. Ratio, in geometrical progression, 434. Ratio and proportion, 218-236. Rational numbers, 321. Rational polynomial, 161, 162. Rational term, 161. Rationalizing denominator, 352. Rationalizing factors, 349, 350. Real numbers, 331. Reciprocal, 190. Rectangle, 1, 124. Reduction, of fractions, 168. of radicals, 333. Remainder theorem, 494. Removal of parenthesis, 59. Repeating decimals, 437. Root, cube, 5. extraneous, 363. of equation , 8, 156. of radicals, 355. square, 5, 133, 279. Roots and coefficients of quadratic equation, 384. Roots, nature of, 385. Rules and formulas, 123. Series, defined, 427. infinite geometrical, 437. sum of terms of, 428, 435. Sign, radical, 133. Signed numbers, addition of, 18, 20. defined, 18. division of, 27. multiplication of, 25. subtraction of, 22. Signs, in division, 27, 93. in fractions, 171. in multiplication, 26, 69. Similar radicals, 311. Similar terms, 35. Simple equations, 104-131. defined, 107. Index 503 Simultaneous equations, 250-278, 305- 314. denned, 251. Solution of problems, 116. Square, completing the, 294. Square root, 5, 133, 279-289, 35(3-361. Substitution, elimination by, 259. Subtraction, 50-66. defined, 16, 22. of fractions, 179. of imaginary numbers, 369. of like monomials, 50. of polynomials, 54. of radicals, 341. of signed numbers, 22. of unlike monomials, 52. Subtrahend, 22, 23. Surd, conjugate quadratic, 350. defined, 332. quadratic, 350. Symbols, of deduction, 5. representing numbers, 1. representing operations, 2. Symmetrical equations, 407. Synthetic division, 496. Table of logarithms, 461-463. Term, absolute, 290. defined, 32. integral, 161. last, in progressions, 428, 434. rational, 161. Terms, collecting, 41. dissimilar, 35. of a fraction, 167. of a proportion, 219. similar, 35. sum of, in progressions, 428, 434. transposing, 110. Third proportional, 220. Transposing terms, 110. Triangle, area of, 1. Trinomial, 32. Type forms in multiplication, 78-87. Unknown number, 7. Unlike monomials, addition of, 40, subtraction of, 52. Unlike terms, 35. Use of formulas, 124. Value, absolute, 20. Variable, 448. Variation, 448-453. constant of, 448. direct, 448. inverse, 451. joint, 452. notation, 449. Vinculum, 58. Writing algebraic numbers, 9. Zero exponent, 318. LOAN DEPT. L (C?795sl0)476B General Library . -YB 17276 YB 17277 M306039 QA THE UNIVERSITY OF CALIFORNIA LIBRARY mm\%: