UC-NRLF *B S31 SSE '^Hm ipiiiii!i:i!p'iiiiyi,l 'I' !^ii II llitifl III ! nil 'n Hi i;!ilM!i!i|!'||!!l;!;iilli!!||! V ^JLChv^ ' I UUAJL^,^ COPYEIGHT, 1915, BY AMERICAN BOOK COMPANY All rights reserved SECOND COURSE IN ALGEBRA E. P. 2 PREFACE This book is intended to follow Milne's "First Year Al- gebra/' or its equivalent, to provide for the division of Elementary Algebra into two courses. The general plan and scope of the book have been deter- mined by the recommendations of leading mathematical asso- ciations throughout the country and by a careful study of courses of study in many states and cities, including the requirements of the principal colleges and of the College Entrance Board. As the second course in algebra is usually taught for a half year in the third year of high school, and the first year stu- dent is rarely able to retain all that he has learned of algebra when a year intervenes between the courses, the book begins with a thorough review of first year algebra. In the treat- ment of all review topics, the details of development and explanation are omitted ; but the essentials are given, includ- ing the restatement of all important laws, principles, and rules. By referring the student to the Glossary for review definitions, the massing of definitions at the beginning of chapters is avoided. The applications in the review are new and somewhat more difficult than those in the "First Year Algebra," to provide for the exercise of increasing mathemat- ical power on the part of the student. In the chapters containing the requirements of the second course, the new principles are most carefully developed and the explanations are full and clear. These chapters are fol- lowed by a general review and by supplementary subjects for optional study. 893304 4 PREFACE Each topic is accompanied by a large number of exercises for practice?. They provido sufficient work for classes desiring to devote a whole year to the subject. For a half year's work, every alternate cxeieise laay be omitted, at the discretion of the teacher. Equations and problems are especially emphasized. The problems are based on interesting facts gathered from a vari- ety of sources, including physics, geometry, and business. A few traditional problems are included for the purpose of familiarizing the pupil with them in case they appear on examination papers as well as for their disciplinary value. The formulae and applied problems are easily within the com- prehension of the students for whom they are intended. Functionality has received brief but sufficient attention in the chapters on Graphic Solutions, where its utility is apparent. Publisher's Note. — As Dr. Milne did not live to finish the manuscript of the " Second Course in Algebra," the com- pletion of the work was intrusted to his assistant, Mr. Charles R. McKenzie, who for many years was associated with Dr. Milne in the writing of his mathematical works, and who is intimately acquainted with his methods and ideals. To Mr. McKenzie's valuable experience gained through this close asso- ciation is due the successful completion of the book in entire accord with the author's plans. CONTENTS PAGR Signs and Symbols ... 8 Introductory Review 9 Notation and Definitions 9 Positive and Negative Numbers 13 Addition 14 Subtraction . . . . . . . . . .16 Parentheses . .18 Multiplication 21 Division 27 Equations and Problems . .33 Transposition in Equations 34 Factors and Multiples . . . . . . . . .41 Factoring .41 Monomial Factors .41 Factoring Binomials . . . . . . . .42 ^ Factoring Triyiomials ........ 44 Factoring Larger Polynomials 50 Summary of Factoring . 65 Equations solved by Factoring 58 Highest Common Factor 59 Lowest Common Multiple 60 Fractions 61 Reduction of Fractions 62 To Integers or Mixed Numbers 62 "^ To Lowest Terms 63 To Lowest Common Denominator 64 Addition and Subtraction of Fractions 65 Multiplication of Fractions 67 Division of Fractions 69 Complex Fractions 70 6 6 CONTENTS PAOE Simple Equations . 73 One Unknown Number 73 Clearing Equations of Fractions . . . . .74 Literal Equations . . 77 Formulce 84 Ratio and Proportion ......... 87 Ratio 87 Properties of Batios 88 Proportion 90 Properties of Proportions . . . . . , .90 Simultaneous Simple Equations . . . . . . .95 Two Unknown Numbers 95 ' Elimination by Addition or Subtraction . . . .95 Elimination by Substitution 96 Literal Simultaneous Equations 99 Three or More Unknown Numbers 100 Graphic Solutions 107 Linear Functions 107 Plotting Points and Constructing Graphs . . . .111 Graphic Solutions of Simultaneous Linear Equations . 112 Involution and Evolution .117 Involution 117 Binomial Theorem . . . ^ . . ... 118 Evolution 122 Square Boot of Polynomials 123 Sqtiare Boot of Arithmetical Numbers . . . . 125 Exponents and Radicals 129 Theory of Exponents . .129 Meaning of a Zero Exponent ...... 131 Meaning of a Negative Exponent . . . . . 131 Meaning of a Fractional Exponent 132 Radicals 138 -f^ Beduction . . . . . . . . . . 139 Addition and Subtraction . . . . . . . 142 Multiplication 143 Division . 145 Livolution and Evolution . 146 Bationalization 149 Badical Equations . .152 Imaginary Numbers . . . , . . . . . 157 CONTENTS 7 PAGE Quadratic Equations 161 Pure Quadratic Equations . 161 , Affected Quadratic Equations 163 ^ Literal Equations 168 Radical Equations 169 Formulce 174 Equations in the Quadratic Form 176 Simultaneous Equations Involving Quadratics . . . , . 179 Graphic Solutions . . . . 193 Quadratic Functions 193 Graphic Solutions of Quadratic Equations in x . . . 193 Graphs of Quadratic Equations in x and y ... 196 Graphic Solutions of Simultaneous Equations Involving Quadratics 200 Properties of Quadratic Equations . . . . . . 203 Nature of the Boots . . 203 Belation of Boots and Coefficients . . . . . 206 Formation of Quadratic Equations ..... 206 Number of Boots 208 Factoring by Completing the Square . . . , . 209 Interpretation of Results ........ 211 The Forms a x 0, -, -, - . . . . . . .212 00 Progressions 215 Arithmetical Progressions . .216 Fielding the nth Term . 216 Finding the Sum of n Terms . . . . . . 217 Inserting Arithmetical Means . . . . . . 220 Geometrical Progressions 223 Finding the nth Term . 223 Finding the Sum of a Finite Geometrical Series . . 225 ' Finding the Sum of an Infinite Geometrical Series . . 226 Inserting Geometrical Means 228 General Review 231 Supplementary Topics . , . 245 Cube Root 245 Variation ^ . . . 251 Logarithms 250 Complex Numbers . . 269 Glossary 275 Index 285 SIGNS AND SYMBOLS 4-, sign of addition, read ^plus ' or * increased by ' "I also signs — , sign of subtraction, read ' minus ' or * diminished by" j of quality. ±, ambiguous sign, read ^plus or minus.'' q=, ambiguous sign, read ^ minus or plus.'' \ signs of multiplication, read ' times ' or * multiplied by.'* -=-, sign of division, read ^divided by ' = , sign of equality, read ' is equal ^o ' or ' equals.'' =, sign of identity, read ' is identical with.'' >, sign of inequality, read ' is greater than.'' <, sign of inequality, read ' is less than.'' =5^ , read ' is not equal to. ' ^, read 'is not identical with.'' 5^, read ' is not greater than.'' -jt, read ' is not less than.'' : , sign of ratio, read * is to. ' : : , sometimes used between the ratios of a proportion, read ' equals ' or 'as.' .-., sign of deduction, read ' therefore ' or ^ hence.'' *.-, sign of deduction, read ' since."* • .., sign of continuation, read ' and so on ' or, ''and so on to.'' 0, parentheses [], brackets {}, braces — , vinculum I , vertical bar ^, root, or radical, sign, read ' square root of. ' ^, ^, etc., read ' cube root of,^ '•fourth root o/,' etc., respectively. V— 1, or 1, symbol for the imaginary unit. \ji, factorial sign, read ''factorial w,' n being any integer. •oc, sign of variation, read ' varies as.'' 00 , symbol of infinity, read ' infinity.'' 0, symbol of an infinitesimal number and of absolute zero, read ^ zero.'' ri, read '• r-sub one"* r', read ^ r-prime\ ^2, read ^r-snb tivo.'' r^', read '• r-second\ rs, read ' r-sub three.'' r'", read ' r-third.'' TT, symbol for the ratio of the circumference of a circle to its radius, read 'pL' /(x), F{x)^f{x), symbols of functions of x, read 'function ofx,^ ' large F function ofx^'' and ' f-prime function ofx,^ respectively. 8 signs of aggregation. SECOND COURSE IN ALGEBRA INTRODUCTORY REVIEW 1. In this chapter, as well as in all chapters that are en- tirely or partly review, the student should refer to the Glossary for definitions of terms unfamiliar to him, noting especially terms printed in black-faced type. The algebraic signs and symbols used in this book are explained on page 8. NOTATION AND DEFINITIONS EXERCISES 2. Read ana tell the meaning of each algebraic expression : 11. -yjx. 16. pg + ^'^• 12. V2rs. 17. Ix^-^if, 13. 3a'hG\ 18. a?-2ah + h\ 14. {l + tf. 19. {a-^h)(T-s). 15 ?_^ 20. r^ + ^t'-'SrtK ' y ^' 21. c^-^x'^^x-l. 22. How many terms has each of the above expressions? Point out the monomials ; binomials ; trinomials ; polynomials. 23. ^ame the numerical coefficient in each term in exercises 1-21. Name the coefficient of a; in : 24. 3 07. 25. Ix. 26. 2 a?x. 27. a}hx, 28. ^rs^^x. Which coefficient of x is numerical? Which coefficients of X are literal ? mixed ? 9 1. r + s. 6. xy. 2. a — n. 7. Z • V. 3. 2x3. 8. 4:X. 4. z-^t. 9. 5y\ 5. x-y 10. S2 10 INTRODUCTORY REVIEW Give the numerical coefficient in : 29. X, 30. 4a6. 31. 1 x^if. 32. im^. 33. ^nHx, Name the exponent of ?/ in : 34. yK 35. 4.y, 36. 2xf. 37. xY- 38. 1 y^. State the difference in meaning between : 39. 2iKand x^. 40. 3iK and a?, 41. 4aj and x^. 42. 5 a; and aj^ 43. Write two similar monomials ; three dissimilar mono- mials. In xy + x'^y'^ -\-ml — 3xy — 2 mx -f 4 a^y — zy^, which terms are like ? which are unlike ? 44._ What is the value of 1^ ? of 1^ ? of 1^ ? of Vl ? of ^1 ? of Vi ? How do these powers and roots of 1 compare ? 45. What is the value of X 2 ? of x 0? of 0^ ? of VO ? Represent algebraically the : 46. Sum of m and n ; sum of the square of a and the cube of h. 47. Difference of r and t\ difference of two times r and three times t. 48. Product of x and y in three different ways ; product of the sum and the difference of x and y, using parentheses. 49. Quotient of u divided by v in two ways ; quotient of u 4- V divided by it — v. 50. Find the cost of 6 apples at y cents each. 51. Grace is a years old. How old will she be in I years ? 52. George has m chestnuts and John has n chestnuts. How many more chestnuts has George than John ? 53. How long is the side of a square whose perimeter is t feet ? 54. If y pounds of tea cost b cents, find the cost of x pounds. 55. Express in brief form a -f a + a + — to 8 terms ; to 7i terms. INTRODUCTORY REVIEW 11 Order of Operations 3. It is agreed among mathematicians that : When only -f and — occur in any expression^ or only x and -f-, the operations are to he performed in order from left to right. Unless otherwise indicated, as by the use of parentheses : When X, -^, or bothy occur in connection with +, —, or both, the indicated multiplications and divisions are to be performed first. EXERCISES 4. Find the value of : 1. 5__34.4 + 2-3 + 6. 7. 6x8-4. 2. 6-2 X 7-f-3 X4--2. 8. 6 x (8 - 4). 3. 8-3-5 + 7 + 9-4. 9. 12-4+7x5. 4. 18-6x8x2-12x5. 10. 9-3x2 + 8-4. 5. 8+2x4-10 + 7-9. 11. (12-5)x 6-3 + 11. 6. 9-3 + 8-T-2 x5-18. 12. (12 - 5) X 6 -(3 +11). Numerical Substitution EXERCISES 5. When a = 3, 6 = 4, c = 5, n = 2, find the value of : ' 1. 8 6. 5. 2ab\ 9. V2bn. 13. aVbd". 2. Sac. 6. {^anf. 10. (bcf. 14. V'4 6Vn. 3. 5(m. 7. f6cl 11. ¥c\ 15. c'^ + 7t^-\ , 6ac ^ 5ab^ ,^ a^ft^ ,^ a^ + b^ o. — • 1/0. • lb. — • an 3 en 4 a^n When x= 6, y = 3j z =0, r = 2, s = |, evaluate : 17. rx -\- yz ^ rs — xz. 21. ^ xr'^ — ^ y^z -\- ^ s"^. 18. sx^ — r'^s + xyz + xy^. 22. 4^9?/2 — ^ x^r'^ — ii x^y'^z. 19. 12^ + 7^y^ + 5xy-^3s. 23. 5 a??/^ — 2/ VrV + 1 a;s2;. 20. 40- + ^)^r+lx4.. 24. x+fr±£iY^_±^'Vs'- 2a; X \ r J\ x"" 12 INTRODUCTORY REVIEW 25. The area (A) of any rectangle is equal to the product of the base (b) and the altitude, or height (h). Write the formula for the area of a rectangle in terms of its base and altitude. Find A when b = 6 and /i=4 ; when 6=12 and ^=4i. Note. — Since the algebraic form is concerned only with the number of units in A, 6, and h, in this and similar exercises the principles stated refer only to numerical measures. 26. The area of a triangle is equal to one half the product of its base and altitude. Write the formula. Find A when 6 = 15 and y^ = 10 ; when b=20 and h = 7. 27. The area of a circle is equal to tt times the square of its radius (7-). (tt = 3.1416, approximately.) Write the formula and find A when r = 6', when r = .25. 28. The volume (F) of a rectangular solid is equal to the product of its length (I), breadth (b), and thickness (t). Write the formula and find V when Z = 7i, 6 = 4, ^ = 1|^ ; when I = 4.5, b = 2.4, t = .7. 29. The hypotenuse (c) of a right triangle is equal to the square root of the sum of the squares of the perpendicular sides (a and b). Write the formula. Find c when a = 3 and 6=4; when a = 6 and b = 8. 30. Interest (i) is equal to the principal (p) multiplied by^ the rate (r) multiplied by the time (t) in years. Write the formula and find i whenp = 650, r = .06, and t = 2i. 31. The material removed from the bed of a river in cutting a channel through a bar consisted of s cubic yards of sand and r cubic yards of rock. The cost of removal, per cubic yard, was c cents for sand and d dollars for rock. Express the total cost (T) by a formula. Find (T), if s = 300,000, r = 2000, c = 12, and d = 4^. INTRODUCTORY REVIEW 13 POSITIVE AND NEGATIVE NUMBERS 6. Including zero, the scale of algebraic numbers is written : ... , -5, -4, -3, -2, -1,0, +1, +2, +3, +4, +5, ... There are as many negative numbers below zero as positive numbers above it, zero being neither positive nor negative. Positive and negative numbers may be added or subtracted by counting forward or backward, respectively, along this scale. EXERCISES 7. Find the algebraic sum, and give its absolute value : 1. 5 2. 5 3. -5 4. -5 5. -4 6. 8-8 8-8 2 10 -3 -2 8. -6 9. 20 10. -13 -9 16 -7 -13 Sept. 1 8P.M- 8A.Mr-> 11-30. In exercises 1-10, subtract the lower number from the upper one ; the upper number from the lower one. The tide gauge shown in the margin is graduated to feet and tenths of a foot. It was nailed to a dock with its zero set at an average stage of low water. By means of it the height of the water in a river was found to vary as follows : Sept. 1, 8 A.M., 3.5 ft. ; 1 p.m., - 1.2 ft. ; 8 P.M., 4.3 ft. ; Sept. 2, 3 a.m., - .8 ft. ; 9 A.M., 5.3 ft. 31. How far did the water fall from 8 A.M. to 1 P.M., Sept. 1 ? 32. How far did the water rise from ^-^ "" 3 a.m. 1 P.M. to 8 P.M.? iP-M- 33. How far did the water fall from 8 P.M. to 3 A.M. next day ? 34. How far did the water rise from 3 a.m, to 9 a.m. ? Sept. 2 -9 A.M. 14 INTRODUCTORY REVIEW ADDITION 8. Law of order, or the commutative law, for addition. Since 3 + 5 = 5+3 and in general a-{-b = b + a, Numbers may be added in any order, 9. Law of grouping, or the associative law, for addition. Since 4 + 3 + 7 =(4 + 3)+ 7 = 4 +(3 + 7) = (4 + 7) + 3 and in general, a + b-{-c=^{a + b)-\- c=:^a+{b + c) = {a + c)-{-b, The sum of three or more numbers is the same in whatever manner the numbers are grouped. Addition of Monomials EXERCISES 10. Add: 1. 7 a! 2. Sr 3. — 5an^ 4. -&«' 5. 2x 2x -2r r aii^ -4 6f -5bi? -3y 9x — 4 aw' 2x-3y To add similar monomials, j^nc? the algebraic svm of the nu- merical coefficients and annex the common literal part. When the monomials to be added are dissimilar, they cannot be united into a single term, but their sum may be indicated as in exercise 6. 6-30. Add the monomials in exercises 6-30 on page 16. Simplify : 31. ^x-\-^x — ^x + x. 34. 3ac — 5ac + 8ac. 32. 96-4& + 76 + 56. 35. .4a2-1.5a2 + 8a2. 33. ^a+3a— 2ia + a. 36. 5Vm +9Vm — |-Vm. 37. S a^b ^-^a^b -11 a^b- 2 a^b + ^a^b. 38. l|ajV-|a;V-ly^ajy + 3-^a^?/2 + ajy. 39. ?>{xyY - 3(xyy - 15(xyy + 4.(xyy + 13(a^)2. 40. 4Va6c + 2lVa&c — 8Va6c + 3Va6c — 6Va6c. 41. 2(x - 1) - 13(aj - 1) + 5(x - 1) + 10(a; - 1) + 6{x - 1). 42. {a — i») + 15(a -x)-{- 7(a — x) — 3(a — x)—2{a — x). 43. 3x(x^-2x + S)-x(x''-2x-i-3) + 2x(x''-2x-^S). 44. B-Vb^c + 19^/b^ - iWb^ + 24 V6 - c - 17 V6^. INTRODUCTORY REVIEW 16 Addition of Polynomials EXERCISES 11. Add: 1. 3x — 2y 2. 2r — 3s + 5« S, a^ -j- 2 ab + b^ -x+5y Sr-4:S- t -3ab-b'^ 2x-Sy -Sr + 2s 2a^-f- ab + ¥ 4:X-5y 2r-5s + 4:t 80^ :^ Rule. — Arrange similar terms to stand in the same column. Find the algebraic sum of each column, and write the results in successio7i with their proper signs. 4. 36 + 4c b. 21+ m- n e. 3c^-4:cd+ d'^ — b + 2c — Z-f4m + 5n 4.c^ + ocd 5b-Sc -5m-2n - c^ -{- 2cd - 6d^ 7. 4 a -f 3 &, - 5 6. 10. 5 x^ -\- y\ y^-3x^. 8. 2x — 4:y, —3x. 11. 2m+ 71, 3 n — 6m. 9. 3d + 4c, 2c-5d 12. 2 a^b + c'', 2 c'^ - a^b. 13. 4.r-5t,2r-s + 3t,2s-3r, and s + 2t. 14. a — i 5 + c, I a + ^ 6, I a — ^ c, and | ?> — | c. 15. Va + Va6, 2.Va — Vb + 2-Vab, and 2 V6 - 3 Va6 - Va. 16. xi-3(a + l)-y, -{a -{-l)-2x-\-4:y, and 3a;-4(a + l). Simplify the following polynomials : 17. 4.a-b-\-2c + 3b-d-3a-\-3c+2d-4tb+5a-4.c-3d. 18. 3af-22/* + l+3?/''-4aj« + 2/^-6 + 2af + 4-22/* + af. 19. (a + c)x — (b — d)y — 2{a+ c)x + 3(b — d)y -\- 4.{a + c)x -{b-d)y. 20. 2ay — 3ac — 4: ay -\- 4: ac — 6 ay + 5 ac + 11 ay — 4: ac — ay. 21. 2c-7d + 6n-{-llm-3c-5n + Sd-3m-{-10c + 7n -2d-Sc + 4.d-3n-Sm-6n+m-3d-\-2m. 22. 5 am — 3a2m2-h 4 - 4 am + a'^m'^- 2 + 5 + a^m^ -6+3 am + 2 a^m^ _3ct^_3_^3^^_l_2_ a^m^. 23. 6Vx — 5 Vaj?/ + 3 V.^ — 4 Va; -f 6 Va;?/ — Va; — Vy + 3 V.y — 2Vxy + Va; + 2^xy — 3Vy + 6\/x + 4:Vxy + Vy. 16 INTRODUCTORY REVIEW SUBTRACTION Subtraction of Monomials EXERCISES 12. Subtract the lower monomial from the upper one : 1. 2. 3. 4. 5. 12 a Sb 3d2 -7c'x S{a-b) 5a -46 2d^ -2&x - 2(a - b) ~Ya "126 3^2 -2# -5c^x ^la-b) Rule. — Consider the sign of the subtrahend to be changed, and add the result to the minuend. 6. 7. 8. 9. 10. 7x 7x 7x 7x 7x 5x 6x '7x Sx 9x 11. 12. 13. 14. 15. 4 a X a — m -1 1 -2 h -2n 16. 17. 18. 19. 20. a^b mux --2a¥ -Vr - 5(aj + y) -a^b 2 mux -4.a¥ 3Vr 9(x-^y) 21. 22. 23. 24. 25. 5xy -Sb 5 n V 2sn 2cd 27. — 421? 2(a + 5) 29. i^-y) 26. 28. 30. Sa"" - 15 b^c" — 7 xY -13VS -3(a + 6) -2a^ 9 6V -Uxy — 5Vx -10(a + 6) 31-55. In each of the exercises 6-30, subtract the upper monomial from the lower one. INTRODUCTORY REVIEW 17 Subtraction of Polynomials EXERCISES 13. Subtract the lower expression from the upper one : 1. a-\-b-{-c 2. 3m — 2n— p^ 3. x'^-{-2xy— y'^ a — 6 4- c — m + n — 6 p^ —x'^ — Sxy+ y '^ 2 6 4 m — 3n + 5j92 2x^ -\-^xy — 2y'^ Rule. — Arrange similar terms to stand in the same column. Consider the sign of each term of the subtrahend to be changed, and add the result to the minuend. 4. a-\-b a-b 1 1-x- -a;2 6. X -y x + y 9. X 2x-x^-\-4: 6. 10. — a-\-m a — m 7. 11. 2r- s r-2s 8. 5 — X + 1 a-\-x — 2 12-19. In exercises 4-11, subtract the upper expression from the lower one. 20. From a^ + 1 subtract 1 — a + a^ — a^ + a^ 21. What number subtracted from a -^ x will give a-j-x? 22. Take Ax'"" + 2 x'^y'' + 5^" from 7 o;"^ 4- 2 x^'y'' + 9 y\ 23. To what must r^ — 4 s- be added to produce 3 s^ — r^ ? 24. From 5x — 2y subtract the sum of 2x — y and x — 2y. 25. Take 6 m" + 11 m'n^ + bn^ from 10 m' 4- H "rn'n^ 4- 8 n'. 26. From the sum of 1 4- a; and 1 — x^ subtract 1 — x-\-x^—o?. 27. What number added to a^ — ab — ac -\- b'^ — c^ will give ? 28. From the sum of a—b — c and a + b + c subtract the sum of a — b + c and a + 6 — c. 29. From the sum of 3 a;^ _ 2 a; + 1 and 2x — 5 subtract the sum oi X — x^ + 1 and 2 ic^ — 4 a; + 3. If r = c^ - d\ s = c2 4- d2, ^ = c^ 4- 2 cr^ + d\ and ?i = 2 cd, 30. r — s 4- ^ 4- i^ = ? 32. r — s 4- ^ — ?/ = ? Z\. r + s — t-\-u — ? 33. 5 — r — 21+ ^ = '^ milne's sec. course alg. — 2 18 INTRODUCTORY REVIEW PARENTHESES Removal of Parentheses 14. When numbers are included by any of the signs of aggre- gation, they are commonly said to be in parenthesis, in a paren- thesis, or in parentheses, 15. The sign + before a parenthesis indicates that the terms in parenthesis are to be added and the sign — , that they are to be subtracted. Hence, Principles. — 1, A parenthesis preceded by a plus sign may be removed from an expression without changing the signs of the terms in parenthesis. 2. A parenthesis preceded by a minus sign may be removed from an expression, if the signs of all the terms in parenthesis are changed, EXERCISES 16. Simplify each of the following : 1. 54-(~a). 8. x + v-{y-z). 2. x-\-{y — z). ' ' 9. a -f c — (a -f-d). 3. l—{r—s). ' 10. a— 6— (— c+a). 4. m —{m — n). 11. l — {t + v) + {u-\-v). 5. a+(— 6 + c). 12. .5a; — a +(1.5 a; + a). 6. 4c+(d-2c). 13. ^:x? + xy -{y'^ + 2xy + x^), 7. a-(-6-f2a). 14. a + 6 -(2a + 2 6) + (4 6 - a). When an expression contains parentheses within parentheses, they may be removed in succession, beginning with either the outermost or the innermost, preferably the latter. 15. Sim^mj6x~[3a-\4.b-\-(Sb-2a)-Sb\+4.xl Solution. 6a;-[8a-{4 6+(8 6-2a)-3ft}-|-4x] Prin. 1, =6x-[3a-{46 + 86-2a-36} + 4x] Prin. 2, = 6 a; -[3a -45 -86 + 2a + 36 + 4a;] Prin. 2, = Qx -Sa + 4b -\- Sb -2a - Sb - 4:X Uniting terms, = 2x — 6a + 9b. INTRODUCTORY REVIEW 19 Simplify each of the following : 16. aH-2&+(14a-56)-J6a + 66-(a + 46)|. 17. 12a -{4-3 6 -(6 6 + 3 c)+?>- 8 -(5a -2 6- 6)j. 18. 25 - [10 - 11 - 7 -(16 - 14) 4- 8 + 6 - 3]. 19. x^-[x^-{l-x)^-\l + x^-{l-x)-\-x^l. 20. 1 -x-\l-[x-l-{-(x-l)-(l-x)-x']-i'l-xl. 21. '— \Sax — [5xy — 3z'] + z—(4cxy-{-[6z-\-7ax'] + 3z)\. 22. l_{a-[-2a+a2-(a2+a3)-4a2] + [l-(3a + 4a2)]J. Grouping Terms in Parentheses 17. It follows from the principles in § 15 that : Principles. — 1. Any number of terms of an expression may he inclosed in a parenthesis preceded by a plus sign without chang- ing the signs of the terms to be inclosed. 2. Any number of terms of an expression may be inclosed in a parenthesis preceded by a minus sign, provided the signs of the terms to be inclosed are changed. In grouping terms, it is customary to make the first term of each group positive by choosing the proper sign, -|- or — , to precede the group. EXERCISES 18. Group as binomials without changing order of terms : 1. a + b + c-d. 4. x^-y^ -xy -hy'^- 2x^-2 f, 2. a — b — c—d. 5. l — x+x'^ — :i(^ — x^+:xf—:](^+x\ 3. a + b-c-^d, 6. l-2x-~4.x^ + ^^-lQx^-32x\ Group the last three terms as a subtrahend : 7. x'---y^ + 2yz-z\ 9. y^ + v^ -:x^ -\-2 x'^z-z''. 8. c^-h'^-2bd-d\ 10. c2 + 2cd + d2-a^-a3 + a2. Group the terms of like degree beginning with the highest : 11. a?J^a-b-\-b\ 14. x^ -2 xy + y'^ -2x + 2y. 12. 2-x^-{-2xy-y'^. 15. c + & - &d - d^ -\- cd-\-d. 13. r^ + s'- 3rs -hs2 16. a^ + 4:a^b ~¥ -{■ a — ab -^b. 20 INTRODUCTORY REVIEW Collecting Coefficients EXERCISES 19. Add: 1. ry 2. sw 3. — ax 4. {b-2c)v sy — tw -bx (b + g)v (r + s)y (s — t)w — (a + b)x {2 b- c)v 5. —4m 6. -ay 7. 2pq 8. (2c-d)z — bm 2 by -5q {d - 8 e)z 9-16. In exercises 5-8, subtract the lower expression from the upper one ; the upper expression from the lower one. Simplify: 17. (a + c)x+(a~- c)i». 21. (4.-\-c^)v-(2 c^ -\-2)v. 18. (a + c)x-(a-G)x. 22. {a~by +(2b - a)s\ 19. {b-d)y-{b-\-d)y. 23. {a^ + W]x^ - {a^ - b'')x\ 20. (a + 2)2;+ (3 -a);^. 24. {2t^ + l)y''-\-{4.-3t'')y'', 25. Collect the coefficients of x and of y in ax- by —bx—ay. Solution. — The total coefficient of x is (a — 5). The total coefficient ofyis (— a—b)^ or — (a -f- 6). .*. ax — by — bx — ay = (a — &)x — (a + 6)2/. Collect the coefficients of x and of y in: 26. aoj — 6a; + ciy + dy. 31. 2 ex — ay + by — S dx. 27. mx—nx — ry + sy, 32. 3 ax -{-2 ay — by — 2 bx. 28. aa; — 3 a; + d?/ — 4 2/. 33. J.aj + 2 J??/ — ^'.t — jB'^/. 29. nic — n?/ — 2 a; + 5 ?/. 34. 3 rx — 5 my -\-25x — ny, 30. 2ax — 2ay — x — y. 35. 1 px — 4: qy — 12 x -\-l() y. 36. a^a; — ft^y — 2 ax -{-2by -\-x + y. 37. (a^ — l)a; —{a^-\-l)y — 2ax-^4.ay + 2x — 3y. 38. 6^a? — 7^^.v — 3 ^^a; + 3 ^i^.y — 3 bx -^ 3 ny — x — y. 39. (a2 — 4 a + 2)aj +(a2 — 6 a + l)?/ ~ (a^ - 3 a)ic + 8 i/. 40. (5c2-2c?>-(3c2 + 4c^)2/-(4c2 + l)a;+(2c2 + 4)?/. INTRODUCTORY REVIEW 21 MULTIPLICATION 20. Law of order, or the commutative law, for multiplication. Since 2x3 = 3x2, and in general ah = 5a, The factors of a product may he taken in any order, 21. Law of grouping, or the associative law, for multiplication. Since 2x3x5=(2x3)x5 = 2x(3x5) = (2x5)x3 = (3 X 5) X 2, and in general ahc = (ah)c = a{hc) = (ac)h = (hc)a, The factoids of a product may he grouped in any manner. Multiplication of Monomials EXERCISES 22. Multiply : • 1. 2. 3. 4. 5. a — X -2r2 5ah''(^ — 4 0?"" h 22/ -3r3 -2a'h^c -3a;- ah -2icv 6r' -10a^6V 12 x'^'^^ In finding the product of two monomials, apply in succes- sion the following laws for multiplication : Law of signs. — The sign of the product is + when the mxdti- plicand and multiplier have like signs, and — lohen they have unlike signs. Law of coefficients. — I7ie coefficient of the product is equal to the product of the coefficients of the multiplicand and multiplier. Law of exponents. — The exponent of a number in the product is equal to the sum of its exporients in the multiplicand and multi- plier. 6. 4a2 -1 7. — 6 m*n^ - 3 nhn^ 8. ap'^q^ 13. — 4 a'b' - 3 rr-25* ab"xY 14. 10. - 2 ahnhi"^ 8 b'n'm' 11. — x'^yh 5 xy^z* 12. - ah^c' d'"b^''c 15. m'^n'b'^y'' 22 INTRODUCTORY REVIEW 23. When there are several monomials, by the law of signs, — ax—b = -{-ab', — ax —hx —c = -{-abx—c = — abc ; — ax —bx — ex —d = — abc x—d = -\- abed ; etc. Hence, Tlie product of an even mimber of negative factors is positive ; of an odd number of negative factors, negative. Positive factors do not affect the sign of the product. EXERCISES 24. Find the products indicated : 1. (_1)(_1)(-1). 4. {-2xy){-^xy){bx^){-y'^), 2. (_2)(-a6)(-3a2). 5. (-4 6c)6(-3c2)c(-6) (- c). 3. {-a^x){4.bx)\^5a'y 6. (- 23)(- 2^)(5 • 22)(-52 . 2). Multiplication of Polynomials by Monomials 25. The distributive law for multiplication. In general, a{x -{- y + z) = ax -\- ay + az. That is, Tlie product of a polynomial by a monomial is equal to the algebraic sum of the partial products obtained by multiplying each term of the polynomial by the monomial. EXERCISES 26. Multiply as indicated : 1. 2a{^x + 2y). 6. 6m\(om'' -2m}n). 2. -upw-uv), 7. a2"(3 0^ - 10 aY). 3. — 3 6(4c + 3e). 8. aaj2(^^ — a;"-i -f a;^-^). 4. a26c(3a*-4a36). 9. ^tu\u^ + 4.1? -2thi^y 5. 2xy{bx^ —lOxy). 10. —xyz{—xy + yz + 2xz), 11. - 3 yz{f -Z y'^z^ - 3 2/:^^ + 2;^ - 2/^ + 3 fz). 12. abc (a252 _ 2 aV - 2 b'^c' _ a^ _ 4 6^ - c^ - 5 abc). 13. - bc{¥ + c^ - 2>' - c^ 4- &'c2 _ 4 62c + 8 6c2 - 2 be). 14. m'^n^ (m^ — 5 mhi^ — 16 mhi^ + 24 mn^ — n^^). 15. a;""32/"'"^X^^2/'"~^ — ^ iK^-"^/"'""^ + 10 x^-^'y'^'^ — 5 aj^-^"^/^-"*). INTRODUCTORY REVIEW 23 Multiplication of Polynomials EXERCISES 27. 1. Multiply a^ — ai/ + 2/^ by a + 2/- TEST (When a = 2 a^ — ay + y d + y a; PROCESS ,2 a^y + ay^ a^y — ay^ -\- y^ y 7 5 = 35 and y = 3) Rule. — Multiply the multiplicand by each term of the multi- plier and find the algebraic sum of the partial products. Test. — To test the result, assign to each letter any value, and observe whether for these values, Product obtained = midtiplier x multiplicand. It is usually most convenient to substitute 1 for each letter, but since any power of 1 is 1 , such a value does not test the exponents. Multiply as indicated, and test : 2. 3. 4. 5. 6. 7. 14. 15. 16. 17. 18. 19. 20. 21. ;3a + 4)(a + 2). 2x + l){3^-x). 2a + 4)(4a-3). ;6 6 + l)(2 6-4). ^c-{-2d){2c + d). 2a;-3a)(3a;-4a). x^"" + 2 a;"2/"* + y^'^)(x'' — y"^). 5x-5x'-i-10){12-^30x + 2x'). 4.x-3x' + 2a^)(3x-10x^ + lU). 2 a' -3b'- ab) (3 a^ - 4 ab - 5 W). 8. (ab-15)(ab + 10). 9. (3a-{-cd)(4:a + cd). 10. (x' + x + l)(x — l). 11. (f + by-b')(y-b). 12. (a + b + c)(a + b-c), 13. (x-y + z){x + y-z). a^ + b'^ -\- G^ — ab — ac— bc)(.a + 5 + c). ^a+ Y 1 _^ a;c-l^a+l _|_ l)(a^«- ^+1 — X' .a+y-l ^ J[)^ 1 2,2a+l . \z'^ + ^2a-i)(2 2;2«-i + 2 ^2«-2 _!_ 2 ^i^^'-s). 24 INTRODUCTORY REVIEW 28. When polynomials are arranged according to the ascend- ing or the descending powers of some letter, processes may often be abridged by using the detached coefficients. EXERCISES 29. 1. Expand (2a^~3a^ + 3a;-|- 1)(3 x + 2). FULL PROCESS 2aj4-3aj3+3a;4-l 3 a; + 2 DETA CHED COEFFICIENTS 2 -3 +0 +3 + 1 3 +2 6 -9 +0 +9 +3 4 -6 + + 6 +2 6i»5~5aj^-6cc3H-9a;2 + 9ic-h2 (^ x^-b x!"-^ a? + ^ x^-^^ x+2 Since the second power of x is wanting in the first factor, the term, if it were supplied, would be x'^. Therefore, the detached coefficient of the term is 0. The detached coefficients of missing terms should be supplied to prevent confusion in placing the coefficients in the partial products and to avoid errors in writing the letters in the result. Arrange properly and expand, using detached coefficients : 2. (x + x^ + l^x%x-l). 3. (o^ + 10 - 7 oj - 4 x^){x - 2). 6. (a^ + 4a2_lla-30)(a-l). 6. (4a2-8a + a4-3)(2 + a). 7. (2 m - 3 + 2 m^ - 4 m^){2 m - 3). 8. {x + x^-5)(x''-3-2x). 9. (&2 + 55_4)(_4_^252-36). 10. (f-5y + 2y'-{-S){2y^ + y + l). ' 11. (4 ^3 -I- 6 - 2 71^ + 16 n - 8 n^ + n')(n + 2). 12. (7 + 5 a;2 _ 4 o.'S 4- 3 oj* - 2 x^ + x^){x^ + 2 a: + 1). 13. {l + x + 4.x' + 10a^-\-4:ex' + 22x^){2x^-i-l-Sx). INTRODUCTORY REVIEW 25 Special Cases in Multiplication 30. Show the truth of each of the following formulas by- actual multiplication, and state the corresponding principle in words : Formula 1. {a + by=a'' + 2ab + b\ Applications. (x + 3)2 = x'^ -\- ^x-\- ^. Also, 142 = (10 + 4)2 = 102 + 2 X 10 X 4 -h 42 = 196. Formula 2. {a — by= a} — 2ab + b"^. Applications. (2 — yY = 4 — 4 ?/ -f 2/2. Also, 192 = (20 - 1)2 = 202 - 2 X 20 X 1 4-12 = 3(51. Formula 3. (a + 6) (a - 6) = a^ - 6^. Applications. (x -f 5) (x — 5) = x^ — 25. Also, 32 X 28 = (30 + 2) (30 - 2) = 302 «. 22 = 896. Formula 4. (jr + a) (jr + 6) = jr^ -f (a + b)x + ab. Applications. (x + 2) (x + 5) =: x2 + 7 x + 10. Also, (2/ + l)(2/-4) = 2/"^-3?/-4. Also, (n - 2)(/i - 3) = n2 - 5 n + 6. Formula 5. {a+ b -^ cY= a} + b''^ c''+2 ab-\^2 ac-{^2 be. Application. (x — y-^Zzy^ = x^ -\- y'^ + 9 z^ - 2 xy -{- 6 xz — 6 yz. EXERCISES 31. Expand by inspection : 1. (x + yy. 9. 212. 2. (r-sy. 10. 182. 3. (a + 4y. 11. 322. 4. (x-sy. 12. 432. 5. (5 + 2/)^. 13. 282. 6. (a2+62)2. 14. 522. 7. {x^-fy. 16. 272. 8. (r4-l)^ 16. 342. 17. (22)(18). •18. (r + s)(r — 5). 19. (^+2)(t+3). 20. (3r+4)(3r + 4). 21. (ab-\-cd)(ab —cd). 22. (x^^^ + y^)(x^^^ — y^), 23. (a4-& + c)(a + 6-c). 24. (^a-'b + c){a-h-\-c). 26. (3-22/2)^. 34. (.2c^-5y. 27. (a"" + 6^)2. 35. (ix'^ + ay. 28. (1-Sxyy, 36. (Im^-^ny 29. (2a^ + ,5y. 37. {abc-^dy. 30. (r - .2 s)2. 38. (ix-^yy. 31. (^m_2^n)2. 39. (a + 5 - cy. 26 INTRODUCTORY REVIEW Expand by inspection : 25. (a;- + r)^. 33. (l + r^s^y. 41. (a; + 3)(a; -4). 42. (2ii;+l)(2a;+2). 43. aa-2)(ia-4). 44. (aa?-l)(aa; + 3). 45. (xy + 4.)(xy-4.), 46. (a2"-6^)(a2" + 6"'). 47. (3r+2)(2r + 10). 32. (4a + .3&)2. 40. (x + y-^zy. 48. {5x-,12){5x+12). 49. (5-c-d)2. 67. (2a + 2/>(2a + a;). 50. (Sxy-\-2a^z^y, 68. (22/ + ;^)(32/-22;). 51. Qrs-ir+i)2. 69. (2/V-8)(^V+5). 52. (ac + M-ce)2. 70. {5r^ + 2 s)(2r^ - 5s). 53. (m""^2 _ ^6^c+i^2^ Yl^ (3a;" + m")(3a;~ + 7i"^). 54. (2.5 4- 12p2g2r3)2. 72. (r4- M^)(r- M^). 55. (2a + 3&4-4)2. 73. (5 cdx + 1)(5 cdx - 6). 56. (x'^-^-y^+^y, 74. (3 ?)aj3 + 7)(3 + 7 6a^). 57. (a26V H- c?V)2. 75. (adV -10) (ad^a^ -3). 58. (a;2-32/'-2^4)2. 76. (x-+' + Sy)(x^-^^ -2 y). 59. (|mV + fpY)^- 77. (^ + ?/ + 'y)(^-t^-'y). 60. (3 a^ + 4 a6 + 6)2. . 78. (a"*6" + aj^)(a'»2>" — a;^). 61. (4de/-ia5c)2. 79. (a^^-^ -2/)(2aJ'*"^ + 32/). 62. (5rV + 1.5r«-^)2. 80. (Ta^a:^ _^ g^n^^j ^2^_ 6;2ny 63. (18 - 2 a + 3 6c)2. 81. {a^-'' + b^-'')(a^-^ + b^~^), 64. (ic+5 6)(a;-5 6). 82. (4. a^ - 3 y^)(2 az' + y''). 65. (3a; + 4)(3a; — 5). 83. (xf'y^ -ho(^y'')(x''f — x^y""). 66. (2a+6)(3a-26). 84. (a + 6 + c + d)(a + 6-c + c?). INTRODUCTORY REVIEW 27 DIVISION Division by Monomials EXERCISES 32. Divide as indicated : I. a) -ah 2. -5x)25xy'^ 3. 5'')5'^ 4. In finding the quotient of two monomials, apply in succes- sion the following laivs for division : Law of signs. — The sign of the quotient is + when the dividend and divisor have like signs, and — when they have unlike signs. Law of coefficients. — Tlie coefficient of the quotient is equal to the coefficient of the dividend divided by the coefficient of the divisor. Law of exponents. — The exponent of a number in the quotient is equal to its exponent in the dividend minus its exponent in the divisor. Since a number divided by itself equals 1, a^ -^ a^= a^-^= a^ = 1 ; that is, a number whose exponent is is equal to 1. (Discussed in § 175. ) 5. 2^)21 10. -x)^. 15. 4 g ) - 8 s^. 20. 8 a'b^ ^ - 4: a''b\ 6. 22)2^. 11. z^)^. 16. -2n^)67i\ 21. - 20 by -t- 5 b^ 7. 3^)3^. 12. 2 )4 m . 17. 7l' )-Ul\ 22. _6ay---9ay. 8. 52j5^. 13. -2>f. 18. 2 7rr )4 Trrl 23. - 4^ x'^z' -^ 32 x^'z', ^ oW^ ^^ Sxyz j^ 4a^5^c^ 24 ^^'(^-y)\ 33. The distributive law for division. Since (a + 6) a; = ao^ + 6a?, (aic + bx) -r- .t = a + 6 ; that is, 7%e quotient of a polynomial by a monomial is equal to the algebraic sum of the partial quotients obtained by dividing each term of the j)olynomial by the monomial. 28 INTRODUCTORY REVIEW EXERCISES 34. Divide: 1. - xy )ax^y - 2 xy'^ 2. 3 ax ^)^ aV — 12 a^x^ + 6ax^ — ax + 2y ■ 3aa?- 4a2a^ + 2.T2 ' 3. 6 aW - 9 a&3 by 3 a6. 7. - a - 6 - c - d by -^ 1. 4. 4 a^y + 2 a;y by 2 x'y'^, 8. - a + ^^^ — a^c by — a. 5. a6c^ — 2 a^^^c by — a6c. 9. xHj — a?^/^ + x^y^ by i a??/. 6. 9 xhjH + .3 a;?/2;2 by .3 xy. 10. c^d — 3 cd^ ^ 4 &d^ by - cd. 11. 34 a^o^y - 51 a~+2^y - ^% a^+Vi/^ by 17 a^x^- 12. 8 a76^+^ - 28 a'^h^'^'' - 16 a^6^+2 ^ ^4^x+i ]^y 4 ^4^3^ 13. 2 a2(6 - cy - 3 a6(& - c)^^ 2 5c(6 - c) by (2> - c). • 14. 3(a; — 2/)-3a;(a;— ?/)2 + 4aj2(aj — 2/)^by (a; — ?/). 16. a2^+^?>^+2 _ ^2x+3^^+4 _^ ^2^+25^+6 _. ^2x+l^z+8 ]^y a^^^^^^ Division by Polynomials EXERCISES . 35. 1. Divide 2 a^- ha^h + ^ aW - 4 a^^ + ^^ by a^- ah + 62. PROCESS , .^> TEST a-— ao-[-tr — 7-^7 = — 1 2a2-3a&+52 2a'-2a^b + 2aW • _ 3 a36 +4 a^¥-4 aW v^a^^ (When a = 2 -^a^h + 'daW-:iah^ and 6 = 3.) a262- a63+?>' aW- ab^-i-h' Note. — When a = 1 and & = 1 the test becomes -r- 1 = 0. In gen- eral, -r- a = ; that is, zero divided by any number equals zero. Similarly, the result may be tested by substituting any other values for a and 6, except such values as give for the result -r- 0, or any number divided by 0, for reasons that will be shown in § 283. INTRODUCTORY REVIEW 29 Rule. — Arrange both dividend and divisor according to the ascending or the descending powers of a common letter. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient. Multiply the whole divisor by this term of the quotient, and sub- tract the product from the dividend. The remainder will be a new dividend. Divide the new dividend as before, and continue to divide in this ivay until the first term of the divisor is not contained in the first term of the new dividend. If there is a remainder after the last division, ivrite it over the divisor in the form of a fraction, and add the fraction to the part of the quotient previously obtained. Divide, and test each result : 2. x^ -\- a^y -^ xy'^ + f hj X + y. 3. 6 a2 + 13 a& + 6 62 by 3 a + 2 6. 4. 3 m^ — 4 am^ -f a^m'^ by am — 1. 5. x^ — 4 .T^2/ + 6 ^?/^ — 4 xy^ -{• y^^J x — y. 6. a^ 4- 5 a^x -f- 5 ax^ + cc^ by a^ + 4 ao; + x'^. 7. a^ 4- 5 a^ - a^ + 4 a* +'2 a - 3 a^ + 3 by a - 1. 8. c^ -d^hj c — d. 12. ic^ + 81 by a; — 3. 9. x'^ + y^hj x + y. 13. a^ + 6^ by a + 6. 10. ?^ — s^ by r + s. 14. a;^ — 64 by a; + 2. 11. a^ — b^ hj a — bi 15. m^ + n^ by m^ + n^, 16. x^''-^ -{- y^''-^^ hj x''-^ + y^'+K 17. x^ -{- y^ -^ z^ — S xyz hy x -\- y -[- z. ' 18. a^ _ 6^ + c^ + 3 abc by a^ + 6^ + c^ + ah -ac + be. 19. 3^g m* + I m — f m^ 4- -y- — | m^ by | m'^ — m — |. 20. ^aV-|aa^ + |a^-|aM)y f a;2+^a2-|aa;. 21. a?" + ?/" by a; 4- 7/ to live terms of the quotient. 22. - X^'^^Y' - 2 a;2r+3^2«+l _ ^2r+5^2.+2 |)y _ ^y-1 _ ^r+2^«^ 30 INTRODUCTORY REVIEW Divide, using detached coefficients : 23. 3c^ + 4:a:^+7x-\-6hjx — 2. PROCESS 1 + 4+ 7+ 6 1-2 1-2 1+6 + 19 6+ 7 = x 2 + 6a;+ 19 6-12 19+6 19-38 44, remainder a^ + 3x''- 4a; + lby a; + 2. a4 + 2 a^ + 3 a2 + a-2 by a-1. 21 a;^ + 4 - 8 oj'^ + 6 a; - 29 a^ by 3 a; - 2. 27. y^+7y^l0y^-f-{-15hjy'-2y^3. 28. 7 aj3 + 2 a;* - 27 a;2 + 16 - 8 a; by a;2 _,_ 5 ^ _ 24. 25. 26. 29. a5-fa^+ 29tt3_||^2^5^_^by^__ 3. 30. a* — Ibya — 1. 33. a;^- 5 a;+4 by a;^— 2 a;+l. 31. a;^ + lbya;— 1. 34. aj^+aj^+a^+lbyaj^-ar^+l. 32. aj^-^i^ by a;+|. 35. a;^+8 a;+7 by a;2+2 a;+l. 36. Synthetic division. — This is an abridgment of the method of division by detached coefficients that is most useful and easily applied when the divisor is a binomial of the form x±a, For example, in the process at the top of the page, since the first number in each partial product is the same as the number directly above, it may be omitted and the terms of the dividend need not be brought down. The process may then be written more briefly thus : 1+4+7+6 -2 1 + 6 + 19 6 12 19 44, remainder INTRODUCTORY REVIEW 31 We may write this process more compactly and further shorten the work by omitting the first term of the divisor and writing the second term with its sign changed^ which will give all the partial products with changed signs so that we may add them to the dividend instead of subtracting them. Also, since each partial product now consists of but one term, we may write all the partial products in the same horizontal line under the dividend, thus : Dividend l + 4-|.7+6[2_ Partial products 2 + 12 + 38 Quotient 1 + 6 + 19 | 44, remainder That is, the quotient is x'^ + 6x + 19 and the remainder, 44. EXERCISES 37. Divide by synthetic division : 1. x^+a^-3x^-lTx-30\)jx-3', hj x+2. Solutions l-fl-.3-.17_30]^ l4.1-_3-17-30 |-2 3 + 12 + 27 4- 30 -2 + 2+ 2 + 30 1 + 4+9 + 10 ' 1_1_-1^15 In the first case the quotient is x^ + 4x^ -}- 9x -{- 10 and in the second it is x^ — cc2 — a: — 15. The division is exact in both cases. 2. a^ + 4 a;2 + 5 a; + 2 by aj + 1. 3. 1 + 2 a; + 3 a?2 + 4 a;3 by 1 + a?. 4. 5^ -12v + Sv^-{-4hjv + 2. 5. yi+3y^-4.y' + Sy-24.hjy-3. 6. 5 a^ + 2 a'* - a^ - a2 + 2 a + 3 by a - 1. 7. x' ---x^-2x'-a^+3x^-10x'i-4.x''-36hj x-2, 8. t'-^2t' + ^\f + it'+^^^t + ihjt^i, 9. a^ + 1 by a + 1. 13. a^ - 3 a;^ - 4 by a: - 2. 10. z^-32hyz-2. 14. 4:y^ -3 f - fhj y-1. 11. a^- 256 by a + 4. 15. m^-lQm-B by m + 2. 12. u^+2A3hju + 3. 16. a^ -38 a + 12 by a-2. 32 INTRODUCTORY REVIEW Special Cases in Division 38. Show the truth of these divisibility principles for positive integral values of n, by substituting such values and actually dividing : Principles. — 1. x^ — y"" is always divisible by x — y, 2. x"" — y"" is divisible by x + y only when n is even, 3. x"" + 2/"* is never divisible by x — y, 4. X" + y"" is divisible by x + y only when n is odd. Proofs of these principles are given on page 54. 39. The following law of signs may be inferred readily : When x — y is the divisor, the signs in the quotient are plus. When X -\- y is the divisor , the signs in the quotient are alter- nately plus o>nd minus. 40. The following law of exponents also may be inferred : The quotient is homogeneous^ the exponent of x decreasing and that of y increasing by 1 in each successive term. EXERCISES 41. Write out the quotients by inspection : 1. (c3 + cZ3) -J. (c -f- d). 7. (x^-64:)^{x-i-2). 2. (a^ - b^) -^(a- b). 8. (x^y^ + a') -- {xy + a). 3. (r^ + s^) ~ (r+ s). 9. (m^ + n^) -r- (m + r?). 4. (1 + a^) ^ (1 + a). 10. (a^ + 128) - (a + 2). 5. (a;5 _ y^^ -^{x- y). 11. {y' - 1000) -^ (y - 10). 6. (x^ — l)^(x + 1). 12. {x^^ + y^z^) -^ (x'^ + y^)' 13. By Prin. 4, find an exact binomial divisor of a^ + ^^• Suggestion. a^ 4- x^ may be written as the sum of two cubes thus, Find exact binomial divisors :' 14. a^ — ml 18. x^ + /. 22. a^ — b\ four. 15. 53 + x\ 19. x'^ + a\ 23. a^ - 1, five. 16. x^ — a^ 20. a^^ + ^^l 24. a^ — b^, six. 17. c^ + nl 21. a^ - 27. 25. a^^ - 6^^ five. INTRODUCTORY REVIEW 33 EQUATIONS AND PROBLEMS 42. Write an equation and point out its first member; its second member. 43. The following axioms are constantly used in the solution of equations and problems : 1. If equals are added to equals, the sums are equal, 2. If equals are subtracted from equals, the remainders are equal. 3. If equals are multiplied by equals, the products are equal. 4. If equals are divided by equals, the quotients are equal. 5. Numbers that are equal to the same number, or to equal numbers, are equal to each other. 6. The same powers of equal numbers are equal. 7. The same roots of equal yiumbers are equal. In the application of axiom 4, it is not allowable to divide by zero, or any number equal to zero, for the result cannot be determined (§ 283). EXERCISES 44. 1. Solve X — 2 = 3 by adding 2 to each member (Ax. 1). 2. Solve ic + 8 = 10 by use of axiom 2. 3. Using axiom 3, find the value of oj in \x = 5. 4. Apply axiom 4 to the solution of 5 a; = 30. 5. Solve f a; = 12 in two steps, first finding the value of -J- x by axiom 4, and then the value of x by axiom 3. Solve and give the axiom applying to each step : 6. 3a: = 9. 13. \r = 1.5. 20. |m = 8. 7. 407 = 6. 14. ia; = 2.5. 21. f w; = 9. 8. |a; = 3. 15. 2/ + 3 = 10. 22. f. = 15. 9. \x = 5. 16. 'y - 2 = 15. 23. 5n-l=9. 10. 5aj = 10. 17. 4 + aJ = 20. 24. 4/i4-3 = 5. 11. 3aj=16. 18. w; - 4 = 16. 25. i6 + 2 = 8. 12. |a; = 12. 19. 2 ;2 + 3 = 8. 26. ia; + 2 = 6. milne's sec. course alg. — 3 34 INTRODUCTORY REVIEW Transposition in Equations 45. In solving the equations on page 33, the student may have discovered that the effect of applying axioms 1 and 2 has been to make a term disappear from one member of the equa- tion and appear in the other member with its sign changed. That is, Principle. — Ajiy term may be transposed from one member of an equation to the other, provided its sign is changed. EXERCISES 46. 1. Solve 6x - 3(x - 6)= 4.(2 X -1)-^ 2, for x. Solution. 6x — 3(a: — 6)= 4(2a: — 1) + 2. Expand, 6ic — 3x + 18 = 8z - 4 + 2. Transpose terms, Qx — Sx — Sx =— 18 — 4'-f 2. Unite similar terms, — 5 x = — 20. Divide both members by — 5, x = 4. Verification. — Substituting 4 for x in the given equation, we have, 6 . 4 - 3(4 - 6) = 4(2 . 4 - 1) + 2. Simplify each member, 30 = 30, an identity. Hence, 4 is a true value of x and satisfies the equation. Rule. — Remove signs of grouping, if there are any. Transpose terms so that the unknown numbers stand in one mem- ber and the known numbers in the other. Unite similar terms and divide both members by the coefficient of the unknown number. Find the value of x, and verify the result, in : 2. Sx-4: = 5. 9. 2(ic-l)=12-5a;. 3. 5-i-lx = S. 10. 16 = 5x-(3x + l). 4. 2^_8 = _.2. 11. |a; = 15-i(a; + 3). 5. 2x-\-6 = S-x. 12. 3x = 5x—4.(x-3). 6. 1.5x-7 = 5-\-x, 13. 5-2(x+l)=6-^4:X. 7. 24.-2x = 3x-6. 14. 5(2 - x)+ 6 = 2x- 5. 8. 2x-^x = 6 + {x. 15. 30-a; = 20 + 3(a;4-2). 1 INTRODUCTORY REVIEW 35 Solve and verify : 16. 3a;- 2(3-0;)= 9. 22. 2^(^ - 5)- ^2 ^ ^2 _^ 30. 17. 4(a;-2)=3(a;-l). 23. 5x-3(x - i)= ^.x -\-7, 18. 3r = 2(l-r)+18. 24. 4(a; - 5)- 3(a; + 6)= 0. 19. 5-3x-7-{-6x = 0. 25. x(x-2)=:{x-3y-\'9, 20. 4s + 5-fs = |s-5. 26. 3(2a;-4)=4(a:-5)+32. 21. (x-3){x + 2)=x'^ — 7, 27. 3x — x'^= x(l -- x)+4:2, 28. (a; + l)'+(i» + 3)2 = 2(a;2 + 9). 29. |a; + a;(a; — i)=.^(a; — 5)+ 10. 30. (x 4- 2)2 + (a; _ 3)2 = 2 (a; + 4)2- 1. 31. 5aj - 24 + a;2 - 65 -3(.T-2) = (a; + 3)2. 32. 17 a; -(8 a; -9)- [4 -3 a; -(2 a; -3)]= 30. 33. (a; + 2) (a; + 1) (a; + 6)- 9 a;2 = a:3 ^ 4(7 ^ _ ly 34. (x 4. l)(a; - 1)- a;2 +(2 a; -f 1)2 = 4 (a; + 2)2 + 8. 35. (x - 5)2 4- 2[3 a> -(a; + 2)2 + 5]= 3(a; + 4)- a;2. 36. 3[2 a; + 5 - 2 ^a; - 6 + 5a;J + a;2] = a'(3a; - 13). 37. (2a; - 3)2- 4(1 - a;)2=: 2[a! + 6 - 3(a; - 8 + 4)]- a;. 38. (4 - a;)2- 2[8 -(x + 1)2- 3aj]= 3(4 - a;)2- 4(1 - 3a;). Solve for x : 39. aa; + 16 = a2-4a;. 43. ax - 0^= 2 ab + ¥- bx, 40. da; + 9a2 = c^2_3^^ 44 3x_l2a=4a2— 2aa;+9. 41. a;(l-3c)+9c2 = l. 45. a(x-\-a) + b{b-x)=2 ab. 42. ca; — 9 = c2 -f 6c — 3a;. 46. ax — c^—a^+ac-\-a^c—cx. 47. a{x — a)— 2ab = — b{x — b). 48. (a2 + x) (52 -\-x) = {ab + xf + (a^ - b'^)\ 49. 4 m^ — 2 TTi^x — 3 mx = 1 — 6m-f9m2 — a;. 50. a^ + cL^x — c^x 4- a;(c2 — 1)+ 2 aa; + a; = a;(a + 1)2— W. 51. c{2x-d)+G\d^-c)+d{x+c)=d'{d + c)--c{d?-x)+c^d?, 36 INTRODUCTORY REVIEW Problems 47. General Directions for Solving Problems. — 1. Represent one of the unknown numbers by some letter, as x. 2. From the co7iditions of the problem find an expression for each of the other unknown numbers. 3. Find from the conditions tivo expressions that are equal and write the equation of the problem. 4. Solve the equation. Solve each of the following problems : 1. What number multiplied by 3 is equal to 54 ? Suggestion. — The equation of the problem is Sx = 54. 2. What number increased by 10 is equal to 19 ? 3. What number diminished by 30 is equal to 20 ? 4. What number decreased by 6 gives a remainder of 17 ? 5. What number divided by 4 is equal to 24 ? 6. What number exceeds ^ of itself by 10 ? 7. What number diminished by 45 is equal to — 15 ? 8. What number is 3 more than i of itself ? 9. If |- of a number is 30, what is the number ? 10. Find three consecutive numbers wlfbse sum is 42. 11. Find three consecutive odd numbers whose sum is 57. 12. Find a number which added to its double equals 12. 13. Find three consecutive even numbers whose sum is 84. 14. Separate 64 into two parts whose difference is 12. Suggestion. — Let x equal one part and 64 — x, the other. 15. Separate 40 into two parts, one of which is 3 times the other. 16. If c times a number is a + 6, what is the number? 17. If I of a number is added to the number, the sum is 30. Find the number. 18. If i of a number is added to twice the number, the sum is 35. What is the number ? INTRODUCTORY REVIEW 37 19. The sum of two numbers is 35 and one number is \ of the other. Find the numbers. 20. If 5 times a certain number is decreased by 12, the remainder is 13. What is the number ? 21. Eighty decreased by 7 times a number is 17. Find the number. 22. If I subtract 12 from 16 times a number, the result is 84. Find the number. 23. If from 7 times a number I take 5 times the number, the result is 18. What is the number ? 24. One number is 8 times another ; their difference is 14 a. What are the numbers ? 25. The sum of a number and .04 of itself is 46.8. What is the number ? 26. What number decreased by .35 of itself equals 52 ? 27. Find two numbers whose sum is 60 and whose difference is 36. 28. The sum of two numbers is 82. The larger exceeds the smaller by 16. Find the numbers. 29. Separate 2 a into two parts, one of which is 4 more than the other. 30. Four times a certain number plus 3 times the number minus 6 times the number equals 7. What is the number ? 31. If 5 times a certain number is subtracted from 5S, the result is 16 plus the number. Find the number. 32. Twelve times a certain number is decreased by 4. The result is 6 more than 10 times the number. Find the number. 33. Three times a certain number decreased by 4 exceeds the number by 20. Find the number. 34. Three times a certain number is as much less than 72 as 4 times the number exceeds 12. What is the number ? 35. Twice a certain number exceeds i of the number as much as 6 times the number exceeds 65. AVhat is the number ? 38 INTRODUCTORY REVIEW 36. Two boys had 350 apples. They sold the green ones for 3 ^ each and the red ones for 5 ^ each and received in all $ 11.60. How many apples of each kind did they sell ? Solution. — Let x — the number of gi*eeii apples. Then, 350 — x = the number of red apples, 3 ic = the number of cents received for green apples, and 5(350 — x) = the number of cents received for red apples. .-. 3x4-5(350-5c) = 1160. Solving, we have x = 295, the number of green apples, and 350 — cc = 55, the number of red apples. Verification. — This solution satisfies the first condition of the prob- lem ; namely, the boys had 350 apples, for (295 + 55) apples = 350 apples. It also satisfies the second condition, for 295 x3^-f55x5^ = 1160 5^, or $ 11.60. Hence, the solution is presumably correct. Solve the following problems, and verify the solutions : 37. John and Frank have $72. John has $12 more than Frank. How many dollars has each ? 38. Charles solved 14 problems, or | of the problems in his lesson. How many problems were there in his lesson ? 39. A house and lot cost $ 3000. If the house cost 4 times as much as the lot, what was the cost of each ? 40. What is the number of feet in the width of a street, if f of the width, or 48 feet, lies between the curbstones ? 41. How long is one side of a square, if the perimeter added to the length of one side is 15 inches ? 42. A and B began business with a capital of $ 7500. If A furnished half as much capital as B, how much capital did each furnish ? 43. Ada is f as old as her brother. If the sum of their ages is 28 years, how old is each ? 44. If f of the number of persons who went on an excursion to Niagara Falls were teachers, and 240 teachers went, what was the whole number of persons who went on the excursion ? INTRODUCTORY REVIEW 39 45. I owe A and B $ 45. If I owe A | as much as I owe B, how much do I owe each ? 46. A rectangle having a perimeter of 46 feet is 5 feet longer than it is wide. Find its dimensions. 47. Twelve years ago a boy was ^ as old as he is now. What is his present age ? 48. In 2 years A will be twice as old as he was 2 years ago. How old is he ? 49. A lawn is 7 rods longer than it is wide. If the distance around it is 62 rods, what are its dimensions ? 50. In a lire B lost twice as much as A, and C lost 3 times as much as A. If their combined loss was $6000, how much did each lose ? 51. A father is 4 times as old as his son. Six years ago he was 7 times as old as his son. Find the age of each. 52. In a business enterprise the joint capital of A, B, and C was $ 8400. If A's capital was twice B's, and B's was twice C's, what was the capital of each ? 53. How old is a man whose age 16 years hence will be 4 years less than twice his present age ? 54. A boy is 8 years younger than his sister. In 4 years the sum of their ages will be 26 years. How old is each ? 55. A prime dark sea-otter skin cost $400 more than a brown one. If the first cost 3 times as much as the second, how much did each cost ? 66. In 510 bushels of grain there was 4 times as much corn as wheat and 3 times as much barley as corn. How many bushels of each kind were there ? 57. In a certain election at which 8000 votes were polled for A ^and B, B received 500 votes more than i as many as A. How many votes did each receive ? 40 INTRODUCTORY REVIEW 58. A had $ 40 more than B ; B had $ 10 more than i as much as A. How much money had each ? 59. A man has $ 1.80. He has twice as many quarters as dimes. How many coins has he of each denomination ? 60. A wagon loaded with coal weighed 4200 pounds. The coal weighed 1800 pounds more than the wagon. How much did the wagon weigh ? the coal ? 61. Mary bought 17 apples for 61 cents. For a certain number of them she paid 5 cents each, and for the rest she paid 3 cents each. How many of each kind did she buy ? 62. A mining company sold copper ore at $ 5.28 per ton. The profit per ton was $ .22 less than the cost. What was the profit on each ton ? 63. The students of a school numbering 210 raised $ 175 with which to buy pictures. The seniors gave $ 1.50 each, the rest $ .50 each. Find the number of seniors. 64. A man has $ 27.50 in quarters and half dollars, having 5 times as many half dollars as quarters. How many coins of each kind has he ? 65. Two boys sold 150 tickets, the reserved seat tickets at 75 ^ each and the others at 50 ^ each. The total receipts were $ 87.50. How many tickets of each kind did they sell ? 66. The length of a classroom is 4 feet more than twice its width. If its width is increased 2 feet, the distance around it will be 120 feet. Find its dimensions. 67. My house is 16 feet deeper than it is wide. If it were 6 feet deeper than it is, the distance around it would be 140 feet. Find its dimensions. 68. A had 3 times as many marbles as B. A gave B 50 marbles ; then B had twice as many as A. How many marbles had each ? FACTORS AND MULTIPLES 48. Review definitions and tell the meaning of : 1. Factor. 7. Degree of a term. 2. Prime factor. / 8. Degree of an expression.^ 3. Factoring. 9. Common factor. - 4. Prime to each other. 10. Common multiple. 5. Rational expression. 11. Highest common factor (h. c. f.). 6. Integral expression.^ 12. Lowest common multiple (1. c. m.). \ FACTORING 49. Until noted farther on, the term factor will be under- stood to mean rational integral factor. Monomial Factors 50. The type form is nx + ny -{- nz = n(x -f / + '^)> in which the terms of the expression have a common factor. EXERCISES 51. Factor: 1. S a'^x — 6 ax^ -\- 9 ax. Solution. 3 a^x — 6 ax^ + 9 ax = 3 ax(a -- 2x + 3). 2. 362 + 361 8. 4ta''x'-Sa^x^ + 6a''x\ 3. 2y^—S y\ 9. 6 mW + 9 mhi^ - 3 mV. 4. 6a2 + 4a6. 10. 8 a6V - 4 aWc^ + 12 a^^^c^ 5. 3x1/2- 6 a^y. 11. 1% r'st^^ 12 rs'^t^-24trht\ 6. 2a2 + 4a3 + 6a^ 12. 201)^0(1^ -imy'c'd^-2U^cd\ 7. 5r^-10rH + 5rh\ 13. 9:x?fz'+21 xYz^-l^:x^yh\ 41 42 FACTORS AND MULTIPLES Factoring Binomials 52. Difference of two squares. The reverse of formula 3 (§ 30) gives the type form, a^ — b^ =(a -\-b)(a —b). Rule. — Find the square root of the two terms, and make their sum one factor and their difference the other factor, EXERCISES 53. Factor, and test each result : 1. 18c2-50; 0^-2/^. Solutions. 18 c^ - 50 = 2(9 c^ - 25) = 2(3 c + 5) (3 c - 5). Z^ - y^=(x^ + y2)(x2 _ y^^) = (x'^ + y^)(x + y)(x-y). Test. — The product of the factors should equal the given expression. Note. — As in the above, sometimes the factors first found may be factored. When told to factor an expression, find its prime factors. 12. 2a^-2y*. 13. 5oi:^ — 5 y\ 14. aj2»+i — a^2n 15. 9b^-(a-xy, 16. {a^-\-x^y-{x-\-2)\ 54. Sum or difference of two cubes. Applying the principles of §§ 38-40, and taking the divisor and quotient for factors gives the type forms, a^ + 6^ = (a + b)(a^ — ab -{- 6^), and a^ - 63 =: (^ __ 6) (^2 + ab + b"). EXERCISES 55. Factor and test : 1. &'\-d^. 2. W-&, 3. Q? + S, 4. 2/3-125. 5. r« + s'; a^-12bh\ Solutions, r^ -{■s^ = {r'^fJ^ {s'^Y={r'^-\-s'^){r^-rH'^+s^) . a9~125 53=(a8)3_(5 6)3=(a3-5 6)(a6-f5aa54-25 62). 6. a^ -f y\ 10. v^ + 27 v. 14. l-{-{a + b)\ 7. x — ccf^. 11. r«4-64s3. 15. 216 a^^" + 64 ly^". 8. aW-(^. 12. (x-yy~S. 16. S(m+ny-{-125n\ 9. aj3" + 64. 13. r3*-729s3^ 17. (x-yy-(x-\-yy. 2. 9-c^ 7. a' - 81. 3. a'-h\ 8. 9W-cH\ 4. ic2_16. 9. Ax^-2oy\ 5. 25 - c\ 10. 9a2_49 2>2. 6. aV-1. 11. 16 a^- 81 5^. FACTORS AND MULTIPLES 43 56. Sum or difference of the same odd powers. Applying principles § § 38-40, as in § 54, gives for fifth powers the type forms, a^ + 6^ = (a + b){a' - a'b + a^b^ - ab' + b% and a'-.b'={a- b){a' + a'b + a'b^ + ab' + 6^. EXERCISES 57. Factor: 1. m' + 32x'', 128ai4-l. Solutions, m^ + 32 x^ = m^ + (2 x)^ = (m + 2x) (m4-2 m^x+4: m^x'^ -S mx^ -\- 16 ik*). 128ai4-l=(2a2)7_l = (2a^~ l)(64ai2 + S2a^^ + 16a^ + Sa^ + 4a* + 2a2 + i). 2. X^ + 2/^ 7. x^ — a?. 12, m^ — m^. 3. a;5 - yK 8. a' + 128. 13. a^ + 512. 4. a'-l. 9. 64-2al 14. 32 + a''?>''. 5. x' + f. 10. a«6 - ab\ 15. 1 _ a'b'^c'K 6. 1 + 2/^. 11. i^Sm _ ^5n^ 16. x'' + 2^3a\ 58. Difference of the same even powers. Solve as in exercise 1, § 53. MISCELLANEOUS EXERCISES 59. Factor each of these binomials, and test the result : 1. aj2 _ y\ 11. 25x''-l. 21. ^2n-2 _ y\m^ 2. 7^-^-^. 12. Sx^-y\ 22. 2x^-2f. 3. f-9. 13. a'W + 1. 23. 243 + x^y\ 4. v'-l. 14. ^3n _ 53n^ 24. 8 a« - 729. 5. n^ + w^. 15. ^16 _ ^8^ 25. 8 a?-^ - 18 &2^ 6. z' - 27. 16. 3x'-3y\ 26. 9a2-(2a-5)2. 7. y'-S6y. 17. 4a2-f 27. 27 + (^ + 2^)3. 8. y^ + aW. 18. ^5a ^ yla^ 28. ^n+l _ y^y^n^ 9. 1-144^^. 19. ^^-tV 29. {a-2by-(a-5y. 0. 128 - 2/V. 20. ^9 _ ^,9^9^ 30. (r-^sf -729 t'v\ 44 FACTORS AND MULTIPLES Factoring Trinomials 60. Trinomials that are perfect squares. Applying the re- verses of formulas 1 and 2 (§ 30) gives the type forms , a^-{-2ab + b'' = {a + b)\ and a'-2ab + b''={a-b)\ It will be observed that these trinomials are perfect squares, for each is the product of two equal factors ; also that a tri- .nomial is a perfect square, if it has : Two terms, as + a^ and -f b'^, that are perfect squares and another term that is numerically equal to twice the product of the square roots of the terms that are squares. To factor a trinomial square : EuLE. — Connect the square roots of the terms that are squares with the sign of the other term, and indicate that the result is to be taken twice as a factor. In factoring, usually only the positive square root is taken. First remove the monomial factor, if there is one. EXERCISES 61. Make a trinomial square by writing the missing term : 1. a;2 + =^ + y\ 4. c2 - 2 cc? + *. 7. * + 4 a6 + b\ 2. a2-^ + 62. 5. x'^-^4:Xy-{-^. 8. ^ — 2pq + ql S. y"^ + ^ -\-z\ 6. r2 — 8 rs 4- ^. 9. =* + 6 a^ + y\ Factor, and test each result : 10. m2-8m + 16. 18. 2x -{- 20a^x -}- dOa'^x. 11. 4 2/ — 4 2/2 + y^. 19. 4aj2« + 8 icY + 4^^^- 12. a2-16a + 64. ' 20. S a'^b -\- 4.0 ab^ -^ 50 b\ 13. Sx'^-{-6xy-\-3 y\ 21. cc^" - 2 x^'y^'z'' -\- ^/^"a;^". 14. 9a;2-42aj + 49. 22. x^ + 2x(x - y) + {x - yf. 15. S6n^-12n\+l. 23. t'' - At{t - 1) -h 4:{t - If. 16. oj^r ^Sx'z + 16 z^. 24. 14(a; - y) + (x - yf + 49. 17. 2a4-4a2&2-^261 25. c'' - 6c{a - c) + 9(a-cy. FACTORS AND MULTIPLES 45 62. Trinomials of the form x'^ -\- px -\- q. Applying the reverse of formula 4 (§ 30) gives, jr^ 4- (a 4-6)jr + a6 = (jr + a){x + b), wliich is in the type form^ jr^ + /?jr + qr, having an x^ term, an X term, and an absolute term. Hence, if a trinomial of this form is factorable, it may be factored as follows : Rule. — Find two factors of q (the absolute term) such that their sum is p (the coefficient ofx), and add each factor ofq to x. EXERCISES 63. 1. Find the two binomial factors oi x^ + 4:X — 21. Solution. — The first term of each factor is, of course, x. The second terms of the factors must be two monomials whose alge- braic sum is -f- 4 and whose product is — 21. Evidently these numbers must have unlike signs and it is seen that + 7 and — 3 fulfill the necessary conditions. Hence, x^ + 4 x - 21 = (ic + 7)(x - 3). Factor, and test each result : 2. a2 + 6a + 8. 7. c^ + c- 30. 3. a;2 + 3a;-10. 8. d^-^7d-60. 4. r2-2r-15. 9. 2b'-6b^-56. 5. y2_^jy_is^ 10. a^^ + 10 a" 4- 16. 6. aj2 + 12 a? -h 20. 11. a;2»+i + 3 a;"+i + 2 0?. 12. Factor 15 — 7i^-\-2n. Suggestion. 15 — n^ -\- 2 n = — ti^ + 2 n + 16 = — (71^ — 2 11 — 15). 21. 2 7/4 + 26?/ -180. 22. 54 a2 — 3 a?/ — 7/2. 23. 4:ax-2ax'^ + 4:Sa. 24. 3a2-15a6-72 62. 25. a;2 — 2(a — n)x — 4 an. 26. 9 ¥x 4- 54 6a; - 144 x. 27. 150 n — 5 naP- — ^5 nx. 20. x^-(a-d)x-ad. 28. 20 6a; + 10 ^^ - 630 a^l 13. 12 + 4.y-y\ 14. ^2 + 18 ^ ^ 56^ 15. x" -abx-2 a^b\ 16. _^2_ig^_l_35^ 17. m^ — 2 mn — 15 n^. 18. _a2-9a + 52. 19. x"^— (c + d,)x-^cd. 46 FACTORS AND MULTIPLES 64. Trinomials of the general form ax'^ -\-bx -\-c. The types of trinomials so far treated are really special forms of the general type. If the general quadratic trinomial ax^ -\-hx-\- c has binomial factors, they are of the forms rx + 1 and sx + v, EXERCISES 65. 1. Factor 2 a;2 - 5 a; - 3. Solution. — If this trinomial is the product of two binomial factors, 2 x2 is the product of their first terms and these terms must be 2 x and x. Since — 3 is the product of the last terms, they must have unlike signs and the only possible last terms are 3 and — 1 or — 3 and 1. These first and last terms associated in all possible ways give : 2x-3 2a;-l 2x-h3 2x+l x + l a; +8 x-\ x-3 Of these we select by trial the pair that will give — 5 x (the middle term of the given trinomial) for the algebraic sum of the cross-products. Hence, 2^2 _ 5x- 3 =(2x + l)(x ~ 3). Observe that : 1. When the sign of the last term of the trinomial is +, the last terms of the factors must be both + or both — , and like the sign of the middle term of the trinomial. 2. When the sign of the last term of the trinomial is — , the sign of the last term of one factor must be + , and of the other — . Factor, and test each result : 2. 2 aj2 - 5 aj + 3. 11. 21 a'' -a- 10. 3. 3 ^2 _ 8 ^ _ 3^ 12. 6x^-10x+4.. 4. 2 a;2 + 7 07 - 4. 13. 15x^-i-22x + 8. 5. 6x'^-lSx-{-6. 14. 10x'--llx^-6. 6. 5x''-13x-6. 15. 16x^-6Sx + 66. 7. 8a;2 + 10a;-3. 16. 10aj6 + 42 0^ + 44. 8. 15 a.2 _ 9 ^ _ 6. 17. 12x^-\-Ux- 40. 9. 6x^ + llx--10. 18. 3x^ + 7 xy-\-2y\ 10. 27 6^-3 62-14. 19. 3x^ + 5xy-2y\ FACTORS AND MULTIPLES 47 When the coefficient of x^ is a square, and when the square root of the coefficient of x^ is contained exactly in the coeffi- cient of X, the trinomial may be factored as follows : 20. Factor ^x^-42x + 40. Solution. 9 x^ ~ 42 x + 40 = (3 x)2- 14(3 x) + 40 = (3x-4)(3x- 10). Factor, and test each result : 21. 4iK2 + 4a;-3. 26. 25^/2 + 15 2/ - 18. 22. 9a;2-9a; + 2. 27. 36 v^ + 12 'y - 35. 23. 4a;2~6a;-10. 28. 32 aj^ + 16 a; - 30. 24. 9a;2H-18a; + 8. 29. 49 a^- 14 a -24. 25. 16a;2-8a;-3. 30. 81 a;^ - 36 a? - 32. When the coefficient of x'^ is a square, and its square root is not contained exactly in the coefficient of x, multiply and divide by the coefficient of x^^ as follows : 31. Factor 4 a;2 — 5 aj — 6. Solution. 4x2-6x-6=(4x2-5x-6)x-= ^^ ^^ - 20 x ->- 24 4 4 _ (.4 yQ^ - 5(4 x) - 24 ^ (4 x - 8) (4 X + 3) 4 4 = i(^:^lKi^±ll = (X - 2)(4 X + 3). 32. Factor 24 aj^ + 14 a; - 5. Suggestion. — When the first term is not a square, it may always be made a square whose square root will be contained exactly in the second term by multiplying the trinomial by the coefficient of x^, or by a smaller multiplier. In this case multiply by 6, and divide by the same number. Factor, and test each result : 33. 4 a;2 4- 19 a; — 5. 39. 9 a^2 _^ 43 a; — 10. 34. 9 ^2 _ 13 ^ _ 10 40^ isa^-9x- 35. 35. 4a2 4-17a-15. 41. 9 aj^ - 10 a;^ - 16. 36. 8a;2 + 22aj + 9. 42. 16 aj^ + 50 a; - 21. 37. 1862 + 466-24. 43. 32 ti^ + 28 n - 15. 38. 3x'^-10xy+3y\ 44. 5 x^"" + 9 x^'y - 2 y\ 48 FACTORS AND MULTIPLES 66. Trinomials of the form a^ + na'^b'^ + b\ By adding such a positive perfect square to the middle term as to make this tri- nomial a perfect square and then subtracting the same number so that the value will not be changed this type form becomes a special case of the difference of two squares (§ 52) whose type form is a^ — b^. Thus, a* + a^b^+ b^ = a^-\-2 a^h^+ 6* - ^252 ^ (^2+ 52)2 _ ^^^2^ whose factors are a^ + ab -\- b'^ and a^ — ab + b^. EXERCISES 67. 1. Factor 4 0^4- 13 a;2 + 9. Solution. 4 x* - ISx^ + 9 = 4x4 - 12 x^ + 9 - x^ = (2x2- 3)2 -x2 = (2x^ + X - 3)(2 x2 - X- 3) = (2x + 3)(x - l)(2x - 3)(x + 1). Factor, and test : 2. x' + xY-^y\ 9. 9 6^-166V + 4c^ 3. 54 _|. 3^2.^ 4 10. 16c^ -17 cW-{-d\ 4. x^ + x^:^-{-z\ 11. y^ - 37 y'^z'' -j- 36 z\ 6. 9x^ -\- 5 xY + y^' 12. 9 a;* - 46 xy + 25 y^, 6. a'-5a^b'' + ib\ 13. 16 a' -{- 15 a'b^ + 9 b\ 7. 4 2/^ + 7 2/V + 4 z\ 14. 25 x^ - 29 xY -f 4 2/^. 8. a'b' - 21 a'b^ -h 36. 15. 36a' - 52aW + 16b\ 68. The method given in § 66 may be used to factor bino- mials of the type form, p' + 4. Thus, p* + 4 =p4 + 4^2 ^4 _4p2_(^p2_^2)2 — 4j)2^ whose factors are p2 4. 2p + 2 and p2 _ 2p + 2. EXERCISES 69. Factor, and test : 1. x'-i-A. 4. m«+4. 7. a^ + 324. 2. 4?/4 + l. 5. x' + 64:X. 8. 2^4 + 128. 3. 26^ + 8. 6. x' + 64.y'. 9. 4 aj^ + 81 ^/^ FACTORS AND MULTIPLES 49 MISCELLANEOUS EXERCISES 70. Factor orally each of these trinomials : 1. a2 + 4a+4. 9. x^-\-x'^ + l. 2. x^ + 3x-{-2. 10. 2y'^ + Sy-2, 3. z'^-i'5z-{-4r. 11. 3'y2_8v-3. 4. 2a;2-i»-l. 12. x'^-lOx + 25. 5. y^ — 6y-{-5, 13. x'^-^xy — 20y\ 6. 9 + 6^2 + a*. 14. 4a;2_|_8a:2/ + 42/^. 7. l+48 + 4s2. 16. m2 + 8mn + 16n2. 8. 3ir2+6iiJ + 3. 16. m^ — 6mn ~167i\ Factor, and test each result : 17. 5^-362-4. 34. 9a^ + 12az'' + 4.z^. 18. 4a;2_f_8a; + 3. 35. 6^ _^ 19 6c + 48 c^. 19. 2y^-y-15. 36. 36 a;^ - 48 a; - 20. 20. c4-8c2 + 16. 37. 4 0^-72 0^2 + 324. 21. 62-126-45. 38. 18a;2_51i»+36. 22. 5c4-5c2-60. 39. cW + 7cW + 12. 23. 4:X^-5xy-{-y\ 40. 4 a^ - 48 a^ - 256. 24. z^ - 10 z'^ -{- 24.. 41. 16 ?/2 + 24 ?/^ - 7 ;^2^ 25. 6a;2_a^^_2?/2. 42. 4 a;2 _ 14 a;^ + 10 2/2. 26. 5a;2_38^_l_2i. 43. 25 64-62c2 + 64c4. 27. 2x'^ + 5xy -^2y\ 44. 2oy'^ - 25yz + 6z\ 28. 9a;2_27aj+18. 45. 9 62 + 49 6c- 30 c2. 29. 16 + 16a + 4a2. 46. 9 6^ - 13 62a;2 + 4 a;^ 30. aV + 3 a2.T2 _ 28. 47. 4 (C^n _p 4 ^n^n ^ ^2n^ 31. 10a3 + 14a2-f 4a. 48. 49 a;^ + 14 o^i/ - 1^ 2/^- 32. ?/2 ~ (a — 6)?/ - ab. 49. (r + s)2 — 4(r + s) + 4. 33. 2;2~(m + 7?)2; + mn. 50. 16 - 24(« - /) + 9(^ - ^)2. milne's skc. course alg. — 4 50 FACTORS AND MULTIPLES Factoring Larger Polynomials 71. Polynomials whose terms may be grouped to show a common polynomial factor. The type form, ax + ay + bx -h by, may be solved as illustrated below. Thus, ax ■{■ ay -{- bx ■\- by = a(x-\- y)+ b{x + y) = {a-\- b){x + y). EXERCISES 72. 1. Factor mx — my — nx + ny. Solution. mx — my — nx-\-ny= (mx — my) — (nx — ny) = m(x — y)— n(x — y) = (m — n)(x — y). 2. Factor ex -\- y — dy + cy -- dx + x. ^ Solution. cx + y — dy -\- cy — dx -{- x Arranging terms, = ex— dx + x ■}■ cy — dy + y =x(c -di-1) -\-y(c-d-\-l) = (x-{'y)(ic-d + l). Factor, and test each result : 3. bc + bx + cx + x\ 15. 2/^ -f- 2/^+2/ + 1. 4. ab -{-ex — ax— be. 16. n^ -\-n^ — 4:n — 4. 5. x^ — xy + 3y—3x. 17. ay^ — b + by'^ — a. 6. bd — ae — be -{- ad. 18. Sm^n—9mv?+am—San. 7. ax — by + bx — ay. 19. 36 ab— IS ae — IS ¥+9 be. 8. ax + 2y + 2x + ay. 20. 15ab'^-9b'^e-35ab-\-21be. 9. 2a + bx^ + 2b + ax\ 21. 16ax+12ay-Sbx—6by. 10. by — bx + 3ax— 3ay. 22. ax'^— ax — axy + ay -\-x—l. 11. abe + aed + bd + a'^e^. 23. xy+x—3f—3y'^—4:y—4:. 12. bx — ¥y + obey — aex. 24. mx—nx—x-my-\-ny-\-y. 13. 6ab + 12b -3a,e-6e. 25. bx'^-b-xy-y+yx'^-bx. 14. 5ax—5ay+3bx — 3by. 26. rx + sx + ry + sy + r -\- s. FACTORS AND MULTIPLES 61 73. Polynomials as special cases of types a^—b'^ and x'^+px-^-q. Many polynomials may be grouped as the difference of two squares. EXERCISES 74. 1. Factor a2 + 2 a6 4- 62 - 1. Solution. a"^ -^ 2 ab + b'^ - I = (a^ -^ 2 ah -}- b'^) -- 1 = {ai-by2-l Factor as in § 52, = (a + 6 + l)(a + 6 - 1). 2. Factor x^ -- y^ — 4:X + 4:. Suggestion, x'^ — y^ -^ 4:X + 4: ={x'^ -- 4:X -^ 4:)^ y^ =(x — 2)^— y^, 3. Factor a^ + 6^ - c^ - 4 - 2 a6 + 4 c. Solution. ^^^2 ^ 52 _ ^^ — 4 — 2 a6 + 4 c Arranging terms, = a^ — 2 ab -\- b^ — d^ + ^c— 4 = (a2_2a6 + 52)-(c2-4c4-4) zz:(a-5)2-(c_2)2 = (a- 6 + c-2)(a-6-c + 2). 10. 9G^-y^-z^-2yz. 11. 4.0^-6^ -(P + 2cd. 12. 25 2/^-1- 4 a-4a2. 13. 9a;2 + 6a; + l-16ay. 14. 6c2 - 9 a2& - &3 __ 6 a62. 15. a&2_4 a' - 12 a^c- 9 ac2. 16. a'-2ab + b^-c' + 2cd-dK 17. a;2-2a?2/ + ?/2-m2 + 10m-25. 18. a2 - 4 a6 + 4 52 _ c2 _ 12 c - 36. 19. a^ + 2a^+a^-a2_2a-L 20. x'^-a'^-\-y^-b^ + 2xy — 2ab. 21. 4ic2^9_-i^2a;+10mn-m2-25n2. ^^ac jtor, and test each result : 4. a2 + 2 a + 1 - 6'. 5. ?>2_25c + c2-l. 6. l + 2c^4-d2_c2. 7. a''-b'-2bd-dh 8. r^ — 2 rs + §2 _ x^. 9. n^ — x^-\-2xy-y\ 52 FACTORS AND MULTIPLES Factor the following polynomials by writing them in the form x^ -\-px + q, x^ and x being replaced by polynomials. 22. Factor ^ x^ -\- A:]^- ^\2 z^ -\-2\x% ^ 14 yz -f 12 xy. Solution. 9 x^ f 4 ?/2 -f 12 2;^ 4- 21 x^ + 14 ?/^ + 12 x?/ Arranging terms, = (9 x'^ + 12 xy -\- 4 y'^) -f (21 x^; + 14 yz) -f 12 ;32 = (3 X + 2 2/)2 + 7 ^(8 X -f 2 ?/) + ^zSz § 62, = (3 X + 2 2/ + 4 0)(3 X + 2 2/ + 3 ^). 23. a2 + 2 a& + ^^ + 8 ac + 8 6c + 15 c^. 24. cc2 — 6 x?/ + 9 ?/2 + 6 a^2; — 18 2/2: + 5 ;2!2. 25. m^ + n^ — 2 m^^ + 7 mp — 7 rip — 30 p^. 26. 9m^ + A:2- +39 m^ + 13 A: + 6 m%. 27. 16 n^ + 55 — 64 n — 16 m + m^ + 8 m/i. 28. 25 a^ + 2/2 + 10 a;^ + 10 ay - • 35 ax — 7 xy, 75. Polynomials factorable for binomial factors by the factor theorem. If a product is equal to zero, at least one of the factors must be or a number equal to 0. Sometimes a polynomial in x reduces to for more than one value of X. For example, x^ — 5 x + 6 equals when a; = 3 and also when x = 2; or when x — 3 = and cc — 2 = 0. In this case both x — S and x — 2 are factors of the polynomial. 76. Factor Theorem. — If a polynomial in x, having positive integral exponents, reduces to zero ivhen r is substituted for x, the polynomial is exactly divisible by x — r. The letter r represents any number that we may substitute for x. Proof. — Let Z> represent any rational integral expression containing X, and let D reduce to zero when r is substituted for x. It is to be proved that D is exactly divisible by x — r. Suppose that the dividend D is divided by x— r until the remainder does not contain x. Denote the remainder by B and the quotient by Q. Then, D=q{x--r)-\-B. (1) But, since D reduces to zero when x— r^ that is, when x — r = 0, (1) becomes = + 72 ; whence, i? = 0. That is, the remainder is zero, and the division is exact. FACTORS AND MULTIPLES 53 EXERCISES 77. 1. Factor :^ - x^ - 9x + 9. Solution. — When x=l, x^-x^-9x-{-9 = l-'l-9-\-9 = 0. Therefore, x — 1 is a factor of the given polynomial. Dividing x^ — x'^ — 9x -{- 9 hj x — 1 gives the quotient x'^ — 9. By §62, x:^^9={x-{-S){x-S). Hence, x^ -- x^ - 9 x -{- 9 =(x ^ l){x + S)(x - S). Notes. — 1. Only factors of the absolute term of the polynomial need be substituted for x in seeking the binomial factors of the polynomial, for if X — r is one factor, the absolute term of the polynomial is the product of r and the absolute term of the other factor. 2. Since when 1 is substituted for x the value of the polynomial is equal to the sum of its coefficients, x — 1 is a factor of a polynomial when the sum of its coefficients is equal to 0. 3. In testing for factors, instead of using ordinary substitution it is convenient to employ synthetic division, for this will show in one opera- tion whether or not there is a remainder and give the quotient of the poly- nomial by the factor being tried. Thus, in exercise 1 in trying the factor X — 1, we have, 1 _ 1 _ 9 4. 9 12 +1 -0-9 1 + 0-9 which shows at once that there is no remainder and the quotient is x^ - 9. 2. Factor 2a^-9a?2_2.'«-f 24. Solution Since the sum of the coefficients is not equal to 0, x — 1 is not a factor. Using synthetic division to try for the factor x — 2, we have, 2-9- 2 + 24[2 -^4-10-24 2-5-12 which shows that x — 2 is a factor and that when this factor is divided out the quotient is 2 x^ — 5 x — 12. By § 64, 2x2- 5x- 12 =(x - 4)(2x + 3). Hence, 2x3- 9x2- 2x + 24 =(x - 2)(x - 4)(2x + 3). 3. Ysictov 3x^-Sx^y + 3xif-\-2f. Suggestion. — When x = i/, 3x3 - Sx^y + 3x2/2 -\-2'if = Sy^ -Sf -\- Sy^ + 21/3 = 0. Therefore, x — ?/ is a factor of 3 x3 — 8 x'^y + 3 xy- + 2 y'^. 54 FACTORS AND MULTIPLES Factor by the factor theorem : 4. a^ + 4a;2-f ir-6. 17. a:^ - 27 x -\- BL 6. a:^ + 2x'^-5x-6. 18. i^_39^_70. 6. x^ + ex'^ + Bx-U. 19. a^-7ab^-\-6b\ 7. 0.-3 — 7i»2 4-7a; + 15. 20. a^ — 21 xi/ -{- 20 y^ 8. c»3~12a;2 + 41aj-30. 21. 6^ _ 552 _ 29 6 + 105. 9. a3 + 4a2-lla-30. 22. a^ + 10 a^ - 17a - 66. 10. ic3-13a;2 + 46a;-48. 23. a^ + 2 x'^y - xy"^ - 2 y\ 11. a^ + 9a2 4-26a + 24. 24. a^ -{- 4:x'^y + 5xy^ + 2f. 12. 2aj3-3aj2-17a;-12. 25. 63 + 1652 + 73^ + 90. 13. a^ - 16 a;2 4-71 a; -56. 26. a:^ - 15 a;2 + 10 a; + 24. 14. 2x^-9x'^-2x-^24.. 27. a;4-h8a^H-14aj2-8aj-15. 15. 2^3 __ 7^2 _ 7^ + 30^ 28. x^-2x^-5x^+14:X+12. 16. ri^ + 12^2 + 4171 + 42. 29. c(^-4:X*-^19x^-2Sx^l2, 78. Proofs of divisibility principles for x" ±/". The prin- ciples laid down by experiment in § 38, and later used in factoring certain binomials may be proved readily by the factor theorem : ' Proof of Prin. 1. — In sc" — y^, substitute yforx; then, for any posi- tive integral value of n, x" — 2/" = J/" — y" = 0. Hence, x — y is Si factor of x" — y"^. That is, x** — y^ is always divisible by x — y. Proof of Prin. 2. — In x" — 1/", substitute — ?/ f or x ; then, x** — i/" = (— VY — y^t which is equal to when n is even but not when n is odd. Hence, x + y is a factor of x^ — y^ only when 71 is even. That is, X** — y" is divisible by x -\-y only ivhen n is even. Proof of Prin. 3. — In x** + ?/", substitute y for x; then, for any positive integral values of n, x" + 2/^* = 2/"* + 2/"» which is not equal to 0. Hence, x — 2/ is not a factor of x" + y^. That is, x'^ + y^ is never divisible by x^y. Proof of Prin. 4. — In x"^ + y", substitute — y iox x\ then, x" + 2/" = (— VY + y^i which is equal to when n is odd but not when n is even. Hence, x + y is a factor of x" -h 2/" only when n is odd. That is, x" + y" is divisible by x -\- y only wheri n is odd. FACTORS AND MULTIPLES 65 Summary of Factoring 79. In the previous pages the student has learned to factor expressions of the following types, MONOMIAL FACTORS Common to all terms, /ijr + /i/ + nz, BINOMIALS a^-bK a" ± b" (when n is odd). a' + 6'. a" — 6" (when n is even, as in a' - -60- a» - b\ p* + ^ (special case of a? — b^). TRINOMIALS a^ ± 2 o6 + b\ ax^ + bx+e. x^ + px + q. a* + /ja^A^ + b* (special case of a^ -¥). LARGER POLYNOMIALS With common polynomial factor, ax + ay + bx -h by. Special cases of types, a^ — 6^ aiid x'^ + px -\-q, • Having binomial factors, by the factor theorem. 80. General directions for factoring. — 1. Remove monomial factors^ if there are any, 2, Determine whether the resulting expression is a binomial, a trinomial, or a larger polynomial, then decide to which type under that head it belongs, and factor by the proper method for that type, 3. Continue as in 2 loith each factor found until the given expression is resolved into its prime factors. Note. — The factor theorem is applicable to binomials and trinomials as well as to larger polynomials and may often be used when other methods fail. 56 FACTORS AND MULTIPLES MISCELLANEOUS EXERCISES 81. Factor orally : 1. 2a -26. 17. x^-y\ 33. b a}h'' -^ ^ a^h\ 2. 0^2 _ 3 ^, 18. a^ - 1. 34. a;2 -f 5 a; + 6. 3. X- — y^, 19. a^ — 8. 35. {x — ?/)2 — z^, 4. a^ - 1. 20. m^ + 1. 36. 4 a^ - 9 61 5. iZ;2_^a7. 21. a2 — 4 61 37. ^2a _ ^2a^ 6. 4 a2 + 4 a. 22. d^ - 9 c?. 38. & -\-% dK 7. 8a^3-2aj2. 23. a^ - a?y. 39. a^" - 61 8. a^—¥. 24. 4 2/^ — 4?/. 40. ic2"+2 _ 1^ 9. c2 — 4. 25. x'^ — 2x + l. 41. ^^2 _^ 2 a;i/ -f- ^/l 10. a^ + ?/l 26. a'^x - 2 a^xK 42. 8 o^V + 10 ci^f. 11. 2/3 - 0^. 27. cc2 + 3 ar + 2. 43. aj^ + 3 a;?/ + 2 1/2. 12. a;2-9. 28. a2-(6 + c)2. 44. 3-^4cX + o?, 13. x^ — x. 29. aj2 — a; — 6. 45. x^ -{- ax -\- x + a. 14. 2/^°-l. 30. aj2_^^_5^ 46. a^-aj + i»-l. 15. a;"+i + a;. * 31. a!2 + a; — 2. 47. a6 — 6a;+ac — caj. 16. 2/2»+i-2/. 32. aj2_2a;-3. 48. a^ + a;2+a; + l. Factor, and test each result : 49. a;^ + a;. 57. a^-256. 66. 12o — ^x\ 50. IP - c\ 58. a^" - a^. 66. 16 m^ + 2. 51. r^-sl 59. ^^ + 322;. 67. z^-^z^ + 1, 52. 1/6 — ;2l 60. 5?/4+20. 68. 2 a;2 + a? — 1. 53. l-a^o. 61. a^^-aU^ 69. a^2 _|. 9 ^j _ 90. 54. a9-6l 62. 7n^ + 7n. 70. 1 +(aj +- 1)^. 55. a^-{-x\ 63. a^^ -f 4 a:. 71. 3 aj2 - 2 a; - 8. 56. x^^ + 1. 64. c^ - 16 c. 72. 15 + 6 a; - 9 a;2. FACTORS AND MULTIPLES 57 73. l-{x + iy. 89. {a-\-by-l. 74. 1000 a^- 27 2/3^ 90. {a + xy-a^. 75. a'b''-\- a?b - 12. 91. llx^ + 2^x- 18. 76. 25 p2^" - 36 i»2p. 92. {x + ^)3 + (i» - 2/)^. 77. 17-16a-a2. 93. (a- 2)^ + (a - 1)'. 78. 6 62_7?>_3. 94. 4a;3_^.^2_g.^_2. 79. 4ta — ^ax — ax^, 95. o^^ _|_ 5 ^ _l_ ^^^ _l_ 5 ^ 80. 12 c2 4- 7 c - 12. 96. x^ - 119 xY 4- 2/^. 81. a;2-92/2 + 6a: + 9. 97. {a+by-{b-cy, 82. 3a;2 + 7a;i/-6/. 98. 3 ab{a + b) -^ a^ -\- b\ 83. aa; - 2; + 2 a - 2. • 99. {x'' - y^y- {x'^ - xyy. 84. aV+2a2aj2 + 9. 100. (a2 + 62-c2)2-4a262. 101. ^2n-2_^^2^2_|_2aj«-%. 102. a?> — bx"" + 0?"?/"* — a?/"*. 85. 60a^ + Sax-Sx^, 86. 9 &4 -t- 21 ^V 4- 25 c^ 87. 25y^-25yz + 6z\ 103. ic^ 4- 15 i»2 _|. 75 ^ ^ 125. 88. 10 a^c 4- 33 ac- 7c. 104. a"^- b^ -{a + b)(a-b). 105. a;2 — ;32_|.^^2_(^2_2a)^4-2a;2. 106. 2 62^ - 3 a¥ + 2 6ma; - 3 abx. 107. a2 4- 52 4- c^ - 2 a& - 2 ac 4- 2 6c. 108. Sa¥x'^ + 4:cdy -4.ab^xy — 3cdx. 109. 9 a;2 4- ^/2 4. 16 ^2 _ 5 ^^ _ 3 ^^^ _|_ 24 3.^. 110. a?2^92/'^4-25;22_5^^_10a;;24-302/:2. 111. 9 a" - 12 ab -4.0" -12 cd-\-4.b^- 9 d^ 112. icy2;2 + aW 4- 1 4- 2 abxyz + 2xyz + 2 ab. 113. Factor 32 — x^ by the factor theorem. 114. If n is odd, factor cc" — a'^ by the factor theorem. 115. If n is odd, factor x"" 4- ?'" by the factor theorem. 116. Factor a^ — 9 x'^y + 27 a;?/2 — 27 y^ by the factor theorem. 117. Discover by the factor theorem for what values of n. between 1 and 20, ic" + a" has no binomial factors. 68 FACTORS AND MULTIPLES EQUATIONS SOLVED BY FACTORING EXERCISES 82. 1. Solve the equation x'^-{-l=2x + 16. Solution. x^ -\- 1 -- 2 x + W. Transpose all the terms to the first member and unite similar terms, a:2_2x-15 =0. Factor the first member, (a; — 5)(x -f 3) = 0. If a product is equal to at least one of its factors is equal to ; that is, X — 6 = or X + 3 = 0, whence, x = 5 or x = — 3. Verification. — Substituting these values of x in the given equation, we find that each satisfies the equation. Solve for x by factoring : 2. a;2-l = 3. 14. a;2-10.T = 96. 3. 0^2^3 = 28. 15. a;2 + 12x = 85. 4. x^ + 35 = 39. 16. 600 = x''-10x, 5. 0^2 _ 50 = 60. 17. 4a;2~862 = 862. 6. a;2-4 62 = 0. 18. Sx'^ + llx = 4.. 7. a;2_9^2^o. 19. x"^ - a^ = 2a + 1, 8. a;2_4o = 24. 20. x'^ + 2hx + b^ = 0. 9. aj2-3a2 = 6a2. 21. 2a;2 __ 1 = 14 _ a;. 10. x'' + 5b^ = 6b\ 22. x^-b' = 4:-4.b\ 11. 0^2 _ 3^^ 40. 23. 3ic2_7^_4,^2. 12. ir2_9^ = _20. 24. 0^2- c2 = c?2-2cd 13. cc2 + 12a^ = 28. 25. 4ic2 _^ 9^_9 = 0. 26. x^-4:X^-\-2x'^ + 4:X-3 = 0. Suggestion. — Factor by the factor theorem. 27. 2a;^-5a^-23i»2 + 36a? + 28 = 4-2a;. 28. ic^-lOaj^-i- 400.^ -800^24. 80a; -32 = 0. 29. 3a;* + 3a;3-47aj2-56a; + 180=7aj-4a^. FACTORS AND MULTIPLES 59 HIGHEST COMMON FACTOR 83. Principle. — The highest common factor of two or more ex- pressions is equal to the product of all their common prime factors. EXERCISES 84. Find the highest common factor of : 1. c2-d2g^ndc2-2cd + dl 2. x'^ + x'f and x^y + xy\ 3. a^ - W and a^ - 2 a6 + ^^. 4. x^ — ?/2, x^ — 2/*, and y'^ — x^, 5. a^ - x\ a^-\-2ax-[- x^, and a^ + a?. 6. x^ + 1 x-m and a;^ _ 12 a? + 35. 7. a^ + aW + ¥ and a^ — ah-\- h\ 8. 1 ~ 4 c2, 2 a - 8 ac\ and 2 c - 1. 9. (a - 5)(6 - c) and (c - rt)(62 _ a?), 10. 16aj2_25and20aj^-9i»-20. 11. 5 a?'* 4- 5 a;2 _|_ 5 ^^^d 5ax^ — 5 ax + 5a, 12. oj^i/ + ^2/^ ^^^ ^ ^^2/ "~ ^ ^2/^ + 2 a;^/^. 13. 6aj2-5aj-6and 9a;2-6a;-8. 14. by — z-^-yz — h and by"^ -\-y'^z — b — z, 15. 6 a;2 - 54, %x + 3), and 30(a;2 -x- 12). 16. 8 a - 8 a2, 12 a(a2 - 1)^, and 18 a^ - 36 a + 18. 17. 9 a\x' - 8 aj + 16) and 3 a^a; + 6 aoj - 12 a^ - 24 a. 18. a;2 _ 4 a^nd aj3 - 10 aj^ + 31 aj - 30. 19. 3a^-12a?2and6aj4 + 30a;3_95^2^24a;. 20. a^b - a'¥ and d^b + 2 a362 + 2 a^^^ + a6^ 21. 4- a^ and a^ -f a^ - 10 a^ - 4 a + 24. 22. {x - xy, (x" - ly, and (1 - xf, 23. (1 - yy and {y + iy(l - y)\f - 7 2/ + 6). 24. x^ — (?/ + zy, {y — a;)^— z^, and y^ — {x — zy. 60 FACTORS AND MULTIPLP:S LOWEST COMMON MULTIPLE 85. Principle. — The lowest common multiple of two or more expressions is equal to the product of all their different prime factors, each factor being used the greatest number of times it occurs in any of the expressions. EXERCISES 86. Find the lowest common multiple of : 1. a'-b'^2iiidia^ + 2ab + b\ 2. r^ — s^ and 7*^ — 2 rs -f s^. 3. 2c^d + 4: cd^ + 2d^ and c^ - d\ 4. a^ — b^ and ax — a -{- bx ^ b. 5. x'^ + 5x + 6 and x^ + 6x + S. 6. a'^-5ab-^4.b^2inda'^-2ab-\-b\ 7. x(a^ — b^), x\a — b), and a^ + ab + b\ 8. 2a + l,4a2-l, andSa^ + l. . 9. x^ — 16,x^ + 4:X + 4, and x^ — 4:. 10. 1 — a^, x^ -\- X, xy — y, and a? -{-1, 11. 3 + 3 a, 2 a — 2, 1 — a^, and 4: — 4: a. 12. x^ + 5x+6, x^-x -12, 2iiidx^ -2 x^S. 13. 15(a?b - ab% 21(a3 - ab^), and 35(ab' + ¥). 14. x^ — Sx-\-15,x^—4.x-5, and a;^ _ 2 a; — 3. 15. xy — 2/^, 0.*^ + xy, xy + ?/^, and x^ + 2/^. 16. 2/^ — x^y x^ + xy + y^, and a?^ — xy. 17. m — n, (m^ — ri^)^, and (m + 7i)^. 18. a^ - b^ and a« + a^^^ _^ 54^ 19. x^ +y^ and a^a?^ — 6y + a^^/^ — 6V. 20. a^-a^ + 1, a^ + \,a^ + a^ + 1, and a^ - 1. 21. a;3 - 7 a; - 6 and a^ - 2 aj2 - 5 aj + 6. 22. a.^-a;2-14a; + 24 andaj3-3aj2- 18a:+40. 23. a^4-5a^2_ig^_72anda^ + 2a;3-25a;2-26aj + 120. FRACTIONS 87. State the difference between the arithmetical and alge- braic notions of a fraction. Define numerator and denominator. 88. Signs in fractions. Operations with algebraic fractions are performed as in arithmetic except that the signs, of which there are three, must be considered. They are the sign of the numerator, the sign of the denominator, and the sign be- fore the fraction. By the law of signs for division (§ 32) : -a_,a, +a_,a, - a __ a , ^^^ +a^_a^ ^^^^^ .^^ _5 ^jj' +2> ^jy' ^h h' -6 h Principles. — 1. Tlie signs of both terms of a fraction may be changed without changing the sign of the fraction. 2. The sign of either term of a fraction may be changed, pro- vided the sign of the fraction is changed. The sign of a polynomial numerator or denominator is changed by- changing the sign of each of its terms. By the law of signs for multiplication, the sign of either term of a fractio7i is changed by chayiging the signs of an odd number of its factors^ and left unchanged by changing the signs of an even number of its factors. EXERCISES 89. Reduce to fractions whose terms are positive : 1. Zll. 3. Mii:. 5. - '^+" • 7. --y-^ — 3 — a- — u — V ~ X — z 2. =^. 4. ~^ 9, 6 — a — b g — a{b + c) y —x — y x-\-y b(— a — b) Show that (b-a){a-b + c) ^ (a-b)(a - b + c) ^ (a - b)(b - c){c -d) (b - a){c - b)(d - c) 61 62 FRACTIONS REDUCTION OF FRACTIONS 90. Define reduction ; mixed number ; integral expression. Reduction to Integers or Mixed Numbers EXERCISES 91. 1. Eeduce — — — ^^— to a mixed number. bz Solution. — Since a fraction is an indicated division, we simply divide the numerator by the denominator until the undivided part of the numer- ator no longer contains the denominator, thus : bz bz bz Keduce to an integral or a mixed expression : 2. y. 4 6a;2/_ 3x 6. 22/ + ^ 2 o. • y 3. -2/. 2 abc ^ ab + d a _ 4 a;2 -|- 6 a? ^- 2x ^ abc + 2 o6 „ 6a;2+19a! + 10 11. •" -r^-^-r^ . 18 a; 12. ^!±^1±2. 19. 13. -^lil^^l?. 20. 14. ^'-2^-8. 21. aj — 4 15. ^^JZ^^±l. 22. 16. -r^^^-r^ ^, 23 a'-{-b ^ -\- y Sx + 2 a4 + 3 a26^ + b' a" + 62 4a;2 4.22.^ + 21 2a; + 4 aj3 + 2aj2_|_3^4.1 a;-f 2 a4 _ a^h + 2 a262 4- 6' a2 + 62 m^ -2 m^n - 3 mn^ -2 n^ m^-n^ x* + 4x^y+6 xy + 4 a^ FRACTIONS 63 Reduction to Lowest Terms 92. What are equivalent fractions ? When is a fraction said to be in its lowest terms ? 93. Principle. — Multiplying or dividing both terms of a frac- tion by the same number does not change the value of the fraction. EXERCISES 94. 1. Eeduce ^^^-^ — — to its lowest terms. Solution. a^~ 2 a6 + 5-^^_ (a- 6)(^^^ ^_ o-j^ Rule. — To reduce a fraction to its lowest terms, factor both terms, and divide both terms by their common factors. Use cancellation wherever possible. Reduce to lowest terms : a^b& 10. 11. 12. 13. 14. 15. I 18. , 24 20 . 5. aWc x'yh oi^yV r a;2- 2xy + y'^ 3a2j-3a6^ d'^ab^ 10 nx + 10 ny 25 nx^ — 25 ny^ /-81 2/2 + 72/-18' 352 _^ 95 _ 54* (g + 5)2 _ 1 a^c + abc + ac x^y^ a rn-\-2rylr 2 a'y^' 17. 8. 9. 4a;2 2x'^2xy aWc — abc^d abc 18. 19. 20. 21. 22. 23. — cc" a26(a + 2 5)^ a6(a2-4 62)2' g?^ + a;y + y^ x^ + f cd^ — c ^5 _ ^2 _ ^4 ^ ^ a^ — ab — a'^b + b^ a'-a^b-a^b^ + b^' 64 FRACTIONS Reduction to Lowest Common Denominator 95. When are fractions said to have a common denominator? their lowest common denominator (1. c. d.)? EXERCISES 96. Reduce to respectively equivalent fractions having their lowest common denominator : <*^ ^ « 1. -, a, and a^-r ' a-1 Solution. — Since the 1. c. m. of the given denominators is a^ — 1, each fraction or integer must be reduced to a fraction whose denominator is a^— 1. Then, _^!_^_^^; ^^a^ a(a'-l) ^nd _^_^ <«+!). Rule. — Find the lowest common multiple of the denominators of the fractions for the lowest common denominator. Divide this denominator by the denominator of the first frac- tion, arid midtiply the terms of the fraction by the quotient. Proceed in a similar manner with each of the other fractions. All fractions should first be reduced to lowest terms. o 2 3 , 2bx 'day x-y ay 2x z X -\-y « r 7 _ aa?2 d^x „ be ab ' ^ ab'^c bc'd a-1 a + 1 a — b 2a a + x . 2 111. , Of 3(a + 5) (a + by a ' c + d 2x—2yx — y a^ 2 a ab y, , — • xx 12. 13. 14. y'^ ' X + 1 ' y -{-1' y^ — l' y—1 ab be cd 05^ — y*^ x^ -{- 2/^' y- — x'^ X^ X x^ x^ + x' + l' 0.-3 + 1' a.3-1 y4-3 y -2 y — 1 y^-3y-\-2' y^^2y-3' y^'-^y-6' FRACTIONS 65 ADDITION AND SUBTRACTION OF FRACTIONS 97. In algebra, subtraction of fractions practically reduces to addition of fractions, for every fraction to be subtracted is added with its sign changed. EXERCISES 98. 1. Find the algebraic sum of — — a 4- x _ x — a a?^ — a- a — X x -\- a c 4 ax a -\-x X — a 4ax,x-\-ax — a Solution. ' =— -H x:^ — a^ a — X X + a x^ — a^ x — a x -\- a _ 4: ax + (x + ay - (x -^ aY- x2 - aP- . 4 ax+x'^+2 ax + «^— y^4- 2 u x—a'^ ^2 - a2 8 ax x2-a2 Rule. — Reduce the fractions to respectively equivalent frac- tions having their lowest common denominator. Change the signs of all the terms of the numerators of fractions preceded by the sign — , then find the sum of the numerators, and write it over the common denominator. Reduce the resulting fraction to its lowest terms, if necessary. Add : Subtract : 2. — and 5. — ^ from — ^. 4 6 8 6 o3&-,-6 ^ —2a n 'dx 3. — and 6. from 4 c 3 c X a 4. Zl^and^l^. 7. ^-I^from^-^. 32/ ^y 2 3 Reduce these mixed expressions to fractions : 8. x-\--. 11. a^ + a^-h?. 14. x-^—'=-^> X by n 1 y^ ^rt , ax -\- c ^^ r — s -\-t 9. y^—^* 12. x-\ ■ 15. r ^— . O XT S 10. ?+&. 13. ^LzJ:+5a. 16. x'-x--^^. c 2 x-\-y milne's sec. course alg. — 6 66 FRACTIONS Simplify : 17. b_^-±^. 21. 2a-3b-^^^±^ be ac 2 a + 3 6 a — & aH-6 a^ + a + 1 a^— a + 1 19. ^ + ^-3+1. 23. --^^^ ^ - ^^ 52 6 a;2-9 aj-3a? + 3 a^ &2 a — b^ a^-i-b^ , a a6-&2 ab-a" 2{a -{- b) a'-b^ b-a 25. _A_ + ._4_ + ^^+^V 26. 27. a — 6 ' a + 6 ' b^ — a^ 1 1.1 a^ + 8 8 - a^ 4 - a^ 5(a; - 3) 2(0? +2) x-1 x^ — X — 2 a? -{-4:X + 3 6 — x — x^ 28. t±^l±l-l+ 2a; Suggestion. — Reduce the first fraction to a mixed number. 29. a^ + 2ab + i a' + b' „^ ajH-l.a; — 1 a?H-2 aj — 2 ^y. • x—lx+lx—2x+2 a -\- X a^ -\- x^ a — x a^ — x^ _4: a?x + 4 aa^ a — a; o? — x^ a-\-x d^ -\-x^ a^ — x^ Suggestion. — Combine the first two fractions, then the result and the third fraction, then this result and the fourth fraction, and so on. c^ab bha __ a?bc ^ (c — a)(6 — c) (b — a)(b — c) (a — b){a — c) Suggestion. — Change the signs of the factors (c — a) and (6 — a), 33. ^-±^^ I ^ + ^ + ?dlA (a — b)(b — c) (c — a)(b — a) (c — b){a — c) FRACTIONS 67 MULTIPLICATION OF FRACTIONS 99. As ill arithmetic, Principle. — Tlie product of two or more fractions is equal to the product of their numerators divided by the product of their denominators. EXERCISES 100. 1. Simplify f ~ ^' X 4a X 4^o X TT^' Solution. .«il=^ x 4 a x -^^ x -^ ?0-K^) 1 g^^h^ ^iib b ' KuLE. — Reduce integers and mixed numbers to fractions. Factor each numerator and each denominator. Cancel factors common to mimerator and denominator. Write the product of the remaining factors in the numerator over the product of the remaining factors in the denominator. Simplify : 2. abx^-. 5. ^xa%. 8. ^x " a 2a^ b'ho a^y 3. 2.x^. 6. ^X^. 9. =i^x^> z2 aV cW ad z^ 11. ?x-^x5. 16. Sa^x^x^^. y z X a + 6 4 ao2 12. ^X-^X-^. 17. ^^x^^^X^^. be ac ab 6ab xy b — a 13. ^^^x^'. 18. ^^±^x4.c^dX^^=^. yz xz a^ c^—d^ 2acd'^ X^^ tly^^X—' 19. -^'~^' . ^' . P'-^ a¥ yH c^x ' p'^-\-q^ (P+^T s(p—qy b^c a^ mx^ ' x+y xy—y'^ x^—y'^ 68 FRACTIONS Simplify : jod., — X ; * <«0. 22. ^ - ^ X ^ "" ^' - 26. ^' . ^Hi^Y+y. x-{-2 16 — 07^ x2_l_aj2/4-2/^ aic^+^y 23^ a^4-a?> ^^ a^ - &^ 2^^ a^^ + ah'' + ^>^ . (a - ^)^ a^ — 5^ c(&(a + &) a^ — ab a^ — W x'-\-2x x''-x-^ a?^ + 3x4-2 ^ a^'^-e.-g+g a;2-3i» aj2 4-4a^i-4' * x'-?>x-10' x'+^x + l' 2^ g'^ + g^ + 2 g + 2 6 ^ x'-2xy ^ ax — 2ay + 2x — 4:y {a-\-by a'^ — b^ a + b a'^ — ab + b'^ 30. 31. 32. 33. 34. 35. a3 4-63 a^ - ab^ (a + &)2 X + y -{- z ^ X — y -{- z {x — ?/)2 — z^ x + y — z X — y — z (x -\- yy — z^ a;2 4- 6 a; + 8 ^ (a; - 1)^ ^ a;2^5a; + 6 ^ a;2 + a; — 2 ' a;2 — 4 * a;^ + 3a; — 4* c^ + (^>' (c^ - d-)(c - 3) c^ + c - 2 ^ c3 _ (/3 • (^ _ i)(^c + c/)2 * c^-c-e' a^^¥ a'^2aW + ¥ 2ab a{a^ + ¥) b\a? - ab + ¥) (a -f by x^ + 2x-\-2 ^ x^- 1 ^ a^ + l aj2 — a^ + l a;^-f4 x'^ -^x + 1 1^3aj y 4-9a;2 * 4.f 37. a6 \/ a6 \/a2 — 62' a—bj\ a + bJ\a^-\-b\ 38. fl ?^ + ^ Yl ^±^— ^ 2/^+72/4-loA 2/^ + 7^ + 12; 39 I ^' + ^ ^^ + ^' _ 1 Yl _ ^izil^'^ -J^ FRACTIONS 69 DIVISION OF FRACTIONS 101. What is the reciprocal of a number? of a fraction? X 1 Write the reciprocal of 2 : of f : of a ; of - ; of - • y n 102. As in arithmetic, Principle. — Dioiding by a fraction is equivalent to multiply- ing by its reciprocal. EXERCISES 103. 1. Divide ?^±A' by t±^b + b^. a^-b^ ■> a-b SoLUTiox. gl+^^ «!■+«?> + 6^ = ^liL&!x «-^ a'^—b'^ a — b a^ — b'^ a- -{- ab -\- b'^ - (^^)ia^-CLb + b-^) ^ n>-^ _ a- - ab + b'^ Or^)(iJ^=^6) a^ + ab-{-b^ a^ -{■ ab -\- b-^' Rule. — Reduce integers and mixed numbers to fractions. Take the reciprocal of each divisor and proceed as in multijjlication. Simplify : 2. 1^^. 3. 1^^. 4. 2^1. d b'^ a a^ 2ab ^ ia^b -„ 3 xy 9 x^y^ 12a^6 . 4: ax 25 ac ' TKc^' 13. rs — s^ s^ (r + sy • ^2_52 a^ + a\ a'^ ^ax-{- aj2 a' - x" a — x a?-xz^ . i^-zf 7. ^^2xy. 14. o (a? + zy xh — ^ 8. 6^,.12m«\ 15. fa^^\^fb^ ^ 5 ax 15 a^ \ ' bj \ a' 9 g + ^ . a^-b\ ^^ a2 + 3a_4 g^-ie * 4 a ' 2 62 * • (j2 __ 1 ' a' + a' 10. (^ + y)' . ax + ay ^^ x^-y' . x^-\-y\ x^ — y^ ' x — y ' x'^-^2xy-{-y^ ' x^-\'Xy 11. (8?/ + 4)-f-?Xdll. 18 ^^^a?4-m^ , mV-ma^ 3 a; m^x—m^ m^x^—x^ 70 FRACTIONS Simplify : ^^ a^ + 27 , a4-3 a^-27 ' a2+3a + 9' 20 y^ 4-6^-7 , / + 4y-21 y' + 3y-A ' 2y + S 2 a' -\- a -15 , 2a^-3a-5 3a2-a-2 ' 3a'-7 a-6' a^ + b''^ — c^-{-2ab , a 4- ?> + c ^ a^ - 6^ - c'-^ + 2 &c * a - & + c * 23. (r^ + i + 2)^(. + ^| 27. (a^ + on-^i ■ ; ' — )• \ 4:X ax— bx J Complex Fractions 104. Since a complex fraction is only an expression of un- executed division, it may be simplified by performing the division. EXERCISES 105. Simplify: a + b 1 2 14-? 3 "^ 5. 15-24-a a 6a 1-5 a 1 X 3m m — ir + ^ a^ + .V «= d * 6. y ^ X m i_i 2/ ^ FRACTIONS 71 Simplify : I4.I + I 1 J 2 9 X x^ Q^ x-\-yx — yox — y X x^ y'^ — ^x^ 5_-_5_^ 24 x^'\-{a^lS)x-^a'b 2 ^ a;^-(a + 6)a; + a6 9-30? ' * a;^-5^ x x^ — a? ?>xyz X y z 11. yz + xz-rxy ^ 1 1 1 1 X y z A complex fraction of the form is called a con- tinued fraction. h + d H — Every continued fraction may be simplified by successively simplifying its last complex part by multiplying both terms by the last denominator. 12. T— • 16. 1+— ^ x+1 ^ 1+^ 0:4 ^ a x—l a 13. 5 17. C-1+ a + l+ ^ 1 + -^ a + 1-?: 4 — c a 14. -J-^ 18. 14 a * -, , , 2c 1+C + 1 — a c ?/ 1 15. 19. a + y -\ '^ — a — 1 2+1 y-1 a 72 FRACTIONS MISCELLANEOUS EXERCISES 106. Reduce to lowest terms : 10 a;2 + 23 a; + 12 3. a' + b' a* + a'b^ + b\ ex — cd oc^ + Sx--^x-\-3 cx + 3x — 3d — cd Simplify : 2y-l 2y + l 1 - Aif f 1 4_ ^ V ^ _ 2x'^ -{-2ax — a^\ xJ\x-{-a x'' + 3ax + 2ay' / m — 3?A /-| 4n \ /m ^ ^^^ m-\-n J\ m + 7ij ' \7i m x-\-3y) V ^ + ^y) 9. ir_L_v viijv_. 11 {a + 1)2 (a+iy a 1 (a 1 (a + iy 1 1 1-T^ 1 c? m^ — 7?^ A 2 10. z: 12. 1 2— ^ 4-. ^ 13. 1 — a; 6 — cc x\ a i/ xy x-y^ xy — 1 oi^y^ — x^y'^ i4_2,4 xy + 1 xy x^y'^ SIMPLE EQUATIONS ONE UNKNOWN NUMBER 107. Review the definition, explain, and illustrate : 1. Numerical equation. 6. Equation of condition. 2. Literal equation. 7. Root of an equation. 3. Integral equation. 8. Solution of an equation. 4. Fractional equation. 9. Equivalent equations. 5. Identical equation (identity). 10. Simple equation. 11. Give two other names that are applied to simple equations. 12. When is an equation said to be satisfied ? 108. By the axioms in § 43, if the members of an equation are increased or diminished or multiplied or divided by the same or equal numbers, the two resulting members are equal and form an equation. But it does not necessarily follow that the equation so formed is equivalent to the given equation. For example, if both members of the equation z + 2 = 6, whose only- root is ic = 3, are multiplied by x — 1, the resulting numbers, (x -f 2) (x— 1) and 6(x — 1), are equal and form an equation, (x + 2)(x-l) = 5(x-l), which is not equivalent to the given equation, since it is satisfied by X = 1 as well as by X = 8 ; that is, the root x = 1 has been introduced. In applying axioms to the solution of equations we endeavor to change to equivalent equations, each simpler than the preced- ing, until an equation is obtained having the unknown number in one member and the known numbers in the other. 73 74 SIMPLE EQUATIONS 109. The following principles serve to guard against intro- ducing or removing roots without accounting for them : Principles. — 1. -//" the same expression is added to or sub- tracted from both members of an equation^ the residting equation is equivalent to the given eqiiation. 2. If both members of an equation are multiplied or divided by the same known number, except zero, the resulting equation is equivalent to the given equation, 3. If both members of an integral equation are mxdtiplied by the same unknown integral expression, the resulting equation has all the roots of the given equation and also the roots of the equa- tion formed by placing the multiplier equal to zero. It follows from Principle 3 that it is not allowable to remove from both members of an equation a factor that involves the unknown number, unless the factor is placed equal to zero and the root of this equation is preserved. Thus, if x — 2 is removed from both members of the equa- tion (x — 2) (a: + 4) = 7(ic — 2), the resulting equation x + 4 = 7 has only the root x = 3 ; consequently, the root of x — 2 = 0, removed by dividing by the factor x — 2, should be preserved. , Clearing Equations of Fractions '^ EXERCISES no. 1. Solve ?-^^- 1^^15 = 3-^. 4 6 2 Solution. — Multiply both members of the equation by the Led., which in this case is 12, to clear the equation of fractions, obtaining, 3(3x-. 5)-2(7x- 13)=36-6(x-f-3). Expand, 9x - 15 - 14x + 26 = 36- 6x - 18. Transpose, etc., x = 1. Verification. — When x = 7, the given equation becomes — 2 = — 2, an identity; consequently, the equation is satisfied for x = 7. KuLE. — To clear an equation of fractions, multiply both members by the lowest common denomiyiator of the fractions. 1. Reduce all fractions to lowest terms and unite fractions that have a common denominator before clearing. 2. Discover extraneous roots by verification, and reject them. SIMPLE EQUATIONS 75 Solve, and verify each result : 5. ^ + 5 = 24. 7 Sx 7 X X 5x _1 ' "^"^16 2 16 "8* 2x 5 X 4a; a;_l * 15 25 9 6'"9' 3a; 7a?^lla? 8a; 3 '4 12" 36 9 2* . 15x . 5x 11 a; , 19 a; o 10. ^_!^ + ^=16. 3 5 4 11 5a;-6 4a;+7_l , 3a;-4 2. 5. 3. l-'o _26 3* 4. 1+2. = 26. 5 10 2 ' 6 12 a; + l a;-2 a; + 3^^ ■ 6 5 10 13 3y±4 y-3 4-2y_. 13. —4—+^ g--5. ,. 3a; — 5 a;+l 2a;, 5a; — 11 2 4 7 6 15. 10^JL§_6^=10(a;-l). 16. .7 a; + .24 = .08 a; + 9.2- .02 a;. Suggestion. — Clear of decimal fractions by multiplying by 100. 17. .375 - .25 a; + .625 ==,5x-,6 + .05 x, 18. .18 X - 28.4 - .06 x = .35 - .2 a; - 9.55. n + 4 , 2 — 2n_n + l 10 .3 .6 .2 .3 2 a; X o_ 1 a; + 3 a;-5 ""2a;-10 76 SIMPLE EQUATIONS 21. 6r-7 5(r + l) ^l 9r + 6 12r + 8 12* 22. 23. 10g-{-17 5g-2^ 12^-9 18 9 11 g- 8* y — 1 y — ^ __ y — 4 y — 5 2/-2 y-3^y-5 y-6 Suggestion. — Combine the fractions in each member of the equation before clearing of fractions. 2^ 2i«4-l 2aj + 9_a^-3 x-7 X'\-l x-\-5 x — 4: x — S a;+l a; — 1 aj^ — 1 26. 3.1416 X - 17.1441 + .0216 x = .2535. 3?i,-4 /4w n + 2\ 9w Aq «-|-4\ ^'^^ ~~4 VT+^~j=10+(,^^ 2-j- 28. fa-3)^ _ (a; + 4)^ ^^Q_/3^_^5a;+10^ 4a;'. 29. 21 y 21 2a;U-5^ 3a;ri-^', a; — 4 30. l.-2(*^_3)=4-^(..l). 31. 32. 33. (2a; + l)'' (4 a; -1)^ ^15 3(4 a; + 1) .05 .2 .08 .4 17+§ 1+^ 21_i 100^5 a; a; _ a; < ^ ^ ~3~'''~"5 9~"^~l5 ^(a;-4) 4a;-16 _3 5 ^^ 6-5 -I . • SIMPLE EQUATIONS 77 Literal Equations 111. 1. Solve the equation — -^- — = —^ — for x, n m Solution. — Clear the equation of fractions, obtaining mx + m^ = na: -f n^. Transpose, etc., mx— nx = n^ — m^, or (m — n)x = — (m^ — n^) . Divide by m — w, x = — (m^ -f mn -\- n^). Verification. — Let m = 2 and n = \ ] then, 5c== — (4+2 + 1) = — 7, and the given equation becomes —3 = — 3, an identity ; that is, — (m^+wi/i+n^) is the root. Solve for x, and verify each result : 2, W + ax=^a^ + hx, 8. a; — 1 + 4 5 = 6(3 h + x), 9. {x-ay-{x-hf = {a-h)\ b a ^^ h ^ (a + hf-aia + h) ^ a-\-h x 3ax—2b__ax—a_ax_2 ' 3b 2i"~y 3' X — 2ab 1 X — 3g 3. .t(1-3c)+-9c2 = 1 4. ± + ^ = a^ + b\ bx ax 5. n , X ax am —OH — = — m b 6. x-2c 2 ax — 4c a^ c 7. x -i- r r + s 13. X — s r — s ex X abx 14. x(b + o)— 2 a(b + c)= a'^ - ax + ¥ -^ c(2 b -{- c). 15. a(x- a—b)+b(x — a-\-2c)=c(x—2a-{- c)-\- b\ 16. (x ■^G)(x — d)— 2{x + d)(x — c) = 6*2 — (a; — d)(a; — c). 17. + = 0. a(b — x) b{c — x) a{G — x) 18 ^ + ^ | ^ + c . a; + 6 ^a6c,i b a c b c a 19. + -—- — - = a2 -+ 62 _|_ ^2 ^ 2 ab. a -\- b + c a -{- b — c 20. x'^ - ax -bx-^ab _ x^-2bx-^2 b^ c^ X — a x—b X — c 78 SIMPLE EQUATIONS Problems 112. Eeread the general directions given in § 47, and solve : 1. Leo has 3 times as many plums as Carl. If each had 5 more, Leo would have only twice as many as Carl. How many plums has each? 2. Ann paid $3.00 for three books. The' first cost ^ as much as the second and ^ as much as the third. Find the cost of each. 3. Cornstalk paper costs i as much as paper made from rags. A ton of the former costs $ 50 less than one of the latter. Find the cost of each kind of paper per ton. 4. Four wagons drew 38 logs from the woods, one wagon holding 2 logs more than each of the others. How many logs did each wagon hold ? 5. The distance around a desk top is 170 inches. If the desk top is 15 inches longer than it is wide, how wide is it? 6. A man paid $ 300 for a horse, a harness, and a carriage. The carriage cost twice as much as the harness, and the horse as much as the harness and carriage together. Find the cost of each. 7. I bought 15 books for $ 6.60, spending 30 cents each for one kind and 60 cents each for the other. How many books of each kind did I buy ? 8. A shipment of 12,000 tons of coal arrived at Boston on 3 barges and 2 schooners; Each schooner held 3^ times as much as each barge. Find the capacity o^ a barge ; of a schooner. 9. John has $ 6.75. He has 3 times as many dimes as nickels, and as many quarters as the sum of the nickels and dimes. How many coins has he of each denomination ? 10. John is 15 years older than Frank. In 5 years Frank's age will be ^ John's age. What is the age of each ? SIMPLE EQUATIONS 79 11. George is ^ as old as his father ; a years ago he was ^ as old as his father. What is the age of each ? 12. Harold is n times as old as his brother ; r years ago he was m times as old. Find the age of each. 13. Three pails and 6 baskets contain 576 eggs. All the pails contain -|- as many eggs as all the baskets. How many eggs are there in each pail ? in each basket ? 14. The cost per mile of running a train was 14 cents less with electrical equipment than with steam, or |- as much. What was the cost per mile with electricity ? 15. A rectangle is 9 feet longer than it is wide. A square whose side is 3 feet longer than the width of the rectangle is equal to the rectangle in area. What are the dimensions of the rectangle ? 16. A field is twice as long as it is wide. By increasing its length 20 rods and its width 30 rods, the area will be increased 2200 square rods. What are its dimensions ? 17. The length of the steamship Mauretania is 790 feet, or 2 feet less than 9 times its width. What is its width ? 18. The length of a tunnel was 22i times its width. If the length had been 50 feet less, it would have been 20 times the width. Find its length ; its width. 19. In a purse containing % 1.45 there are \ as many quarters as 5-cent pieces and | as many dimes as 5-cent pieces. How many coins are there of each kind ? 20. The St. Lawrence River at a point where it is spanned by a bridge is 1800 feet wide. This is 180 feet less than f of the length of the bridge. How long is the bridge ? 21. A girl found that she could buy 18 apples with her money and have 5 cents left, or 12 oranges and have 11 cents left, or 8 apples and 6 oranges and have 10 cents left. How much money had she ? 80 SIMPLE EQUATIONS 22. A can do a piece of work in 10 days. If B can do it in 12 days, in how many days can both do it ? Suggestion. — Let x = the required number of days. Then, - = the part of the work both can do in 1 day. 23. A can pave a walk in 6 days, and B in 8 days. How long will it take A to finish the job after both have worked 3 days ? 24. A can do a piece of work in 2^ days and B in 3^ days. In how many days can both do it ? 25. A can paint a barn in 12 days, and B and C in 4 days. In how many days can all together do it? 26. A and B can lay a walk in 8 days, B and C in 9 days, and A and C in 12 days. In how many days can C do the work alone ? 27. One pipe can fill a tank in 45 minutes and another can fill it in bb minutes. How long will it take both to fill it ? 28. A tank can be filled by one pipe in a hours, by a second pipe in c hours, and emptied by a third in h hours. If all are open, how long will it take to fill the tank ? 29. In a number of two digits, the tens' digit is 3 more than the units' digit. If the number less 6 is divided by the sum of its digits, the quotient is 6. Find the number. Suggestion. — Let x = the digit in units' place. Then, x + 3 = the digit in tens' place, and 10(x + 3) + x = the number. 30. The sum of the digits of a two-digit number is 11. 63 added to the number reverses the digits. Find the number. 31. In a two-digit number, the tens' digit is 5 more than the units' digit. If the digits are reversed, the number thus formed is | of the original number. Find the number. 32. In a two-digit number, the units' digit is 3 more than the tens' digit. If the number with digits reversed is multi- plied by 8, the result is^ 14 times the original number. Find the number. SIMPLE EQUATIONS 81 33. A man invests $ 5650, part at 4 % and the remainder at 6 % . His annual income is $ 298. How much has he in- vested at each rate ? 34. A man has | of his property invested at 4 %, i at 3 %, and the remainder at 2 %. How much is his property valued at, if his annual income is $ 860 ? 35. Mr. Johnson had $15,000 invested, part at 6% and part at 3 % . If his annual return was 5 % of the total invest- ment, what amount was invested at each rate? 36. A man desires to secure an income on $12,000 which shall be at the rate of 4^ % . He buys two kinds of bonds which yield 6 % and 4 % , respectively. How much does he invest in each? 37. A bank invests s dollars, part at 6 % and the remainder at 5 %. If the annual income is m dollars, how much is in- vested at each rate ? 38. My annual income is m dollars. If - of my property is n invested at 5 % and the remainder at 6 % , what is my capital ? 39. At what time between 6 and 7 o'clock are the hands of a clock together ? Suggestion. — Let a; = the number of minute spaces passed over by the minute hand after 6 o'clock until the hands come together. Then, -^ = the number of minute spaces passed over by the hour hand. Since the hands are 30 minute spaces apart at 6 o'clock, x — — = 30. 40. At what time between 2 and 3 o'clock are the hands of a clock at right angles to each other ? 41. Find two different times between 6 and 7 o'clock when the hands of a clock are at right angles to each other. 42. "Find at what time between 1 and 2 o'clock the minute hand of a clock forms a straight line with the hour hand. milne's sec. course alg. — 6 82 SIMPLE EQUATIONS 43. I have 6\ hours at my disposal. How far may I ride at the rate of 9 miles an hour, that I may return in the given time, walking back at the rate of 3^ miles an hour ? Suggestion. — Let x = the number of miles I may ride. Then, the equation of the problem is - + — = 6J. 9 3J 44. A steamboat that goes 12 miles an hour in still water takes as long to go 16 miles upstream as 32 miles downstream. Find the velocity of the stream. 45. A motor boat went up the river and back in 2 hours and 56 minutes. Its rate per hour was IT^ miles going up and 21 miles returning. How far up the river did it go ? 46. A yacht sailed up the river and back in r hours. Its rate per hour was s miles going up and t miles returning. How far up the river did it sail ? 47. A train moving 20 miles an hour starts 30 minutes ahead of another moving 50 miles an hour in the same direc- tion. How long will it take the latter to overtake the former ? 48. If an automobile had taken m minutes longer to go a mile, the time for a trip of d miles would have been t hours. How long did it take the automobile to go a mile ? 49. In an alloy of 75 pounds of tin and copper there are 12 pounds of tin. How much copper must be added that the new alloy may be 12^ % tin? Suggestion. — Let x = the number of pounds of copper to be added. Since the new alloy weighs (75 + x) pounds, the equation of the prob- lem is .12^ (75 + x) = 12. 50. In an alloy of 100 pounds of zinc and copper there are 75 pounds of copper. How much copper must be added that the alloy may be 10 % zinc ? 51. In a solution of 60 pounds of salt and water there are 3 pounds of salt. How much water must be evaporated that the new solution may be 10 % salt ? SIMPLE EQUATIONS 83 52. In p pounds of bronze, the amount of tin was m times that of the zinc and n pounds less than - that of the copper. r How many pounds of zinc were there ? 53. It is desired to add sufficient water to 6 gallons of alcohol 95 % pure to make a mixture 75 % pure. How many gallons of water are required ? 54. How much pure gold added to 180 ounces of gold 14 carats fine (i| pure) will make an alloy 16 carats fine ? 55. How much pure gold must be added to w ounces of gold 18 carats fine that the alloy may be 22 carats fine ? 56. A body placed in a liquid loses as much weight as the weight of the liquid displaced. A piece of glass having a volume of 50 cubic centimeters weighed 94 grams in air and 51.6 grams in alcohol. How many grams did the alcohol weigh per cubic centimeter ? 57. Brass is 8f times as heavy as water, and iron 1\ times as heavy as water. A mixed mass weighs 57 pounds, and when immersed displaces 7 pounds of water. How many pounds of each metal does the mass contain ? Suggestion. — Let there be x pounds of brass and (57 — x) pounds of iron. Then, x pounds of brass will displace {x -4- 8f ) pounds of water. 58. If 1 pound of lead loses -^-^ of a pound, and 1 pound of iron loses ^ of a pound when weighed in water, how many pounds of lead and of iron are there in a mass of lead and iron weighing 159 pounds in air and 143 pounds in water ? 59. If tin and lead lose, respectively, ^^y and 2% of their weights when weighed in water, and a 60-pound mass of lead and tin loses 7 pounds when weighed in water, what is the weight of the tin in this mass ? 60. If 97 ounces of gold weigh 92 ounces when weighed in water, and 21 ounces of silver weigh 19 ounces when weighed in water, how many ounces of gold and of silver are there in a mass of gold and silver that weighs 320 ounces in air and 298 ounces in water ? 84 SIMPLE EQUATIONS Formulae 113. A formula expresses a principle or a rule in symbols. The solution of problems in commercial life, and in mensura- tion, mechanics, heat, light, sound, electricity, etc., often depends upon the ability to solve and apply formulse. I. The lateral surface (S) of a circular cylinder is 2 times TT (= 3.1416) times the product of the radius (r) of the base and the height (h), or $ = 2 irrh. 1. Solve for h. Find the height of a circular cylinder whose lateral surface is 942.48 square inches and the radius of whose base is 10 inches. II. The formula for the surface (S) of a rectangular solid in terms of its length (Z), breadth (6), and height (h) is S = 2(/6 4- M + bh). 2. Solve for Z ; f or & ; for h. 3. Find the height of a rectangular solid 6 feet long and 4 feet wide, having a total surface of 108 square feet. III. The formula for the percentage (p) in terms of the base (b) and rate (r) is n = bn 4. Solve for h ; for i\ 5. If nickel-steel is 2.85 % nickel, how many pounds of nickel are there in 2 tons of the nickel-steel ? 6. Out of a lot of 360 brass castings 24 were spoiled. What per cent of the castings were spoiled ? 7. The machines in a shop require all together 16 horse power to run them and are driven by a single motor. If 20 % of the power of the motor is lost through friction, etc., what is the necessary horse power of the motor used ? IV. The formula for the interest {i) on a principal of p dollars at simple interest at r % for t years is / = prt. 8. Solve for p ; for t. What principal will yield $ 480 interest in 3 years 4 months at 6 % ? 9. In what time will $ 4000 yield % 350 interest at 5 % ? SIMPLE EQUATIONS 86 V. The formula for the amount (a) of a sum of money (p) at the end of t years at simple interest at r % is a=p{l+rt), 10. Solve for p ; for t. What principal will amount to $ 828 in 3| years at 4 % ? 11. How long will it take $ 600 to amount to $ 1000 at 6 % ? VI. The formula for converting a temperature of C degrees Centigrade into its equivalent temperature of F degrees Fahrenheit is F — - ^ -4- 32 12. Solve for (7. Express 86° Fahr. in degrees Centigrade. VII. If a steel rail at 0° C. is heated, for every degree it is heated it will expand a certain part of its original length. If E denotes the total expansion, L the original length, T the number of degrees change in temperature, and k the certain fractional multiplier, or coefficient of expansion ; then f = LkT. 13. Solve for k. A steel rail 30 feet long at 0° C. expanded to a length of 30.001632 feet at 50° C. Find the value of k. VIII. The formula for the velocity acquired in t seconds by a body moving with uniform acceleration (a) is 14. Solve the formula for a ; for t. 15. A body starting from rest and moving with a uniform acceleration acquires a velocity of 100 feet per second in 5 seconds. What is the acceleration ? IX. The formula for the space (s) passed over by a freely falling body in any second (t) is s = lg(2t-l), g, the acceleration due to gravity being approximately 32 feet. 16. Solve the formula for t, A brick dropped to the ground from the top of a chimney. How far did it fall during the second second? the third second? • • 86 SIMPLE EQUATIONS X. The formula for the width ( W) in inches of a nut for a bolt of a certain diameter (D) in inches is 17. Find the width of a nut for a |-inch bolt. 18. Solve the above formula for B. What is the diameter of a bolt that will fit a nut If inches wide ? XI. The length (I) of the belt required for two pulleys, each with a radius of r feet, equals the circumference of one pulley plus twice the distance (d) in feet between the centers of the pulleys, that is, , ^ ^(^r + d). 19. Solve the formula for d ; for r. 20. How far apart are the centers of two pulleys, radius lOi inches, if a belt 351 feet long is required ? (Use tt = 3|.) XII. The length (L) of a bar of thickness {T) needed to make a welded ring with a certain inside diameter (D) is L = ir{D+ T). 21. Find the length of a bar i of an inch thick required to make a ring with an inside diameter of 10 inches. (Use tt = 3|.) 22. Solve for D. Find the inside diameter of the ring that can be made from a bar 44 inches long and i of an inch thick. XIII. The cutting speed {S) of a tool is the rate in feet per minute at which the cutting tool passes over the surface being cut. It equals ^^2^ of the circumference (ird) of the piece being cut in inches multiplied by the number (n) of revolutions the cutting tool makes per minute, or 12 23. The diameter of a piece of brass being turned in a lathe is 31 inches. If the lathe makes 120 revolutions per minute, what is the cutting speed ? (Use tt = 3|.) 24. Solve for n. The cutting speed of a lathe in turning a piece of iron 4| inches in diameter was 33 feet per minute. How many revolutions did the lathe make per minute? RATIO AND PROPORTION RATIO 114. Define and illustrate : 1. Ratio; couplet. 4. Duplicate ratio. 2. Antecedent of a ratio. 5. Triplicate ratio. 3. Consequent of a ratio. 6. Reciprocal (inverse) ratio. 115. The ratio of two quantities is the ratio of their numeri- cal measures, when expressed in terms of a common unit T^us, the ratio of 33 ft. to 3 rd., or 2 rd. to 3 rd., is f. 116. One number is said to be greater than another when the remainder obtained by subtracting the second frpm the first is positive, and to be less than another when the remainder obtained by subtracting the second from the first is negative. If a — 6 is a positive number, a is greater than b ; but if a — 6 is a negative number, a is less than b. Any negative number is regarded as less than ; and, of two negative numbers, the one more remote from is the less. An algebraic expression indicating that one number is greater or less than another is called an inequality. 117. A ratio is said to be a ratio of greater inequality, a ratio of equality, or a ratio of less inequality, according as the antecedent is greater than, equal to, or less than the consequent. Thus, when a and b are positive numbers, ^ is a ratio of greater in- b equality, if a > 5 ; a ratio of equality, if a = 6 ; and a ratio of less inequality, if a < 5. 87 88 RATIO AND PROPORTION Properties of Ratios 118. Since a ratio is expressed as a fraction, ratios have the same properties as fractions. Hence, Principles. — 1. Multiplying or dividing both terms of a ratio by the same number does not change the value of the ratio. 2. Multiplying the antecedent or dividing the consequent of a ratio by any number multiplies the ratio by that number. 3. Dividing the antecedent or multiplying the consequent by any number divides the ratio by that number. 4. A ratio of greater inequality is decreased and a ratio of less inequality is increased by adding the same positive number to each of its terms. For, given the positive numbers a, b, and c, and the ratio ?. 1. When a>b, it is to he proved that ^ <^ . > ' b + c b a_j-_c _ g _ c(b — a) Since a>b, b -a is negative, and ^^^, ~ ^; is negative ; therefore, ' o(o + c) P^-f is negative, and (§ 116) f4^ - . b + c b As in 1, since a ?. b -{- c b b -{- c b 5. In a series of equal ratios^ the sum of all the antecedents is to the sum of all the consequents as any antecedent is to its con- sequent. For, given - = - = - = r, the value of each ratio. b d f ' By Ax. 3, a = br, c = dr^ e —fr; whence. Ax. 1, a-\-c-}-e=(b + d-\- f)r ; .•.^±^Jl^=>or ^or^or ^. b + d-{-f b d f RATIO AND PROPORTION 89 EXERCISES 119. 1. What is the ratio of 6 a to 9 a ? of 9 a to 6 a? 2. What is the ratio of ^ to ^ ? i a; to ^ a; ? f .V to | ^ ? 3. What is the inverse ratio of 5 : 8? of -y-? of y^^? 4. Write the duplicate ratio of 2 : 3 ; of 4 : 5. 6. Write the triplicate ratio of 1 : 2 ; of 3 : 4. Reduce to lowest terms the ratio expressed by : 6. A. 9 16* 9. |f. 9a; 12. 12 abc 30 a'bc 15. x + y Sx + 3y 12 18' 10. '^''. 10 b' 13. 25 xY 16. (a + by 18 24* 11. ^^^. 12a;2 14. IS a^bhl 48 a¥d' 17. 2a;2 + 2?/2 oc^ — y^ 8. 18. Two numbers are in the ratio of 4 : 5. If 9 is subtracted from each, what is the ratio of the remainders ? 19. When the ratio is ^ and the consequent is 10 ab, what is the antecedent? Find the value of each of the following ratios : 20. H:4i. 23. Mb:. 6 c. 26. (x^ - 4:) : (a^- S). 21. I ax :^ ay. 24. -.4 a;2 : 10 a;^. 27. (a^ + ¥) : (a'^ + b'^). 22. Ibc'^'.^b'^c. 25. ^xY'.^xy. 28. (a^-l) : (a^+a + l). 29. Reduce the ratios a : b and x : y to ratios having the same consequent. 30. In an alloy of 78 ounces of silver and copper there are 6 ounces of silver. Find the ratio of silver to copper. 31. In a mixed mass of brass and iron weighing 57 pounds, there are 15 pounds of iron. Find the ratio of iron to brass. 32. Given the ratio | and a positive number x. Prove that 2 -\- X 2 ^ > - by subtracting one ratio from the other. 3 -f- ^ 3 Suggestion. — Proceed as in the proof of Prin. 4, § 118. 90 RATIO AND PROPORTION PROPORTION 120. Define and illustrate : 1. Proportion. 4. Mean proportional. 2. Extremes of a proportion. 5. Third proportional. 3. Means of a proportion. 6. Fourth proportional. 121. Since a proportion is an equality of ratios each of which may be expressed as a fraction, a proportion may be expressed as an equation each member of which is a fraction. Hence, it follows that : General Principle. — The changes that may he made in a proportion without destroying the equality of its ratios correspond to the changes that may he made in the memhers of an equation ivithout destroying their equality and in the terms of a fraction without altering the value of the fraction. Properties of Proportions 122. Principles. — 1. In any proportion, the product of the extremes is equal to the product of the means. Thus, if - = -, then, ad = be. b d It follows that a mean 2^'i^oportional between two numbers is equal to the square root of their pi^oduct. 2. Either extreme of a proportion is equal to the product of the means divided by the other extreme. Either mean is equal to the product of the extremes divided by the other mean. Thus, if ^ = -^ then, a= ^^ d = ^, b = ^, and c = ^. b d d a c b 3. If the product of two numbers is equal to the product of two other numbers, one pair of them may be made the extremes and the other pair the means of a proportion. Thus, if ad = be, then, - = -, or a : b = c : d. b d RATIO AND PROPORTION 91 4. If four numbers are in proportion^ they are in proportion by alternation. Thus, if - = -, then, - = -, or a :c = h :d, b d c d 5. If four numbers are in proportion, they are in proportion by inversion. i Thus, if ^ = -, then, ^ =^, or b : a = d : c. b d a c 6. If four numbers are in proportion, they are in proportion by composition. Thus, if « = ^, then, a_±^ = c±d, ^^^^^ a_±_& ^ c±d^ b d b d a c 7. If four quantities are in proportion, they are in proportion by division. Thus, if ^ = ^, then, «^ = ^-^^; also, ^n^ =c_-^. b d b d a c 8. If four numbers are in proportion, they are in proportion by composition and division. Thus, if ^ = ^, then, ^±-^ = t±A^ ova + b-.a^b^c-^rdic-d. b d a — b c — d 9. The products of corresponding terms of any number of proportions form a projjortion. Thus, if «=^, ^ = ^,and ? = ^, then, «^=^. b d I n y ^ bly dnw EXERCISES 123. Find the value of x in each of the following proportions : 1. i:a:=i:f 4^ ^^^-12 2. x:x + l = ():2. ^^ ^ 3. x-S:x = SU. 5 3a^^-5a:^2g^ + 3 6 4 6. Find a third proportional to 2 a and 6 a. 7. Find a fourth proportional to 2^, 4, and 8. life. Find a mean proportional between 4 a and 9 a. 9. Prove the truth of each principle given in § 122. 92 RATIO AND PEOPORTION When a:b =c:d, prove that : 10. a^ : 62 = c2 : d\ 13. a:bc = l:d. ^^ a b G d ^A 11 "• 3^3 = 2=2- ^*- ''•■' = ri- 12. Va : Vc = V^> : VcZ. 15. a? : ab = c^ \ cd. i 16 ^^ = :^ 18 ^~^ = ^ 20 ^ ^ + 3 5 ^ 3?> 62-^eZ' ' c-cZ d' ' 2c + 3d 3d* 17 ^'=:M 19 ?A±^ = ^ 21 ^!±Z=?!+_^' 'a c * * 2a + <^ 6' ' a^ - b"^ c^ - d^' 22. ma + n6 : ma — nb = mc + nd:mc — nd, 23. 2a + 3c:2a-3c=86 4-12d:86-12d. 24. a^ + a26 + ab^ +¥ : a^ = c^ + c'd^ cd^ + d^:(^, 25. a + 6 + c4-d:a — 6 + c— d = a4-6 — c— d: a—b — c+d. 26. If a : 6 = c : d, and if x is a third proportional to a and 6, and y a third proportional to b and c, show that the mean proportional between x and y is equal to that between c and d. Problems 124. 1. Divide $ 35 between two men so that their shares shall be in the ratio of 3 to 4. 2. If brass is composed of 2 parts of copper to 1 part of zinc, how much of each substance is required for 75 pounds of brass ? 3. A line a inches long was divided into two parts in the ratio m : n. Find the length of each part. 4. Two partners gained $ 6000 in business one year. Find each one's share, their investments being in the ratio 1 : 4. 5. Two numbers are in the ratio of 3 to 2. If each is in- creased by 4, the sums will be in the ratio of 4 to 3. What are the numbers ? Suggestion. — Represent the numbers by 3 x and 2 x. 6. Divide 25 into two parts such that the greater increased by 1 is to the less decreased by 1 as 4 is to 1. RATIO AND PROPORTION 93 7. Two trains traveled toward each other from two cities 98 miles apart. If their rates of traveling were as 3 is to 4, how many miles did each travel before they met ? 8. A man divided his estate of $50,000 between two heirs in the ratio of 3 to 7. How much did each heir receive? 9. Divide 16 into two parts such that their product is to the sum of their squares as 3 is to 10. Suggestion. -^ Solve the final equation by factoring. 10. The sum of two numbers is 4, and the square of their sum is to the sum of their squares as 8 is to 5. What are the numbers ? 11. A dock is divided into two parts so that the length of the longer is to that of the shorter as 11 is to 6. If its total length is 850 feet, what is the length of each part ? 12. The freight earnings of two railroads on a trainload of grain were $ 2160. One carried the grain 400 miles, the other 500 miles. Find the earnings apportioned to each road. 13. Find a number that subtracted from each of the numbers 7, 9, 10, and 14 will give four numbers in proportion. 14. What number must be added to each of the numbers 11, 17, 2, and 5 so that the sums shall be in proportion when taken in the order given? 15. If 16 men can do a piece of work in 15 days, how long will it take 20 men to do it? 16. The total receipts of a coal mining company one year were $16,725,000, and the expenses were to the net earnings as 13 is to 2. What were the expenses ? the net earnings ? 17. Prove that no four consecutive integers, as n, n + 1, 71 + 2, and n + 3, can form a proportion. 18. Prove that the ratio of an odd number to an even num- ber, as 2 m + 1 : 2 7^, cannot be equal to the ratio of another even number to another odd number, as 2 a; : 2 ?/ -f- 1. 94 ' RATIO AND PROPORTION 19. The areas of two circles are proportional to the squares of their radii. If the area of a circle is 5 square inches, what is the area of a circle whose radius is twice the radius of the first circle? Y d X i> i ^^- ^^^ formula pd = WD ex- presses the physical law that, when a lever just balances, the product of the numerical measures of the power (^9) and its distance (d) from the fulcrum (F) is equal to the product of the numerical measures of the weight (W) and its distance (D) from the fulcrum. Express this law by means of a proportion. 21. Solve the proportion obtained in exercise 20 for TFand find what weight a power of 60 pounds will support by means of a lever, if d = 8 feet and Z> = 3 feet. 22. A pressure of 50 pounds was exerted upon one end of a 5-foot bar to balance a weight of 200 pounds at the other end of the bar. How far was the weight from the fulcrum ? 23. A farmer has a team, one horse of which weighs 1200 pounds and the other 1400 pounds. If draft power is propor- tional to weight, where shall he put the clevis (fulcrum) on his 50-inch double-tree (lever) ? 24. In the figure (right triangle) the altitude h is a mean proportional between the segments a and h of the hypotenuse. Find the length of 6, if ^, = 8 and a = 10. 25. The following is a simple relation for pulleys belted together : The speed (aS), revolutions made per minute, of the driving pulley is to speed (s) of the driven pulley as the diam- eter (d) of the driven pulley is to the diameter (Z>) of the driving pulley. • Write the proportion, using the letters S, s, D, and d, 26. What is the speed of a driving pulley 10 inches in diameter, if the driven pulley is 12 inches in diameter and its speed is 500 revolutions per minute? SIMULTANEOUS SIMPLE EQUATIONS TWO UNKNOWN NUMBERS 125. Define and illustrate : 1. Indeterminate equation. 4. Consistent equations. 2. Dependent equations. 5. Inconsistent equations. 3. Independent equations. 6. Elimination. 126. Principle. — Any single equation involving two or more unknown numbers is indeterminate. Elimination by Addition or Subtraction EXERCISES 127. 1. Solve the equations 3 a; + 2 i/ = 12 and 2x + 3y = V6, Solution Sx-\-2y = 12, (1) 2x + Sy = lS. (2) Multiply (1) by 3, Ox + 6 2/ = 36. (3) Multiply (2) by 2, 4:X + 6y =26. (4) Subtract (4) from (3), 6 x = 10. (5) ,',x = 2. (6) Substitute (6) in (1), 6 + 2 y = 12 ; whence, y = ^. To verify^ substitute 2 for x and 3 for y in each given equation. Rule. — // necessary, multiply or divide the equations by such numbers as will make the coefficients of the quantity to be elimi- nated numerically equal Eliminate by addition if the resulting coefficients have unlike signs, or by subtraction if they have like signs, 95 96 SIMULTANEOUS SIMPLE EQUATIONS Solve by addition or subtraction, and verify results : 5. 6. 7. 8. 1 3 a: + 5 2/ = 11. 2x + 3y = 19, u-h3v = -2. 4:X — Sy = 5, 5x — 6y = 5^, 4 ic + 2/ = 25, 2y-5x = 24c. rSx-y^U, \2x + 2y = 2S. x + 5y = 7, 4ic + 32/ = ll. 9. 10. 11. 12. 13. 14. 15. 7 s-9v = 6, s + 2v = U, Sx-2y = -^, 2 X -^ 5 y = 6, 5x + 2y==16, 3x-5y = -9, f 7 a - 3 6 = 9, |3a-2?> = l. 5u-i-9v=±60, 4.u-4:V = -S. i 7 s + 12 ^ = 12i 7 x-4.y = 81, 5x + Sy = 52. Elimination by Substitution EXERCISES 128. 1. Solve the equations 3 aj + i ?/ = 8 and 5 x — y = 6. Solution. I 3 a: + 1 2/ = 8, [ 5 ic — y = ^. Solve (2) for 2/, 2/ = 5 ic — 6. Substitute the value of y from (3) in (1), Solve (4), x = 2. Substitute (5) in (3), y = 10 - 6 =: 4. (1) (2) (3) (4) (5) Rule. — Find an expression for the value of either of the un- known numbers in one of the equations. Substitute this value for that unknown number in the other equation^ and solve the resulting equation. SIMULTANEOUS SIMPLE EQUATIONS 97 Solve by substitution, and verify results : 2. 3. 6. lx + y = 6, \2x + y = 10. {x-y = -l, \2x + Sy = lS. lSx-4:y = U, \x-'4: = 2y. I 2 s + 4 ^ = 20, \3s-5t = -3. {5x — y = 5, \3x--2y = -^. [ a; + 6 2/ = 15. 8. 25 = 5 a - 6, 28 = 3 a + 2 6. 4 s -f 3 ^ = 3, 5 ^ - 3 s = 34. ^^ I2x^4.y+U, \sx-7y = 23. 11. 12. 13. 7 x — 5y = 15, 3x + 3y=:9. 2x-3y==-7, 4:X — 5y = — 9. 2-x = 4:y, 3 2/ - 10 = 2(2 - x). MISCELLANEOUS EXERCISES 129. Solve and test, eliminating before or after clearing of fractions as may be more advantageous : 1. 2. 3. X 3~ 11- I- 3^ 2l = 7 = 8. X 3~ .y 2' X 3~ h'- f3a; 4 -'i- = 20, 2^ 3y_ 4 = 17. 4. 5. 6. milne's sec. course alg. — 7 a; 2 3 1 = 0, 2 a!-l 3?/- 1 5 ^ 2 3 6 a; 1 -1 3 x + y = 0, a; ^ +3 = 0. -y X 2 -12 _7/ + 32 4 ' y [8 3a; -2y_ 25. 1 5 98 SIMULTANEOUS SIMPLE EQUATIONS Solve, and test each result : fa;-l 7. 4 x-1 8. + 2/ = 3, + iy = 9. 6 3 9. 10. 7 +x 2a; — V o r .2 y + .5 ^ .49 x-,7 1.5 ~ 4.2 ' .5a;-.2 ^41 1.52/-11 1.6 16 8 Solve the following as if - and - were the unknown num- X y bers, and then find the values of x and y : 11. X y 6__2 \x y 13. = 10. 12. i + - = 30, ^ + -=30. 2/ ^ 14. a; 2/ a? 2/ 8 a; 3 ?/ Qx 11 y = 17. Solve the following as if , , etc., were the un- a; — 1 y-^-l known numbers, and then find the values of x and y : fl 3 15. 1 1-^ = 5, x-1 y+1 2 . 3 17. ■! 16. a;— 1 y + 1 5 3 x—1 y—1 2 1 = 12. = 14, y 2-x 5^ 6 y 2-x + 9. 18. a;— 1 y — 1 = 6. a; y + S' 7 ^ 3 a; y + 3" 10. SIMULTANEOUS SIMPLE EQUATIONS 99 Literal Simultaneous Equations 130. In solving literal simultaneous equations, elimination is performed usually by addition or subtraction for each un- known number. EXERCISES 131. Solve for x and y^ and test as on page 77 : 1. ax+by = m, ax — by = n. a^x -\- cy — 2y y — ca; = 1. ax-\-by =^ r, ax -\- cy = s. bx + cy ^ 2, d c cd a 5 — ^ — 1 [b a""2' 6. 7. 8. 9. 10. 2x + ay =^b^ ax 4- 2y = c. ax— dy = b, mx — ny = b, \ ax + by = c, \bx— ay = d, i x + y = ab(a + 6), \^ + l=2ab. [ a b a . b - + -=c, X y X y 11. Given F^Ma, Find the values of F and a when M=15, s = 72, and t = 6. I = a -{- (n — l)d, 12. Given s = '^(a + l). Find the values of a and I when n = 50, f? = 2, and s = 2500 ; the values of d and a when Z = 50, n = 25, and s = 660. 13. Given I = ar''-^ rl — a Find the values of a and I when r = 2, n = 11, and s = 2047. 100 SIMULTANEOUS SIMPLE EQUATIONS THREE OR MORE UNKNOWN NUMBERS 132. Principle. — Every system of independent simultaneous simple equations involving the same number of unknown numbers as there are equations can be solved^ and is satisfied by one and only one set of values of its unknown numbers. EXERCISES {2x-^y -z=2, (1) 133. 1. Solve the equations 3 a? + ?/ — 2 2; = 8, (2) [a; -2 2/ + 3:3 = 4. (3) Suggestion. — Eliminate z from (1) and (2) by subtraction and from (1) and (3) by addition ; then solve the resulting equations. Rule. — Eliminate one unknown number from any convenient pair of equations, and the same number from a different pair. Solve the resulting equations. Solve, and test all results : (x-{-y-\-z = lS, 2. Ix — y + z^Q, [x-^y — z = 4:. (x — 2y + 2z=:6, 3. \2x — y -\-z = 7, [x-\-2y + 2z = 21. {v-\-x — y=z2y 4. \v — x-\-y = 4:, [x— V + y ■=^. {x + y = % 5. iy + z = l, [z + x=i b, 4aj — 52/ + 32; = 14, 6. a; + 7?/ -2; = 13, 2a; + 5^/ + 52; = 36. fa; + 32/ + 2 = 14, 7. a: + 2/ + 3 ;3 = 16, [^x + y + z = 20. Suggestion. — In exercise 4, subtract each equation from the sum of the equations. 8. 9. v + a; + ?/ = 15, x^y + z = lS, y + z + v = 17, , z -\- V -\- X = 16. Suggestion. — In exercise 8, subtract each equation from J of the sum of the equations. y -\-z + v — x=:22, z + v -\-x— y = 1S, v-{-X'\-y--z = 14, X -\- y -]- z — V = 10. SIMULTANEOUS SIMPL^F; L'QUATiOMS^ Wl 10. 11 12. Solve for x, y, z^ and v : f axy — X — y = 0, i hzx — z — X = 0, [ cyz — y — z =0. lx-\-y--z = 0, \x-y=2b, \^x-{- z = S a-{-b. V -\-x = 2 a, x-\-y = 2a — Zy y + z = a + b, v — z=a + c. 13. I 14 . I 15. { ahxyz -|- cxy — ayz — hzx = 0, hcxyz -\- ayz— hzx— cxy = 0, caxyz + hzx — cxy — ayz = 0. a? 4- 2 .V + 3 2; = 6 + 2 c, x-\-?>y-\-^z — h-{-'6c, v-\-x-\-y^a-\-2h-\'Cy y -\-z^v = a-\-h, Problems 134. To solve a problem by means of a statement involving two or more unknown numbers, there must he as many given co7iditions and as viany equations as there are unknown numbers. Solve and verify the following problems. Find two numbers related to each other as follows : 1. Sum = 14 ; difference = 8. 2. Sum of 2 times the first and 3 times the second = 34 ; sum of 2 times the first and 5 times the second = 50. 3. Sum = 18 ; sum of the first and 2 times the second = 20. 4. The difference between two numbers is 4 and \ of their sum is 9. Find the numbers. 5. New York once owned 186 parks. Of these the number that had an area of less than one acre was 28 less than the number of the larger ones. Find the number of small parks. 6. In Dawson, Alaska, recently, 2 tons of coal and 3 cords of wood cost together $ 68. If 3 tons of coal cost the same as 4 cords of wood, what was the cost of a ton of coal ? of a cord of wood ? , 1^2 SIMULTANEOUS SIMPLE EQUATIONS 7. The sum of 3 numbers is 162. The quotient of the sec- ond divided by the first is 2 ; of the third divided by the first is 3. Find the numbers. 8. A merchant has 100 bills valued at $275. Some are 2-dollar bills and the rest 5-dollar bills. How many bills of each kind has he ? 9. A paymaster has 110 coins valued at $40. Some are quarters and the remainder half dollars. How many coins has he of each ? 10. In a plum orchard of 133 trees, the number of Lombard trees is 7 more than -| of the number of Gage trees. Find the number of each kind. 11. If 5 pounds of sugar and 8 pounds of coffee cost $ 2.70, and at the same price 9 pounds of sugar and 12 pounds of coffee cost $ 4.14, how much does each cost per pound ? 12. A lieutenant of the U.S. navy, receiving $ 1620 yearly, earned $ 150 a month while on sea duty and $ 127.50 a month while on shore duty. How many months was he on land? 13. A farmer bought 80 acres of land for $ 4500. If part of it cost $ 60 per acre and the remainder i as much per acre, how many acres did he buy at each price ? 14. If 8 baskets and 4 crates together hold 8 bushels of tomatoes, and 6 baskets and 8 crates together hold 9| bushels, what is the capacity of a basket ? of a crate ? 15. If 2 is added to the numerator of a certain fraction, the value of the fraction becomes f ; if 1 is subtracted from the denominator, the value becomes i. What is the fraction ? Suggestion. — Let - = the fraction. y 16. The sum of two fractions whose numerators are 3, is 3 times the smaller ; 3 times the smaller subtracted from twice the larger gives |. What are the fractions ? SIMULTANEOUS SIMPLE E'QUATia]SS^ im 17. The sum of the digits in a number of two figures is 9 and their difference is 3. Find the number. (Two answers.) 18. The sum of the digits of a two-digit number is 5. If the number is multiplied by 3, and 1 is taken from the result, the digits are reversed. Find the number. Suggestion. — The sum of x tens and y units is (lOx 4- y) units ; of y tens and x units, (10?/ + x) units. 19. The sum of the two digits of a certain number is 12, and the number is 2 less than 11 times its tens' digit. What is the number? 20. If a certain number of two digits is divided by their sum, the quotient is 8 ; if 3 times the units' digit is taken from the tens' digit, the result is 1. Find the number. 21. Separate 800 into three parts, such that the sum of the first, i of the second, and ^ of the third is 400 ; and the sum of the second, | of the first, and \ of the third is 400. 22. A certain number is expressed by three digits whose sum is 14. If 693 is added to the number, the digits will appear in reverse order. If the units' digit is equal to the tens' digit increased by 6, what is the number? 23. If 10 pounds of chicken feathers and 6 pounds of duck feathers cost $ 2.43, and 16 pounds of the former and 5 pounds of the latter cost $2.37, what is the cost per pound of each kind of feathers ? 24. A 5-dollar gold piece weighs i as much as a 10-dollar gold piece. If the combined weight of 3 of the former and 2 of the latter is 903 Troy grains, what is the weight of each? 25. If Rio coffee costs 20^ per pound and Java coffee, 32^ per pound, how many pounds of each must be bought to fill a 120-pound canister making a blend worth 28 j^ per pound ? 26. If a bushel of corn is worth r cents, and a bushel of wheat is worth s cents, how many bushels of each must be mixed to make a bushels worth b cents per bushel ? 1.04 tSIMULTANEOUS SIMPLE EQUATIONS 27. If a rectangular floor were 2 feet wider and 5 feet longer, its area would be 140 square feet greater. If it were 7 feet wider and 10 feet longer, its area would be 390 square feet greater. What are its dimensions ? 28. The cost of cooking meat for 1 hour averages 2.128^ less by gas than by electricity. If meat can be cooked 4 hours by the former means for .256 j^ less than it can be cooked 2 hours by the latter, what is the cost of each per hour? 29. To burn weeds along a railroad by a gasoline burner costs $16.66 less per mile than to cut them by hand. It costs as much to clear 160 miles by the former method as 41 miles by the latter. Find the cost per mile by each method. 30. Single yarn of imitation silk is put up in three quali- ties, A, B, and C. 5 pounds of A and 2 pounds of B cost $ 8.64 ; 3 pounds of B and 1 pound of C cost $ 5.40 ; 2 pounds of A and 3 pounds of C cost $ 6.72. Find the cost per pound of each quality. 31. The winning baseball team of the National League one year won 44 games more than it lost. If the number won had been 8 less and the number lost 8 more, the ratio of the former to the latter would have been 13 : 9. Find the number of games won ; the number of games lost. 32. A boatman trying to row up a river drifted back at the rate of 2 miles an hour, but he could row down the river at the rate of 12|- miles an hour. Find the rate of the current. 33. A takes 3 hours longer than B to walk 30 miles, but if A doubles his pace, he takes 2 hours less than B. Find A's rate ; B's rate. 34. A and B can do a piece of work in 10 days ; A and C can do it in 8 days ; and B and C can do it in 12 days. How long will it take each to do it alone ? 35. A and B can do a piece of work in r days ; A and C can do it in s days ; and B and C can do it in ^ days. How long will it take each to do it alone ? SIMULTANEOUS SIMPLE EQUATIONS 105 36. When weighed in water silver loses .095 of its weight and gold .051 of its weight. If an alloy of gold and silver weighing 12 ounces loses .788 of an ounce when weighed in water, how many ounces of each are there in the piece ? 37. When weighed in water tin loses .137 of its weight and copper .112 of its weight. If an alloy of tin and copper weighing 18 pounds loses 2.316 pounds when weighed in water, how many pounds of each are there in the piece ? 38. When weighed in water tin loses .137 of its weight and lead loses .089 of its weight. If an alloy of tin and lead weighing 14 pounds loses 1.594 pounds when weighed in water, how many pounds of each are there in the piece? 39. Two pumps are discharging water into a tank. If the first works 5 minutes and the second 3 minutes, they will pump 2260 gallons of water ; if the first works 4 minutes and the second 7 minutes, they will pump 3280 gallons. Find their capacity per minute. 40. A and B together can do a piece of work in 12 days. After A has worked alone for 5 days, B finishes the work in 26 days. In what time can each alone do the work ? 41. If 4 boys and 6 men can do a piece of work in 30 days, and 5 boys and 5 men can do the same work in 32 days, how long will it take 12 men to do the work ? 42. A and B can do a piece of work in a days, or if A works m days alone, B can finish the work by working n days. In how many days can each do the work ? 43. A and B can do a piece of work in a days ; A works alone m days, when A and B finish it in n days. In how many days can each do it alone ? 44. A can build a wall in c days, and B can build it in d days. How many days must each work so that, after A has done a part of the work, B can take his place and finish the wall in a days from the time A began ? 106 SIMULTANEOUS SIMPLE EQUATIONS 45. At simple interest a sum of money amounted to $ 2472 in 9 months and to $ 2528 in 16 months. Find the amount of money at interest and the rate. 46. Mr. Shaw invested $ 8025, a part at 3|- % and the rest at 4 % . If the annual income from both investments was $309, what was the amount of each investment? 47. A man invested a dollars, a part at r per cent and the rest at s per cent yearly. If the annual income from both investments was b dollars, what was the amount of each investment ? 48. A sum of money at simple interest amounted to b dollars in t years, and to a dollars in s years. What was the princi- pal, and what was the rate of interest ? 49. A certain number of people charter an excursion boat, agreeing to share the expense equally. If each pays a cents, there will be b cents lacking from the necessary amount ; and if each pays c cents, d cents too much will be collected. How many persons are there, and how much should each pay ? 60. A mine is emptied of water by two pumps which to- gether discharge m gallons per hour. Both pumps can do the work in b hours, or the larger can do it in a hours. How many gallons per hour does each pump discharge ? What is the discharge of each per hour when a =5, 6=4, and m=1250? 51. Two trains are scheduled to leave A and B, m miles apart, at the same time, and to meet in b hours. If the train that leaves B is a hours late and runs at its customary rate, it will meet the first train in c hours. What is the rate of each train ? What is the rate of each, if m = 800, c = 9, a = lf, and 6 = 10? 52. A man ordered a certain amount of cement and received it in c barrels and d bags ; a barrels and b bags made — of the n total weight. How many barrels or how many bags alone would have been needed ? Find the number of each, if c = 16, d = 15, a = 6, 6 = 15, m = 1, and n = 2. GRAPHIC SOLUTIONS LINEAR FUNCTIONS iT^Ki 135. An expression involving one or more letters is callei^ function of those letters. Thus, 3 a; — 2 is a function of x ; also x + y is a function of x and y. Again, the area of a rectangle is a function of its base and altitude, A = bh; percentage is a function of the base and rate, p = br. 136. The symbol for any given function of x is f(x), read " function of xJ^ Other functions of x in the same discussion may be represented, if desired, by F(x),f'(x), etc., read "large F function of a?,'' "/-prime function of x,^^ etc. Values of f(x) corresponding to particular values of x, as 1, 2, 0, etc., are usually indicated by /(I), /(2), /(O), etc., respectively. Thus, if /(x)=6x + 9, /(I) =6 + 9 = 15; /(2) = 12 4- 9 = 21 ; /(O) = 9. 137. A quantity whose value changes in the same discussion is called a variable ; a quantity whose value remains the same is a constant. Thus, in the formula for the volume of a sphere, F = f Trr^, the volume changes for changing values of r ; then V and r are variables, but tt, whose value remains the same whatever the value of r, is a constant. EXERCISES 138. 1. Evaluate/(a?)=2a;-7 for a;=l; fora;=3; for a;=0. 2. When f(x) = 2ix - 1), find /(I) ; /(2) ; /(5) ; /(8). 3. When f(x) = |(3 - x), find /(O) ; /(3) ; /(6) ; /(12). 4. When F{x) = 1(5 - x), find F{A) ; i^(l) ; F(0) ; F{7). 5. When f{y) = 3(2 - y), find /(O) ; /(3) ; /(15) ; /(20). 6. Evaluate f(u)=^^(u + 8) for ?^ = 1 ; for w = 7 ; f or t6 = 16. I When /'(a.) = .7(^ + 1.5), find/'(0) ; /'(2.5) ; /(|) ; /'(f). 108 GRAPHIC SOLUTIONS 139. Graphical representation. — When related varying quanti- ties in a series are to be compared, it is often convenient and very effective to represent them by a diagram, or graph. The following graph represents the height of water in a cer- tain river above of the gauge from daily observations during the month of September. 11 9 8 t / 's s & / S fs^ 1 a / \. ^ ^ ^ 6 ^ -^ ^ y "v /^ ^ ^ / © ■^ 8 W 1 Dt ,ys of th 5 Monjth { ) ] 5 ^ L ) r J J 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 The horizontal distances represent time in days and the ver- tical distances, the height of water in feet. Thus, on the 19th day of the month the height of the water is repre- sented by the vertical line drawn upward from 19 and is 5 feet. In fact, every point of the irregular black line, or graph, exhibits a pair of corre- sponding values of the two related quantities — days and height of water. From the graph answer the following : 1. How high was the water on Sept. 1 ? on Sept. 23? 2. On what day of the month was the water highest ? lowest ? 3. What was the maximum height ? the minimum height? the range between them ? 4. What part of the month shows the most rapid changes ? 5. Give the time of the greatest change in a single day. Graphs have very many uses. The statistician uses them to present in- formation in a telling way. The broker and the merchant use them to compare the rise and fall of prices. The physician uses them to record the progress of diseases. The engineer uses them in testing materials and in computing. The scientist uses them in his investigations of the laws of nature. In short, graphs may be used whenever two related quantities are to be compared throughout a series of values. The use of paper ruled in small squares, called squared paper or coor- dinate paper, is advised in plotting graphs. GRAPHIC SOLUTIONS 109 90 140. The two graphs given on this page present to the eye the comparative weights of two standard types (" slender " and " heavy ") of boys between the ages of 9 years and 15 years. The scales to which these graphs are constructed are, for vertical distances, 1 space represents 2 pounds, and for horizontal distances, 2 spaces represent 1 year. The vertical spaces for pounds to 49 pounds, and the horizon- tal spaces for years to 7 years are omitted. Graphs may be constructed to any convenient scale and, if desired, the horizontal scale may differ from the vertical scale. 1. From the graph read the standard weight of the slender type of boy at age 9 ; at age 10 ; at age 11 ; at age 12 ; at age 13 ; at age 14 ; at age 15. 2. Eead the standard weight of the heavy type of boy for each age from 9 to 15 inclusive. 3. During what year does each type increase in weight most rapidly ? least rapidly ? 4. What is the difference in weight of the two types at age 9? at age 10? at age 11? at 12? at 13? at 14? at 15? 5. At what age is the difference in weight greatest ? least ? 6. What is the weight of the slender type at 9|- years ? at 12t} years ? at 14|- years ? of the heavy type at 13^- years ? 7. What is the approximate age of the slender type of boy when he weighs 72 pounds ? 97 pounds ? of the heavy type when he weighs 90 pounds ? 98 pounds ? 70 60 " \ ^ — at ^1 y f / / } 1 ) m / u C / '^ J o '^'/ <5- / / is: 1 / 1 j j 1 I P / / / / r j y / / f^ ^ \ 4- ^« in yl iai|-8 10 11 12 13 14 15 no GRAPHIC SOLUTIONS /(x)=2x-3, or y =2x— 3. 141. Let/(a;)= 2 a? — 3. It is evident that we may give x a series of values, obtaining a corresponding series of values of f(x), and that the number of pairs of values of x and f(x) is unlimited. All these values of x and f(x) may be represented by a graph, just as in the preceding illustrations the corresponding values of two vari- ables were represented by a graph. The line AB is the graph of the function 2 a; — 3 or of the corre- sponding equation y = 2 x — 3. Values of x are represented by lines laid off on or parallel to an jr-axis, X'X, and values of f(x) by lines laid off on or parallel to a y-axis, F' Y (usually drawn perpendicular to the a;-axis), the function of x being denoted by y. For example, the position of P shows that when x = S, y = 3 ; the position of Q shows that when x = 4:, y = 5 ; the position of M shows that when x = 5, y = 7 ] etc. Evidently every point of the graph gives a pair of corresponding values of x and /(a;), or 2/. 142. Conversely, to locate any point with reference to two axes for the purpose of representing a pair of corresponding values of x and y, the value of x may be laid off on the i»-axis as an oc-distance, or abscissa, and that of y on the i/-axis as a y-distance, or ordinate. If from each of the points on the axes thus obtained, a line parallel to the other axis is drawn, the intersection of these two lines locates the point. Thus, to represent the corresponding values x = 3, 2/ = 3, a point P may be located by measuring 3 units from to M on the z-axis and 3 units from O to JVon the ?/-axis, and then drawing a line from M parallel to OF, and one from iV parallel to. OX, producing these lines until they intersect. 143. The abscissa and ordinate of a point referred to two perpendicular axes are called its rectangular coordinates. GRAPHIC SOLUTIONS 111 Plotting Points and Constructing Graphs 144. By custom positive values of x are laid off from the zero-point, or origin, toward the right, and negative values toward the left. Also positive values of y are laid off upward and negative values clowmvard. The point A in the figure may be designated as ^ the point (2, 3),^ or by the equation A =(2, 3). Similarly, B = (-2, 4), (7 = (_3, -1), andi>=(l, -2). The abscissa is always written first. Y B^^ '^A x' X 3-2- .0 1 2 3 4 c^^ -i -2 -3 '^D t\ EXERCISES 145. Draw two axes at right angles and locate these points ; 1. ^ = (2,4). 5. (0, -4). 2. B=(-3,2). 6. (-2,0). 3. C = (l, -2). 7. (10,8). 4. X»=(_l, -1). 8. (-5,11). 13. Construct the graph of the equation 2y — x = 2. Solution. — Solving for y, we have y = l(x-h2), in which we substitute values for x and determine corresponding values of y as tabulated below. The points whose coordinates are given in the table are then plotted. 9. (-4,-6). 10. (12, - 9). 11. (-6,12). 12. (-7, -8). Y ■^ X ?/ Point ' -6 -4 -2 2 4 6 -2 -1 1 2 3 4 A B C 1) E F G r^ ^3 ^ > r >i^ y^ ^ r ^ Xl E x' 1 ^ ^ S X ^ ^ r y- ^ B ^ r y' A line drawn through A^ B, C\ i>, etc., is the graph ot2y — x = 2. 112 GRAPHIC SOLUTIONS Construct the graph of : 14. f{x)=^x-^. 16. /(a;)=2i»-f-3. 18. 2y = x. 15. f(x)=^2 — x, 17. y = 2 — ^x. 19. x + 2y — — ^. 146. It is now evident that the graph of a simple equation in two unknown numbers is a straight line. For this reason a simple equation is sometimes called a linear equation, and the corresponding function, a linear function. 147. Since a straight line is determined by two points, to plot the graph of a linear equation, plot two points and draw a straight line through them. To find where the graph intersects the aj-axis, let 2/ = ; to find where it intersects the iz-axis, let a; = 0. Thus, in 1/ = ^(x + 2), page 111, when 2/ = 0, a;=— 2, locating O; when X = 0, 2/ = 1, locating D. Draw a straight line through C and D. If the points plotted as just illustrated are near together, for the sake of accuracy plot points farther apart. In any case check the work by plot- ting a third point and determining whether it lies on the graph. EXERCISES 148. Construct the graph of : 1. y=zx-l. 6. 2 a; -5 2/ = 10. 11. 5aj-2/ = 2i. 2. y — 2x=z2, 7. 4a^ + 32/ = 12. 12. 2x — ^y=-2. 3. 3 2/ + i» = 6. 8. 6 + 3a; =2 2/. 13. |ic-|2/ = 3. 4. 3 a; — ?/ = 9. 9. 3a7-f6 2/=0. 14. .2 a; + .5 2/ = 1. 5. x+2y = -S. 10. 2x-y—4t = 0. 15. .4cc +.6^= - .8. Graphic Solutions of Simultaneous Linear Equations 149. Let it be required to solve graphically the equations 'y = 2+x, (1) y = ^-x. (2) Using the same axes, we construct the graph of each equa- tion as shown on the next page. GRAPHIC SOLUTIONS 113 We desire to discover for what values of x and y both equa- tions are satisfied. When a; = — 1, ?/ = ^B = 1 in (1) and AC=1 in (2). Similarly, the values of y in the two equa- tions differ for every value of X except i» = 2 ; when a; = 2, y = MP = 4 in both equations. The required values of x and y, then, are represented graphically by the coordinates of P, the intersection of the graphs. "^ —" N / \ n>^ y \ / >y / ^ \ R / \ ?. / \ / A M M / \ / 150. Let the given equations be ~ N \ \ ^ ')S f^ N> V X^. O/ >^ ''s 'V K f V \ \ \ J x + y = 7, 2x + 2y = U. (1) (2) If we try to eliminate either x or ?/, we hnd that (1) and (2) are just alike. Since y = 7 — x in both (1) and (2), the values of y are the same for each value of x. The graphic analysis, like the al- gebraic analysis, shows that the equations are indeterminate, for their graphs coincide, 151. Let the given equations be 6-x, (1) X. (2) N \ \ \ \ \ \ \ /'r \ ^ L"' \^ % \ \ C^i \ \ \ \ \ \ \ \ For every value of x the values of y in (1) and (2) differ by 2, and the graphs are 2 units apart ver- tically. In algebraic language, the equa- tions cannot be simultaneous, that is, they are inconsistent. In graphical language, their graphs cannot intersect, being parallel straight lines. milne's sec. course alg. — 8 114 GRAPHIC SOLUTIONS 152. Principles. — 1, A single linear equation involving two unknown numbers is indeterminate, 2. Two linear equations involving tivo unknown numbers are determinate, provided the equations are independent and simul- taneous. TJiey are satisfied by one, and only one, pair of common values. 3. The pair of common values is represented graphically by the coordinates of the intersection of their graphs. EXERCISES 153. 1. Solve graphically the equations 2 a; 4-31/ = 12. Y y S^ ^^ L ^v ,^^ s ^^ S p^ ~Sh^^^ ^ A ^^ ^ ^^ A ^^ X J^ ^^ y ^ Y' Solution. — On plotting the graphs of both equations, as in §§ 145-148, it is found that they in- tersect at a point P, whose coordi- nates are 1.8 and 2.8, approximately. Hence, x — 1.8 and y = 2.8. The coordinates of P are esti- mated to the nearest tenth. Note. — In solving simultaneous equations by the graphic method the same axes must be used for the graphs of both equations. Construct the graphs of each system of equations. Solve, if possible. If there is no solution, tell why. x + y==4,, ^ I3x + 2y=:7, x-y = 2. x + y = 2, y — x= 6, ic + 22/ = 4, 2x-y=3. 'x = y^2, a; = 2/ — 3. 2. 6. 4. 5. 8. 9. 2y-x = S. l2x=S-\-y, 2 2/ = 4 aj — 6. Sy + 2x = 4., 3x + 2y = l. 2y = 3x, aj -f 4 y = 14. GRAPHIC SOLUTIONS 115 Solve graphically as instructed on page 114 : 10. 11. 12. 13. 14. 15. U7j = 10-2x, I 4a; + 32/ = 14, \2x-y = 0. I a; + ^ 2/ = 3. 2x-h3y = 6, Sy = 3-x. f 4 1/ — » = 4, [^x = 2y-2. ly = 3(x-l), 18 = 3(2/ + 2x). 16. 17. 18. 19. 20. 21. Sx = 4:y, 3x-4:y = 9. a; - 2 2/ = 4, 2y + 6x = S. 2y = S(x-2\ 9a; = 6(1/ + 3). 2a; + 42/=-8, X — 3y = — 4:, 2x + 3y = % 62/ + 4a; = 18. f3a; + 22/ = 12, 2y-x=12. 22. During a certain month (July 1-31) one year the aver- age daily maximum temperature for ten cities in the United States was as follows: 80°; 80°; 82°; 82°; 78°; 80°; 81°; 84°; 84°; 84°; 86°; 86°; 85°; 86°; 90°; 89°; 91°; 89°; 87°; 86°; 83°; 82°; 80°; 81°; 82°; 82°; 82°; 82°; 85°; 85°; 84°. Draw the graph with each horizontal space representing 1 day, and each vertical space 1 degree of temperature. 23. On I^ovember 1 of each year from 1909-1913, the whole- sale price of wheat per bushel was as follows: 1909, $1.23^; 1910, $.96 ; 1911, $.991 ; 1912, $ 1.06 ; 1913, $.98. Draw a graph showing the comparative prices for the five years, letting 4 horizontal spaces represent 1 year and each vertical space 2 j^. 24. The cotton crop of Texas given in million bales for years 1907-1913 was as follows: 1907, 4.07; 1908, 2.31; 1909, 3.91 ; 1910, 2.65 ; 1911, 3.14 ; 1912, 4.27 ; 1913, 4.88. Draw a graph showing the variation in the crop for these years, with 2 horizontal spaces representing 1 year and each vertical space ^ of a million bales of cotton. 116 GRAPHIC SOLUTIONS 25. The charge for sending parcels of merchandise weighing from 1 pound to 50 pounds not more than 50 miles by mail is 5 ^ for 1 pound and 1 ^ for each additional pound. Draw a graph showing the charges on parcels weighing from 1 pound to 50 pounds, letting each horizontal space rep- resent 2 pounds and each vertical space 2 ^. 26. Letting two horizontal spaces represent 1 year and each vertical space 1 inch, construct two graphs showing the com- parative heights of two standard types (" slender '^ and " heavy ") of boys between the ages of 9 years and 15 years : Age 9 10 11 12 13 14 15 Slender 53 64i 56 58.i 60 63 m Heavy 61 b^ 54i 56| 58f 60^ m\ 27. Draw two graphs showing the comparative chest girths in inches of boys of the two types mentioned in exercise 26 : Age 9 10 11 12 13 14 15 Slender 23 23^ 24^ 26 26^ 27 27| Heavy 25 26 26J 28 29" 30 32 28. Draw a graph showing the amount of interest at 6 % due on $ 1000 for different periods of time between 1 month and 2 years. (Use a convenient scale.) 29. At 7.00 A.M., Mr. Cox started for a town 18 miles dis- tant, walking at the rate of 4 miles per hour. After walking for 2 hours he rested for a half hour. Draw a graph show- ing at what time he reached his destination. 30. Train ISTo. 1 started from A at 9.00 a.m., traveling toward C, 120 miles away, at the rate of 50 miles per hour. At B, a station halfway between A and C, the train was detained 18 minutes, but it made no other stop. At 10.00 a.m., train Ko. 2 started from C, traveling toward A at the rate of b^ miles per hour and making no stops. Using any convenient scale, draw a graph showing where the trains met. INVOLUTION AND EVOLUTION 154. Define power ; involution ; root ; evolution. INVOLUTION 155. The following laws for involution, which, are the direct consequences of the corresponding laws for multiplication, are applicable in finding powers of monomial expressions : Law of signs. — All powers of a positive number are positive; even powers of a negative number are positive, and odd powers are negative. Thus, 22 = 4 ; (- 2)2 = 4; and (- 2)3 =- 8. Law of exponents. — The exponent of a power of a number is equal to the exponent of the number multiplied by the exponent of the power to ivhich the number is to be raised. Thus, (22)3 ^ 22x3 ^ 26 = 64 ; also (32)2 ^ 32x2 ^ 34 ^ gl. Distributive law. — Any power of a product is equal to the product of its factors each raised to that power. Any jioiver of the quotient of two numbers is equal to the quo- tient of the numbers each raised to that power. Thus, (2.3)2=:(2.3)(2.3)=2.2.3.3=22.32; and (f)2 = f . f = ^ . 32 EXERCISES 156. Raise to the power indicated : 1. {x^yzy. 6. (abcy. 11. (-ir- 2. {aW&)\ 7. (2 xyy. 12. (_ 62)2n+l^ 3. (- 3 ah)\ 8. {-2l'm'dy. 13. (-a'-fpz^y. 4. (2axh/y. 9. (_a2a;"i/«-i)2. 14. (_a"-l^n-2^)3. 5. (-Ga'^a^y. 10. (-x^fz''~y. 117 15. l-^ct-byj. 118 INVOLUTION AND EVOLUTION Raise to the power indicated ; 16. fl^Y. 19. f-^X. 22. f^^Y ■2^Y. 20. r-2^Y 17. -7^ • 20. -^^ . 23. , ^3yJ \ x^yj \^af»+" 18. -^Y. 21. (-^^'^. 24. (- Binomial Theorem 157. By actual multiplication, we have Expansion of (a + x)** Binomial Coefficients (a+a;)o = 1 1 la+xy = a-}-x 1 1 (a+x)^ = a^-{-2ax-\-x^ 12 1 la+xy = a^ + Sa^x-hSax^-\-x^ 13 3 1 (a+xy = a^-\-4a^x + QaH^+4ao^+x!^ 14 6 4 1 (a+x)5 = «54-5a*x-f 10a3x2 + 10a2x3 + 5aa:4+x5 1 5 10 10 5 1 The triangular array of coefficients is known as Pascal's triangle. Each number is the sum of the one directly above and the one to the left of that. 158. From the expansions given above, it will be observed that for SLiij positive integral value of n in the expansion of (a+ jr)" : 1. The first term is a"*, the last term is jr", and the number of terms is n-{-l. 2. The exponent of a decreases 1 in each succeeding term; x appears in the second term and its exponent increases 1 in each succeeding term. 3. The coefficient of any term may he found by multiplying the coefficient of the preceding term by the exponent of a in ^ai term, and dividing this product by the number of the term. \^^ I 4. All terms are positive, if both a and x are positive, and alternately positive and. negative, if x is negative. Note that the coefficients of the latter half of the expansion are the same as those of the first half, written in reverse order. ^!)• INVOLUTION AND EVOLUTION 119 EXERCISES 159. Write by inspection the expansion of : 1. {x — a)\ 3. (b + yy, 5. {r + sy. 7. (u -vy. 2. {x-ay. 4. {c-dy, 6. (r + ^y. 8. {m-ny. 9. Expand (3 a - 6^)^ Solution /> (3 a - 62)4 :=t (3 ay -/4(3 a)8(&2) 4. 6(8 a^ib'^y - 4(3 a)(62)3 4-i|;52)4 = 81 a* - 108 0^62 + 54 ^254 __ 12 a¥ + bK Test the result by giving the letters numerical values. Expand, testing the results in exercises 10-15 : 10. (0^ + 2)^ 20. (1-3 2//. 30. 11. (a-sy. 21. {i + xyy. 12. (m-pny. 22. (2/' - 10)^. 31^ (^^-^Y. 13. (ax-hyy. 23. {1-2 by, ^^ ^^ 14. {ax + 2yy, 24. (2a-3c)3. 32. (v-^- 15. (2a+2>c)^ 25. (a2a. + 4y. ^^ (3 _ f^y 16. (a;2-5 2/)^ 26. (1 - 3 a)^ * V^ 17. (3c + d2y. 27. {ia^-hdy, ^^- (3^ + ^^ 18. {2ah-cy. 28. (im + ^ri/. 19. (a;3 ^ 2 2/;2)3. 29. {2x + \yy, 36. Expand (a + 5 - c)^ ^ Suggestion. (a + 6 — c)^ = (a 4- 5 — c)^, a binomial form. Expand this and then expand each binomial factor in the terms of the resulting expansion. 37. Expand (r-'S — t + vy. Suggestion. [r — s — t -\- v)'^ = (r — s — i — v)^, a binomial form. Expand : 38. {a-h^cy, 4L {x + *dy-2zy. 39. {x^y + 2y, 42. (a + & + c + c?)l 40. {h-c-ay. 43. (a-]-h-'X-yy, 35. (--I)' 120 mVOLUTiON AND EVOLUTION 160. The product of the successive integers from 1 to n, or from n to 1, inclusive, is called factorial n, written [n, or n !. |_5 = 1x2x3x4x5, or 5x4x3x2x1; \n = l .2.3...(n-2)(n- 1)^1, ot n{n - l)(n - 2)... 3 . 2 . 1. 161. Finding the rth term of the expansion of (a + xy. From the expansions given in § 157 and the observations in § 158, it is evident that the following powers of (a 4- x) may be written : 2-1 {a -\- xy = a? -^ 2 ax + - — -a;-. 1 • Zi (a -\-xy = a^ -i-S a}x -[-—^ax^ + ' ' a.^. / . x4 4.^ 3 ,4. 3 22, 4. 3. 2, ,4. 3. 2-1, If the law of development revealed in the above is applied to the expansion of any power of any binomial, as the nth power of (a + x), the result is {a-^-xy = a-+na--''x + ^^^'' ~ ^-^ a"-V + ^'^^ "~ ^)^^ ~ ^^ a^-V .... I This is known as the binomial formula. From the binomial formula, it is evident that in any term : 1. Tlie exponent of x is 1 less than the number of the term. 2. The exponent of a is n minus the exponent of x. 3. The number of factors in the numerator and in the denomi- nator of the coefficient is 1 less than the number of the term. Hence, the formula for the rth term of the expansion may be written : n{n-V){n-2)...{n-T + 2) ^,-,+,^.-,_ Any term of the expansion of a power of a binomial may be obtained by substitution in this formula for the rth term. In the expansion of a power of the difference of two numbers {a — x)", since the exponent of x in the rth term is r — 1, the sign of the general term is — if r is even, and -f if r is odd. \ INVOLUTION AND EVOLUTION 121 EXERCISES 162. 1. Find the 12tli term of (a - by\ Solution 12th term ^ M • 13 . 12 . 11 . 10 . 9 . 8 . 7 . 6 . 5 > 4 , 1.2.3.4.5.6.7.8.9.10.11 ^ ^ = - ^^ • ^^ • ^^ a^ftii = - 364 a«6ii. 1.2.3 Or, since there are 15 terms, the coefficient of the 12th term, or the 4th term from the end, is equal to that of the 4th term from the beginning. .-. 12th term = - liliill? a^ftn = - 364 a^b^K 1.2.3 Without actually expanding, find the : 2. 4th term of (a + 2y\ 5. 20th term of (1 + xy\ 3. 8th term of (x - yy\ 6. 18th term of (1 - 2 xf^ 4. 5th term of (x - 2 yy\ 7. 13th term of fo:^ - -Y- 8. Find the middle term of (a + 3 by. / 1^X10 9. Find the sixth term of ( a; + - • 10. Find the middle term of (- - -J- \y X. fa h\^ 11. Find the two middle terms of f -- h a 12. In the expansion of (x^ + xy^^ find the term containing x^^. Solution. — Since (^2 + cc)" = fx^M + i'\l^^== o^^'^/'l + -^y\ every term of the series expanded from M 4- - j will be multiplied by x^^. IW V Hence, the term sought is that which contains - ) , or — ; that is, the \xl x^ 8th term, which is the same as the 5th term. 1.10.9.8 /ly^ 33,^,,^ 1.2.3.4 \x] 13. Find the coefficient of a^ in the expansion of (a? + ay, 14. Which term contains a^^¥ in the expansion of (a — by^? 122 INVOLUTION AND EV^OLUTION EVOLUTION 163. Define and illustrate : 1. Index of a root. 3. Square, cube, and fourth root. 2. Odd and even root. 4. Real and imaginary number. 164. Since evolution is the inverse of involution, in general: TJie nth root of a is a number of which the nth power is a. 165. Every number has two square roots, one positive and the other negative. Thus, \/25 = + 5 or — 5, often written together ± 5 or T 5. 166. Just as every number has two square roots, so every number has three cube roots, four fourth roots, etc. The cube roots of 8, found later, are 2, — 1 + V— 3, and — 1 — V— 3. The present discussion is concerned only with real roots. 167. A real root of a number, if it has the same sign as the number itself, is called a principal root of the number. The principal square root of 26 is 5, not — 5 ; the principal cube root of 8 is 2 ; of - 8 is - 2. 168. These laws for evolution follow from the corresponding laws for involution (§ 155) : Law of signs. — A7i even root of a positive number may have either sign. An odd root of a number has the sa.me sign as the number. Law of exponents. — The exponent of any root of a number equals the exponent of the number divided by the index of the root. For the principal root, v^ = 26-3 ^ 22 = 4. Distributive law. — Any root of a product is equal to the product of that root of each of the factors. Any root of the quotient of two numbers is equal to the root of the dividend divided by the root of the divisor. For principal roots, y/Wd^ = v^ • Vc? = 5 a : and -x/— = -^ = - • INVOLUTION AND EVOLUTION 123 EXERCISES 169. Find the principal roots indicated : 1. -yJafW. 7. Vl44a^. ^^ 5/ - 1024 g^ 2. a/^V^8. 8. ^-21 a^h\ 32 a;^2/'^ 3/ (x^ 3. v'^Wi^. 9. -■V'd^a^^hK ^^' \~^^^^' 4. ^Z^;;?V«. 10. ^^(-x^yy. 15. ^'p^^^"" 5. -\/a^x^y\ 11. -VaiWn^24 16. Square Root of Polynomials EXERCISES 170. 1. Find the process for extracting the square root of a^-\-2ab + b\ PROCESS a'^-}-2ab + b^ \a-{-b a^ Trial divisor, 2 a Complete divisor, 2 a + b 2ab-^¥ 2 a5 + 6^ Explanation. — Since a^ -\- 2 ah -{• IP' is tlie square of (a + 6), we know that the square root oi cfi -\-2 ah + 1P \^ a -{-h. Since the first term of the root is a, it may be found by taking the square root of a^, the first term of the power. On subtracting a^^ there is a remainder of 2 a& + h'^. The second term of the root is known to be 5, and that may be found by dividing the first term of the remainder by twice the part of the root already found. This divisor is called a trial divisor. Since 2 a?) + 6^ jg equal to 6(2 a + 6), the complete divisor which mul- tiplied by h produces the remainder 2 a6 + 52 jg 2 a + 6 ; that is, the com- plete divisor is found by adding the second term of the root to twice the root already found. On multiplying the complete divisor by the second term of the root and subtracting, there is no remainder ; then , a + 6 is the required root. 124 INVOLUTION AND EVOLUTION Since, in squaring a-\-b+c, a + b may be represented by x^ and the square of the number by x^ + 2 xc + c^, the square root of a number whose root consists of more than two terms may be extracted in the same way as in exercise 1, by considering the terms already found as one term. 2. Find the square root of a?^ 4- 4 a.^ — 6 ic^ _ 20 ^^ + 25. PROCESS x' + 4c x^ -6x^-20x + 25\x^ + 2x-5 x' 2aj2 4.0^- 6x'' 2x''-\-2x 4:X^-{- 4a;2 2 a;2 + 4 a; -10x'-20x-j'25 2x^-{-4.x -5 -10x''-20x-\-25 Explanation. — Proceeding as in exercise 1, we find that the first two terms of the root are x^ + 2x. Considering x"^ -f 2 cc as the first term of the root, we find the next term of the root as we found the second term, by dividing the remainder by twice the part of the root already found. Hence, the trial divisor is 2x2 + 4 X, and the next term of the root is —5. Annexing this, as be- fore, to the trial divisor already found, we find that the complete divisor is 2x2-f 4x — 6. Multiplying this by — 6 and subtracting the product from — 10 x2 — 20 X 4- 25, we have no remainder. . Hence, the square root of the number is x^ + 2 x — 6. EuLE. — Arrange the terms of the polynomial with reference to the consecutive 'powers of some letter. Extract the square root of the first term, write the result as the first term of the root, and subtract its square from the given polynomial. Divide the first term of the remainder by tiuice the root already found, as a trial divisor, and the quotient will be the next term of the root. Write this result in the root, and annex it to the trial divisor to form a complete divisor. Multiply the complete divisor by this term of the root, and sub- tract the product from the first remainder. Continue in this manner until all the terms of the root are found. INVOLUTION AND EVOLUTION 125 Extract the square root of : 3. 16 a;^ + ^4 a;2 -I- 9. 6. 4:X^ + 2x7j + ^y^ 4. l + 50a3 + 625a^ 7. ^ d' - ^ d'n^ -]- ^ 7i\ 5. 9 if + 60 yz -{- 100 z''. 8. (a + 6)2 - 4 (a + 2>) -f 4. 9. 16-\-16x-20x''-12a^ + 9x\ 10. a« + 12 a^&4 - 16 a''¥ - 4 a^^^ _^ ^g 58^ 11. a^-2a26 + 2a2c2-26c2 + &' + c^ 12. 4 a2 _ 12 a& + 16 ac + 9 62 + 16 c^ - 24 be. 13. 9aj2 4-25 2/' + 9;22_3o^^_^13^;2_30^2;. 16. x^ + 2x-l---^-. X x'^ 73m2 3m 9 50 10 16* 19. ^-s - f r^ + 2^ r^ +■ f r^ - -^/r^ + -U^ _^ 25 ^2_ 5^ + 9^ Find the square root to four terms : 20. 1 - a. 22. x^ — 1. 24. y'^ + 3. 21. a^ + l. 23. 4 -a. 25. a^ + 2 6. Square Root of Arithmetical Numbers 171. Compare the number of digits in each number and its square root : Number Root Number Root Number Root 1 1 I'OO 10 I'OO'OO 100 81 9 98'01 99 99'80'01 999 Principle. — If a number is separated into periods of two digits each, beginning at units, its square root will have as many digits as the number has periods. The left-hand period may be hicomplete, consisting of only one digit. 14. f +15 + 9n\ 4^2 15. S-f— '■ 18. 4m« 4m=' 19 m< , 3 m' 9 3 15 ' 5 126 INVOLUTION AND EVOLUTION 172. If the number of units expressed by the tens' digit is represented by t and the number of units expressed by the units' digit, by u, any number consisting of tens and units may be represented hj t -\- u, and its square by {t + uy, or t^ + 2tu-\' u\ Since 25 = 20 + 5, 252 = (20 + 5)2 = 202 + 2 (20 x 5) + 52 = 625. EXERCISES ^ 173. 1. Extract the square root of 5329. FIKST PROCESS 53'29|70 + 3 <2 = 49 00 SECOND PKOCES 53'29|73 49 2 < = 140 M= 3 4 29 4 29 140 3 143 4 29 2 < + M = 143 4 29 Explanation. — Separating the number into periods of two digits each (§171), we find that the root is composed of two digits, tens and units. Since the largest square in 53 is 7, the tens of the root cannot be greater than 7 tens, or 70. Writing 7 tens in the root, squaring, and subtract- ing from 5329, we have a remainder of 429. Since the square of a number composed of tens and units is equal to {the square of the tens) -^ (tioice the product of the tens and the units) + (the square of the units), when the square of the tens has been sub- tracted, the remainder, 429, is twice the product of the tens and the units, plus the square of the units, or only a little more than twice the product of the tens and the units. Therefore, 429 divided by twice the tens is approximately equal to the units. 2x7 tens, or 140, then, is a trial, or partial, divisor. On divid- ing 429 by the trial divisor, the units' figure is found to be 3. Since twice the tens are to be multiplied by the units, and the units also are to be multiplied by the units to obtain the square of the units, in order to abridge the process the tens and units are first added, forming the complete divisor 143, and then multiplied by the units. Thus, (140 + 3) multiplied by 3 = 429. Therefore, the square root of 5329 is 73. In practice it is usual to place the figures of the same order in the same column, and to disregard the ciphers on the right of the products, as in the second process. INVOLUTION AND EVOLUTION 127 Since any number may be regarded as composed of tens and units, the foregoing processes have a general application. Thus, 346 = 34 tens + 6 units ; 2377 = 237 tens + 7 units. 2. Find the square root of 137,641. Solution 13'76'41 [371 Trial divisor =2 x 30 = 60 Complete divisor = 60 + 7 = 67 Trial divisor = 2 x 370 = 740 Complete divisor = 740 + 1 = 741 4 76 4 69 7 41 7 41 Rule. — Separate the number into periods of two figures each, beginning at units. Find the greatest square in the left-hand period and write its root for the first figure of the required root. Square this root, subtract the result from the left-hand period, and annex to the remainder the next period for a new dividend. Double the root already found, with a cipher annexed, for a trial divisor, and by it divide the dividend. The quotient, or quotient diminished, tvill be the second figure of the root. Add to the trial divisor the figure last found, multiply this complete divisor by the figure of the root last found, subtract the product from the dividend, and to the remainder annex the next period for the next dividend. Proceed in this manner until all the periods have been used. The result will be the square root sought. 1. When the number is not a perfect square, annex periods of decimal ciphers and continue the process. 2. Decimals are pointed off from the decimal point toward the right. 3. The square root of a common fraction may be found by extracting the square root of both numerator and denominator separately or by re- ducing the fraction to a decimal and then extracting the root. Extract the square root of : 3. 5776. 6. 86,436. 9. 4.5369. 12. 11.0224. 4. 9604. 7. 8.0089. 10. 864,900. 13. .633616. 5. 6241. 8. 0^6^.^^. 11. 576,081. 14. .994009. I 128 INVOLUTION AND EVOLUTION Find the square root of : 15 2_8 9. 17 6.2,5 -iq 5 2.9. oi 409 6 16 i-96. 1ft i6_9 OO 961 99 1089 J.D. 7 2 9- -'■°* 3 6 1' ^^' TT9 6' ^^- "TOYS' Extract the square root to four decimal places : 23. 8. 25. yV 27. 2.5. 29. .7854. 24. 7. 26. 3^. 28. 3.6. 30. .41265. 31. Find, to the nearest tenth of a rod, the side of a square garden that contains 2 acres. 32. How many rods of fence are required to inclose a square field whose area is 10 acres ? 33. The legs of a right triangle are 12 feet and 15 feet. Find, to the nearest foot, the length of the hypotenuse. Suggestion. — Since the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides, if x represents the hypotenuse, 7i^ = 12^ + 15'^. 34. What is the length, to the nearest tenth of a foot, of the diagonal of a 5-foot square ? 35. Two vessels sailed from the same point, one north at the rate of 15 knots an hour, the other east at the rate of 20 knots an hour. How far apart were they after 6 hours ? 36. The length of the hypotenuse of a right triangle is 18 inches and the length of one side is 14 inches. Find, to the nearest inch, the length of the other side. 37. A rectangular field is 40 rods long and 25 rods wide. Find, to the nearest tenth of a rod, the length of a path extending diagonally across the field. 38. A 30-foot ladder leans against a wall, with the foot 5|- feet from the wall. Find, to the nearest hundredth of a foot, the height of the top of the ladder. EXPONENTS AND RADICALS THEORY OF EXPONENTS 174. Thus far the exponents used have been positive integers only, and consequently the laws of exponents have been obtained in the following restricted forms : I. or X a'' = a"*"^" when m and ii are positive integers. II. a"* -7- a" = a!^~"' when m and n are positive integers and m is greater than n. III. (oT'Y = a""" when m and n are positive integers. IV. a/ a"" = a"*^" when m and n are positive integers and m is a multiple of n. V. (a^)'' = a^'b'' when n is a positive integer. These laws may be proved as follows : I. Let m and n be any positive integers, and let a be any number. By notation, a"» = a • a • a ••• to m factors, and a" = a . a • a ••• to n factors ; . •. a"* • a^ = (a • a • a ••• to m factors) (a • a • a ••• to 7i factors) by assoc. law, , = a - a - a -" to (m -^ n) factors by notation, = a"*+". II. Let m and n be positive integers, m being greater than n, and let a be any number. By notation, a"» = a • a • a ••• to w factors, and a^ = a ' a ' a ••' to n factors ; . a^_ a ' a ' a '•' to m factors a" a ' a ' a •" to n factors Remove equal factors from dividend and divisor. Then, a"» -^ a" = a • a • a ••• to {m — n) factors by notation, = «»»-»». milne's sec. course alg. — 9 129 130 EXPONENTS AND RADICALS III. Let m and n be positive integers, and let a be any number. By notation, (a"»)« = a"» • a"» • a^ .-• to n factors by law I := Qm-\-m-k-m-{- • • • to n terms by notation, = a"»". IV. Let r and s be positive integers, and let a be any number. By law III, {a^y = a"-'. (1) Indicating the sth root of each member, Ax. 7, we have {/Ja^s := i> . - - , r. , --^ h ••• to jj terms ^ ^. Also, Since a* • a* ••• to jp factors = a* * = a*, a^ is the j9th power of a Q'th root of a. The numerator of a fractional exponent imtli positive integral terms indicates a power arid the denominator, a root. The fraction as a whole indicates a root of a power or a power of a root. The fractional exponent with the meaning just given comes directly from § 174, law IV, when the restriction that m is a multiple of n is re- moved ; thus, nj — - — 179. Any fractional exponent that does not itself involve a root sign may be reduced to one of the forms — or —S.. q q "I _4 Thus, 8 ^ =: 8 \ p p p By §§ 176, 168, dis. law, a ? = J- = i^ == f ^V p P \aj a« a« EXPONENTS AND RADICALS 133 EXERCISES 180. Find a simple value for : 1. 3-1 3. (-4)-'. 5. A. 7. (_|)-s. Q-2 9. Find the value of 2^-3 • 2^ + 5 • 2^ - 7 • 2^ + 4.2-1-2-2. 10. Find the value of x^ — 3x^ + 4:X^ + x'^ — 5x-'^ -\- x-^ when a; = i ; when a; = — i ; when a? = 1. 11. Which is the greater, (- i)-^ or (i)^? (- ^)-4 or (i)^? Write with negative exponents : 12. l-f-5. 14. 1^2". 16. m-T-bn\ 18. c^d -^ aV. 13. 1 -f- a2. 15. a -r- 0^. 17. c ^ aV. 19. am^ -^ bx"". Write with positive exponents : 20. 3y-\ 24. xy-h-\ 28. a"^-^". 21. 2aa;-i. 25. a-^WG-\ 29. 5 a^^-V. 22. 5a;-V. 26. 2 b^c~'^d, 30. 3 a?-«if/"^;2;-2^ 23. 3a-i&-2. 27. 3aj-V^^^- 31. 6P'm"''n-'^. 32. 4 0^3 - 2 a;2 + 5 aji - 6 a;« + 3 a;-i - 5 a?-2. 33. 2 a^ - 12 a2 - 16 a + 3 ao + 2 a-^ - 7 a-'\ IT Write without a denominator : 35. ^^. 38. — i 41. — . 44. . ^ 36. '^. 39. ?i^. 42. ^. 45. ^'. 6^71^ x'^ y b~^ 37. A. 40. 2^^. 43. r^Y- 46. "■ b'^^y \mj (ab) 134 EXPONENTS AND RADICALS 47. Find the value of 27*. \ Solution. — By § 178, 27* = ( \/2iy = S'^ = 9. Simplify, taking only principal roots : 48. 4*. 51. lei 54. 8"l 57. 81*. 49. 4*. 52. 16*. 55. 16~*. 58. 64"*. 50. 9-K 63. 25\ 56. (-8)*. 59. ( - 32)"l 60. Which is the greater, 64"* or (-^)i ? 64* or (gV)""^ ^ 4 2 61. Find the value of x~^ — 4 aj"^ + 4 when x = — y^^. 62. Express ■\/x~^y~'^z'^ with positive fractional exponents. -1 _4 2 Solution. vic-i?/-*^^ = x^^V 3;23 — _? — 14 x3?/3 Express with positive fractional exponents : 63. Vc^. 66. {-yJ'xyf, 69. V^V^r^- 64. -Vab. 67. (\/m7r)^. 70. ^^oF^fc^, 65. V^^ 68. (A/a6)3. 71. 2-\/{a + hy, Express roots with radical signs and powers with positive exponents : 72. aj*. 73. ai 74. y^. 77. c^d~i 80. ah^c~^. 83. a;^ ^ 2/"*' 84. Simplify "v^^ + a?* + 8* + 3 a?* - 5"^^ - VW\ 85. Simplify 4^/^ + 5 a^^ - 3 x~^^ + 2^^ - 8* - 2 x^. 75. a*6t. 78. m^n^. 81. a* --6*. 76. aj*2/3. 79. a*rt 82. ci-^dl EXPONENTS AND RADICALS 135 / 181. Operations involving positive, negative, zero, and fractional exponents. Since zero, negative, and fractional exponents have been defined in conformity with the law of exponents for mul- tiplication, this law holds true for all exponents so far encountered. J J For the proofs of the generality of the other laws of exponents, see the author's Aiademic Algebra. J EXERCISES 182. Multiply: 1. a by a~\ 4. a^b^ by d^b^. 7. n'^ by an^. 2. a^ by a°. 5. m^n by m^n-^. ^- ^"* " ^J ^"^^* 12 11 4 3 Tn-\-n m — n 3. x^ by x^. 6. a^b^ by a'W^. 9. a ^ by a ^ . 10. Multiply x^y~^ -{-x'^ + x'^y'^ + x'^y'^ -f- y'^ by x^y\ 11. Multiply 2/" + iK~V^^ -H i»"V"^^ + i«~V^^ ^7 a?"^/"'*- 12. Expand (a*6"i + l + a"*6i)(a56-i-.l + a"*5^). SOLUTION a^6' ■i + 1 + a'M a^6" 4_ 1 + a-h^ a*6- -^ + ah~ ^ + a»6» ah- ■i- 1 -a -ib^ -f a^^o + a *6^ + a"*6 a^5-i + 1 +a"^b Expand : 13. (a^ + b^)(a^-b^-). 16. (x^ - x^y~^^ -i- y~i){x^ + y'^), 14. (a;^ + 2/*)(a;^ — 2/^). 17. (l — x-]-x^)(x-^ + x-^-\-x-^). 15. (a;*_4)(a;* + 5). 18. (a-^ + H + c*)(a-i+ 6~* + 2c^). 136 EXPONENTS AND RADICALS Divide : - 19. n^hjn\ 22 rHy rA 25. x' + x'y'' + y' ^J x^y\ 20. n^hjn\ 23. s'^hj s~\ 26. a-^ — a-'^h -\-¥hj a-'^b. 21. n'^hjn-^. 24. a;^"" by a;^-\ 27. .^'^ — 2 ax^ + a^a;— a^ by aV, 28. Divide h'^ + 3 a"* - 10 a-^^ by ah'^ - 2. Solution a*6-i - 2 a ^ 4- 5 a-ift 6-i + 3a"^- 10a-i& 5a"^ - \Oa-^h Divide : 29. a — h by a* — 6^ 32. ic — 1 by a?* + a;i + 1. 30. a-\-h by a^ 4- fti 33. x^ — 2 -[- x~^ by a;^ — a:"^. 31. a2 + 52 ]Qy a* + &*. ' 34. 3 - 4 oj-^ + a;-2by a;-^ — 3. Simplify the following : 35. (a*)2. 38. {-a?)\ 41. Vo"^. 44. (_ ^i^ a^o)"'. 36. (a-*)«. 39. (-a2)4. 42. Va^V'- ^^- We^"*^')"^- 37. (a-^)2. 40. {-a^)-\ 43. Va'^^-l 46. (i m-i/r^)^ Expand by the binomial formula : 47. (a* - h^y. 49. (a'^ - &t)3^ 51. (a"i + 1)1 48. (a* + h^Y' 50. (a;"* - y^)\ 52. (1 - a;^)^ Extract the square root of : 53. & + 4 6~2c2 + 4 c - 2 &^ — 4 c^ -f 1. 54. a;2 + 2a;^H-3a; + 4a?* + 3 + 2aj"i + a?-^ 55. a;2 + 2/ + 4 ;$;-2 _ 2 a;^/^ 4- 4 xz'^ — 4 .'?/*;s-^ 56. a + 4 6^ + 9 ci - 4 a^ft* + 6 a^c^ - 12 bh^. EXPONENTS AND RADICALS 137 57. Factor 4 x~^ — 9 2/~^ and express the result with posi- tive exponents. Solution 4 a:-2 - 9 y^ = (2 x'l + 3 y-^) (2 x'^ - 3 y-^) = (? + ? V? - 5") . \x yl\x yl Factor, expressing results with positive exponents : 58. a-'^-b-\ ^1. 3^-x-\ 64. a^ + 2 + a-^. 59. d-x'^, 62. 21-b-\ 65. 6^-8 + 16 6-^ 60. 16 -a-*. 63. 6-3_^?/-3. 66. 12—x'^ — x\ Solve for values of x corresponding to principal roots : 71. aj~i = 12. 75. 07^ = 243. 72. \x^ = 25. 76. a;'^-a« = 0. 73. 2a;-t = ^V '7'^- a** -64 = 0. 74. ia;^ = 108. 78. 0^-^-27 = 0. 83. f^r^'- 87. ^--^---^^"^ 67. x^ = 5. 68. a;* = 9. 69. x^ = S. 70. x^ = 16. Simplify, expres /2-2\2 79. i,2-J ■ 80. $)-"■ 81. (^s-y. V^2-2/ 82. ( 9-' y^ 91. 2-1 X 2-2 4-2 X 4-3 92. 32* + 125* 811 + 216t 93. a\ 8. -yi62. 15. v'GlOp. 22. (27 c^ - 27 c^)^. 140 EXPONENTS AND RADICALS Simplify : 23. vV + 46^ + 45. 25. (3 am^ + 6 am -f 3 a)^. 24. V5x^-10xy + 5y\ 26. (5 a' -\- 10 a^x -{- 5 cfx'')\ 27. Reduce VyV ^^ i^s simplest form. Solution. — To make the radicand integral and thus simplify it, we must multiply both its terms by a number that will make the denominator a per- fect power corresponding to the index of the radical, in this case, a perfect square. The smallest multiplier that will accomplish this is 3, thus : ''' ^^- 33. J^. 36. VS- 30. Vf. 34. ^^ 31. ^|. ' 37. ^1^. \3 62 35. xy"^ 32. VtV * \ 962 * \ 125 2^ . (a4.6)j^±^. 41. (^±^.fz±i:. ^ ^ ^\a-h ' a-b \(a-by 39 .2 ^_ . 2v /a; — 2-2/ >io /1 3\ /J — i^ + i»' 40. T^\h-?: — - ' 42. (1 — af)\ In general it may be proved that a*^"* = a* ; that is, A number having a fractional exponent is not changed in value by reducing the fractional exjyonent to higher or lower terms. 43. Reduce VT^y to its simplest form. Solution. ^/A xY = ^/'(^xy^ = (2 xy^) ^ = (2 xy^)i = V^^, Simplify : 44. v/49. 47. ^400. 50. -v/25"a^. 53. ^a^^V/. 45. -^27. 48. -v/625. 51. -v^27^. 54. ■\/125 .tV. 46 a/100. 49. , and Va — 6. 18. Va 4- bj Vcfc^ + ^^ and Va — ^. 19. Which is greater, V3 or V2 ? V4 or V2 ? 20. Which is greater, V5 or V3? 2V3 or 3V2? Arrange in order of decreasing value : 21. ViO, V3, and V2. 24. ^/^, V3, VS, and V2l. 22. V2; Ve, and ^33. 25. y/l, V2, VB, and Vl3. 23. V3, Vt, and V25. 26. 2V3,3V2,2V5,and ViO. Addition and Subtraction of Radicals 191. Principle. — Only similar radicals can he united into one term by addition or subtraction, EXERCISES 192. Add: Subtract: 1. 6Viand4V8. 7. |V24 from VSI. Solution Solution 6Vi = 3V2 \/8r = 3\/3 4v/8 = 4\/2 \V2i=:lV^_ Sum = 7\/2 Difference = 2jv/3 2. V32 and V72. 8. -|V3 from V|. 3. V48 and VIOS. 9. 6V| from Vl28. 4. V8, Vl8, and V324. 10. Vl92 from 2 Vsi. 5. Vie, 2Vi, and V54. 11. 3V| from 2^36. 6. lOVJ, V80, and v/i25. 12. V^=^32 from Vl08. EXPONENTS AND RADICALS 143 Simplify : 13. ^135-^625 + ^320. ^^ j^_ [x.lx M. n+n^m ' "'^•'^^'^'^^ 15. -v/864-^4000+-y32. 19. J^-a/I-Ja. ^yz ^xz ^xy 16. 2V75-3V72 4-5V12. , ,_ , / / 20. sJ^+2J-^-J-^-. 17. V(aj + 2/)2a- V(aJ-2/)'«. ^'4^2 Msb^ ^27 6^ 21. ^Z:24+3^^=^375+-v/"^=^. 22. V-96x'+2^3a^--\/5x+V^0x'. 23. V3 2/3^242/2 + 482/-V32/'-36?/2 4-1082^. 24. (f)^ - (1)-^ + VCSr + VT:35 - il9 2/4-2/V6)(V9 2/-2/V6). 33. (A/7c+V57;2)(A|7c-V5^). 34. (A/l4a; + a;V27)(A/l4aj- W27). EXPONENTS AND RADICALS 145 Division of Radicals 195. In division, when one fractional exponent is subtracted from another, the exponents must be expressed with a common denominator. When one radical is divided by another, the radicals must be expressed with a common root index. EXERCISES 196. 1. Divide2\/iby 4V2; v"^ ' by /l2 - 6 V3 = V9 - \/3 = 3 - V3. Find the square root of : 2. 5-2V6. 6. 11-6V2. 10. 3 - 2 V2. 3. 8+2V15. 7. 22 + 8V6. 11. 6 + 2V5. 4. 9-2Vi4. 8. 24-8V6. 12. a'-^b + ^a^b. 5. 11-2V30. 9. 31 + 12V3. 13. 2a-2VG^^\ 202. Square root of binomial quadratic surds by conjugate relations. Principles. — 1. The square root of a rational number cannot be partly rational and partly a quadratic surd. For, if possible, let Vy = y/h dr m, Vy and y/h being surds. By squaring, y = h ±2 mVh + wi^, and ^/i^^ y-'m'^-h ^ 2 m which is impossible, because a surd cannot be equal to a rational number. Therefore, Vy cannot be equal to y/h ± m. 148 EXPONENTS AND RADICALS 2. In any equation containing rational numbers and quadratic surds, as a + V& = a; -f- V^/, the rational parts are equal, and also the irrational parts. Given a -\-\^h = x -\- Vy, (1) Since a and x are both rational, if possible, let a = x ±m, (2) Then, x ±m -{-y/h = x-\-Vy, (3) and Vy = Vh ± m, (4) Since, Prin. 1, equation (4) is impossible, a = x ±m is impossible ; that is, a is neither greater nor less than x. Therefore, a = x, and from (1), -s/h =Vy. Hence, if a + Vb = x -\- \/y, a = x and VS = Vy. 3. If a + -y/b and a — V^ are binomial quadratic surds and \a -f V6 = 'Vx -\- -y/y, then \ a — V^ = Va? — Vy. To exclude imaginary numbers, suppose that a — Vb is positive. Given V a + Vb = Vx -f Vy, Square, Ax. 6, a -\- Vb = x + 2Vxy + y. Therefore, Prin. 2, a = x-\- y and Vb = 2Vxy ; whence. Ax. 2, a — Vb = x -\- y — 2Vxy. Hence, Ax. 7, v a — Vb = Vx — \/y. EXERCISES 203. 1. Find the square root of 21 + 6 VlO. Solution. — Let Vx-\-Vy = V2I + 6 VlO. (1) Then, Prin. 3, Vx- Vy = V2I - 6 VlO. (2) Multiply (1) by (2), x--y = V441 - 360 = VSl, or X - ?/ = 9. (3) Square (1), Ax. 6, x +2Vxy + ?/ = 21 + 6 VlO. Therefore, Prin. 2, x -f 2/ = 21. ' (4) Solve (4) and (3), x = 15, 1/ = 6. .-. Vx = Vl5, V2/=V6. Hence, from (1), V2I 4- 6 VIO = Vl5 + V6. EXPONENTS AND RADICALS 149 Find the square root of : 2. 26 + 10 V6. 8. 16 + 6V7. 14. 2 + V3. 3. 19 + 6V2. 9. 21-8V5. 15. 6+V35. 4. 46 + 30V2. 10. 47-12Vri. 16. 1+IV2. 5. 35 - 14 V6. 11. 56 + 32V3. 17. 2 + fV6. 6. 11+6V2. 12. 35-12V6. 18. 18 - 6V5. 7. 24-8V6. 13. 56-12V3. Rationalization 19. 30 + 20 V2, 204. If it is required to find the approximate value of — ~, V5 we may divide 1 by the approximate square root of 5, using long division, but it will be more accurate and a saving of labor to change the fraction to an equivalent fraction having a rational denominator, thus, 1 ^ 1V5 ^yg V5~V5.V5~ 5 ' and divide V^ by the simple and rational divisor 5. 205. Define rationalization ; rationalizing factor ; rationalizing the denominator. EXERCISES 206. Rationalize the denominator of : 1. ^- 3. 4^. 6. A^. 7. AI±^. Va' y^^if \2aar' \(a-2f ^f \2aa? V(a-2/ 2. _^. 4. i^. 6. ^^- 8. Jl^^^^- -^ah(? V12 Va;+.v ^ '«^+2 Taking v2 = 1.414, V3 = 1.732, and VS = 2.236, find, to the nearest hundredth, the value of: 9. A. 11. X. 13. -2L. 15, 1« V2 V5 V32 10. A- 12. ^. 14. -^^. 16. V3 V8 V27 -v/324 150 EXPONENTS AND RADICALS 207. Since, § 30, 3, (Va + V6) ( Va - V6) = ( Va)^ - {Vly = a — b, the product of any two conjugate surds is rational. [ Hence, Principle. — A binomial quadratic surd may be rationalized by multiplying it by its conjugate. EXERCISES _ 208. 1. Eationalize the denominator of — ::r-— — ^• V7+V3 Solution \/7-V8 ^ (v/7 -V3)(V7-.V3) ^ 7-2\/21 +3 ^ 5-V2I \/7+V3 (V7 + >/3)(\/7-\/8) 7-3 2 ^ Rationalize the denominator of : 4 2. 3+V2 c Va-V6 2^V3 2V3 + 1 3V3-2V2 4V2+6V3 4. V3 -V2 V3 + V2 R 2- -V2 «>. 6- 3v'2 11. 12. g Va + Vic Va — Vaj ^ a-f2V5, a-2V6 Vrf '+a+l-l Va ^+a+l+l + 6 - Va - 6 Va + 6 + Va - 6 Vic' 5_2-Vaj2 4-2 10. ^-V^^. 13. x+^/x'-l Vx^ -2+^x^ + 2 Reduce to a decimal, to the nearest thousandth : 14. -^-. 16. ^^^. 18. ^^^. 2 + V3 2-V2 3-V6 15. ^-. 17. ^1+^. 19. ^-^^ 3-V5 V3-V2 4+2V3 EXPONENTS AND RADICALS 151 V2--V3-V5 20. Rationalize the denominator of _ _ _ V2+V3+V5 Solution. V2-- V8 - V5^ ( V2 ^ V5)^ V3 ^ (V2 - V6) + V3 V'2+V3-fV5 (v/2+\/3) + V5 (\/2+\/3)-V6 ^ 2-2Vi0 4-5-3 _ 4- 2VI0 2 4-2\/6 + 3-5 2V6 ^ 2^\/i0 ^ 2\/6-2\/15 ^ Ve-Vis V6 6 3* Rationalize the denominator of : 21. V2-V5-V7. 23^ V3 + V2 _ V2+V54-V7 V3+V2-V6 22. 1 24. 2V\-3V3 + 4V5 V2+V3 + V5 * V2+V3~V5 25. Rationalize the denominator of — — -^z, or Solution. — By § 38, Va 4- \^, or a^ + 5^, is exactly contained in 1 2. the sum of any like odd powers of a 2 and h^^ and also in the difference of any like even powers of a^ and h^. The lowest like powers of a^ and h^ that are rational numbers are the sixth powers, which are even powers. 1 2 Hence, the rational expression of lowest degree in which a^ + h^ is ex- 1 2 actly contained is (a^)^— (6^)6, or a^ — b^, 1 2 Dividing a^ — 6* by a^ + 6^, we find that the rationalizing factor for 5 34 18 10 the denominator is a^ — a^bi + a^b^ — ab^ + a^b^ — b"^'. Multiplying both terms of the given fraction by this factor, we have y/a-^Vb^ aS + b^ «' - ^* Rationalize the denominator of : 26. „5^„ • 28. -J^^-. 30 ^^ Va — V ^ \/o? — Vft* Va — V. X 27. -YT^-- 29. y^ + ^ - 31. _J^. 152 EXPONENTS AND RADICALS Radical Equations 209. When the following equations have been freed of radicals, the resulting equations will be found to be simple equations. Other varieties of radical, or irrational, equations are treated later. EXERCISES 210. 1. Solve the equation ■\/x— 5 + s/x = 5. Solution. \/x — 5 + Vsc = 5. Transpose Vic, Vx —5 = 5 — Vx. Square, Ax. 6, a; — 5 = 25 — 10 Vx + x. Transpose and combine, 10 Vx = 30. Divide by 10, y/x = 3. Square, ic = 9. Verification. \/9 — 5 -f\/9=\/4 + \/9 = 2 + 3 = 5; that is, 5 = 5. 2. Given \14 +V 1 +Va:-h8 = 4, to find the value of x. Solution. Vi4 + Vl + V^TS = 4. Square, 14 + Vl + Vx + 8 = 16. Transpose,, etc., v 1 -f Vx -f 8 = 2. Square, 1 -h Vx + 8 = 4. Transpose, etc.. Vac + 8 = 3. Square, a: + 8 = 9. .•.x = l. Verification. V 14 + \/l 4-VH- 8 =Vl4 + Vl + 3 = \/l4 + 2 = 4 ; that is, 4 = 4. General directions. — Transpose so that the radical term, if there is but one, or the most complex radical term, if there is more than one, may constitute one member of the equation. Then raise each member to a power corresponding to the order of that radical and simplify. If the equation is not freed of radicals by the first involution^ proceed again as at first. I EXPONENTS AND RADICALS 163 Solve for x, and verify each result : 3. VS - 2 = 10. 13. 1 + 2 Vx = 7 - Vi. 4. 3 + 2 Vaj = 15. 14. Vx - 21 = V^ - 1. 5. 3V2a;-4 = 32. 15. ^x'-ll + l^x. 6. Va? 4- 11 = 4. 16. Vo; — 16 = 8 — Vx. 7. Vx + 5 = 13. 17. Va; — 15 + Vi = 15. 8. ■Vx + d'^ = c. 18. V2 a; - V2 ic - 15 = 1. 9. ■\/x-2 = 2. 19. VS + 2 = Vo; + 32. 10. Vx + b^ = a. 20. Va? + 4 = 4 — Voj - 4. 11. -v^^ + 6 = a. 21. ■\/x — 5+ Vx~-f7 = 6. 12. V4a;-16=:2. 22. VaJ ~ V4a; - 21 = 0. 23. V9a;+8+ V9a;-4 = 0. 24. 2Va;2 + a; + l=2(2 + a;). 26. 3- V3-6aj + 4a;2 = 2a;. 26. V2(l -.t) (3- 2a;) -l = 2a;. 27. V16 aj + 3 -f V16 a; + 8 = 5. 28. Vl + a;Va;2 + i2 = l + aj. 29. a; + V^*^ + V2-h4a;2 = 1. 30. V3(aj + 1) + V3a;-1= V2(6a;4-1). 31. 2Vx - V4a; - 22 - V2 = 0. 32. \/9x^ - 4 V9 aj2 _ 2 + 3a; = 2. 33. \-\/V2a; + 56 = 2. -34. ^7 + \/l + ^4 4- Vl + 2 VS = 3. 35. V3a; + 7 + V4 a; - 3 = V4a; + 4 + V3aj. 154 EXPONENTS AND RADICALS Solve and verify : 36. ^ = V3a;+2 + V3a;-l, V3aj + 2 Suggestion. — Clear the equation of fractions. ■y/x + 5 Vi + 3 Suggestion. — Reduce each fraction to a mixed number and simplify before clearing of fractions. 38. V^^-6^V^-8. 4j_ Vv — 1 V'w — 5 V27+6 ^ V2r4 V2r + 4 V2r + 1 39. V2r + 6^V2r + 2, ^^^ 2V29;4-4 Va; + 1 + 3 2V2a!-4 V« +1 -3 Vm + 1 — Vm - -1 1 Vm + 1 + Vm - -1 2 V'42 + 3 + 2V2 =5. ^^ Vlln+V2n + 3 ^8^ ^^ Vlln-V2u + 3 3 ■ V42;+3-2V^^^ 4^ V V5^-9 -\/V5a;-21 46. Suggestion. — First square both members. ->/V8^4-16 VV8^ + 32 V V8^+4 VV8^+12 Vi-V3 2 Suggestion. — Simplify the first member. 3 47. V2a;-V2: V2a;-7 ( 48. Solve V^+a + ^^-a = 2+ ^'^-"' for a^. Va: -\- a — Va? — a ^ Suggestion. — Rationalize the denominator of the first fraction. , 49. Solve <» + «' + V 2«.T + ^ _; ^ ^, j^^ ^ a + x — V2 aa; + x^ EXPONENTS AND RADICALS 156 Solve for x, and verify ; 50. Vo; + ^x - {a — by = a -h b. 51. a^/x — bVx = a^-]-P — 2ab. 52. V5 ax — 9 a- -{- a= V5 ax. / 77- ^^ /- 53. ■vx-\-3a= =: — Va?. 54. VS-hV^7 + V3^ = Va. Solution. — Factor, ( VI + V2 + V3) Vx = Va. Multiply by 1 + V2 — V3 to partially rationalize the first factor, (1 + 2 V2 + 2 - 3) Vx = Va(l 4- \/2 - V3), or 2V2 . Vx = Va(l + V2 - V3). Square, 8 x =a(H- V2 - V3)2 ; whence, x=-(l+y/2- V3)2. 8 Solve for x, giving the result with a rational denominator ; 55. V2 X + V3 X + V5 X = Vm. 57. Vo; - a + V2(x - a) = Vs a; -f- a V2. 211. The student will have observed that radical equations are freed of radicals either by rationalization or by involution. Thus, V2¥- 6 = (1) V2x+ 6=0 (2) Multiply by V2^+ 6 V2^- 6 2a;-36=0 2a;-36 = .-.a: = 18 .-. x = 18 If the positive, or principal, square root of 2 a; is taken, a; = 18 satisfies (1) but not (2) ; if the negative square root of 2 X is taken, a;= 18 satisfies (2) but not (1). It has been agreed, however, that the sign ^ shall denote only principal roots in this chapter, and because of this arbi- trary convention, our conclusion must be that (1) has the root a; = 18 and that (2) has no root, or is impossible. 156 EXPONENTS AND RADICALS According to this view, when both members of (1) are mul- tiplied by V2 X -\- 6, no root is introduced because V2 x -\- 6 = has no root ; but when both members of (2), which has no root, are multiplied by V2 x — 6, the root of V2 x — 6 = 0, which is a; = 18, is introduced (§ 108). A root may be introduced in this way by rationalization, or by the equivalent process of squaring. Thus, V2^ + 6 = 0. (2) Transposing, we have V2 x = — 6, Squaring, Ax. 6, we have 2 a? = 36. .-. X = 18. Verifying, we have V2 -18 + 6 = 6 + 6:^ 0. EXERCISES 212. 1. Solve, if possible, the equation ■Vx — 7 — Vx = 7. Solution. — Transposing, squaring, simplifying, etc., we have Vx = — 4. Squaring, we have x= 16. Verification. VW - 7 - Vie = \/9 - \/l6 = 3 - 4 ^t 7. Hence, the equation has no root, or is impossible. Solve, and verify to discover which of the following equa- tions are impossible ; then change these to true equations : 2. V2a;+V2a;-3 = 1. 5. V4a; + 5 - 2VaJ - 1 = 9. 3. VSx-i-J -{-■V3x = 7. 6. V4a;-Vi»=V9a^-32. 4. 2Vaj4-V4a;- 11 = 1. 7. V5x — 1-1 :=V5x +16. 8. Vaj -f 1 + V^ + 2 — V4 X + 5 = 0.' 9. V2(a;2 + 3 a; - 5) = (a; + 2) V2. 10 Va? - 5 Va; + 1 ^ Q ^^ Vl9a?+ V2aT + 11 ^2i V^"^^ Vo; -h8 ' ' -\/Wx--\/ 2x^+11 ^* G^- '^- IMAGINARY NUMBERS 213. Our number system now comprises natural numbers, 1, 2, 3, ... ; fractions, arising from the indicated division of one natural number by another; negative numbers (denoting oppo- sition to positive numbers), arising from the subtraction of a number from a less number; surds, arising from the attempt to extract a root of a number that is not a perfect power; and finally imaginary numbers, arising from the attempt to extract an even root of a negative number. In this chapter only imaginary numbers of the second order will be treated. Before the introduction of imaginary numbers, the only numbers known were those ivlwse squares are positive, now called real numbers to distinguish them from imaginary num- bers, wliose squares are negative. 214. Since the square of an imaginary number is negative, imaginary numbers present an apparent exception, in regard to signs, to the distributive law for evolution. Apparently V-1 X V-1 would equal V(- 1)(- 1) = V+ 1 = ± 1. But by the definition of a root, the square of the square root of a number is the number itself. Hence, V^^ x V^=T: =(V"in)2= - 1, not + 1. (A) In this chapter it will be assumed that imaginary numbers obey the same laws as real numbers, the signs being deter- mined by (A), which we call the fundamental property of imaginaries. 167 158 IMAGINARY NUMBERS 215. Powers of V^^. (V^)^=(V^)(V~l) =-1; (V^Tiy = ( V3T)2( v:ri)2 ^ (^ i)(_ 1) = + 1 ; (^/:ri)5 = (v^^y v^^ = (+ 1) v^^ = + v^=t: ; and so on. Hence, if n = or a positive integer, (V^ri)4n+3 ^ _ v^T; (V^^'^+^ = 4- 1. J Hence, an?/ e?;en power 0/ V— 1 ^'s rea? and an?/ ode? power is imaginary. For brevity V— 1 is often written i. 216. Operations involving imaginary numbers. EXERCISES Find the value of : 1. (V"=t:)^ 3. (V^^^. 5. (V"=Ty«. 7. (-0^ 2. (V"=T)^. 4. (V^^^ 6. (V^^^ 8. (-iy, 9. Add V- a^ and V- 16 a\ Solution V^^^ + V-16a* = aV^^n; + 4 a^V^n = 5 a2 V^H^. Simplify : 10. V^^4-V^^^^^49: 13. V-12 + 4V^^. 11. V^=^+V^^64. 14. 5V^=38-V^=^. 12. 2Ar^ + 3V^^. 15. 3V^^^^~V^^^^^80. IMAGINARY NUMBERS 159 Simplify : 16. V— 16aW 4- V— aV — \/— 9 a^aj^. , 17. (V^^ + 3V^^)-|-(V^^-3V^^. 18. {^ — ^ xy — ^ — xy) — (V— 4^??/ + V- 19. V- ic2 -f- V- 4 a;'^ — V- ar*^ + 3 a^V^^. a?!/). 20. V-16-3V-4 + V-lS-h V^^^50 + V-:^5. 21. V^^ + aV^^-V^98-5V^^^^2a2. 22. V1-5-3V1-10 + 2V5-30. 23. Multiply 3 V- 10 by 2 V^. PROCESS 3V^T0x2V^^=3Vl0^ ■lx2V8V-l = 6Vl0^x(-l) = -6V80^=-24V5 Explanation. — To determine the sign of the product, each imaginary number is reduced to tlie form 6 V— 1. The numbers are then multiplied as ordinary radicals, subject to (^), § 214, that V— 1 x V— 1= — 1. 24. Multiply V- 2 + 3 V- 3 by 4 First Solution 4V^^- V'3^=(4V2- V3)V^n;; ... (V^^4-3V^^)(4V^r2-Vir3) = (V2 + 3V3)(4V2 - \/3)(\/=l)2 = (8 + 12\/6-V6-9)(-l)=l- live. Multiply : 25. 3 V^^ by 2V^ri5. 28. 8 V _2-V-3. Second Solution -4V4- 12 Ve 4-3V9+ VG 1 -iiVe Iby V-6'. 26. 4V- 27 by V- 12. 29. V- 125 by V- 108. 27. 2V^^by5V^^. 30. V^^^^M by V-30. 31. V^=^ + V^^ by V^^ - V^=^. 160 Multiply : IMAGINARY NUMBERS 32. V—ab-\- V— a by ^/' — ab — V- a. 33. -\/—xy-\- V— xhj -V—xy-^ V- • aj. / 34. V3:50-V-12by V-8- V-75. 35. V— a+ V— 6 + V— c by V— a+ V- 36. Divide V-12 by V^^. VT2 V- Solution. V- 12 Vi2\/^rT V3 \/4=:2. 37. Divide Vl2 by -3. Solution VT2 \/l2 Vi V-3 VsV-i V31 v^^H" 2\/^3 1 2V-1. 38. Divide 5 by (V-1)^ Solution (v-i)' Cv-iy (V^=n[)3 " Divide : 39. V-18by V-3. 40. V27 by V^^. 41. 14 V"^^ by 2V^^. 46. •2 by 47. (V-l)'by iV-1. 48. (V^=^)-' by (V"^)^^ 42. -V-a^by V-^l 43. 1 by V^^. 49. V4 a6 by V— be. 50. (V^^^by -iV^=T. 44. V8 + 3Vl4by V^^.j 51. (V^^'byCV^^) 45. Vl2 + V3 by V^^ J 53. V^^by 52. V — a +6V — 1 by V — a^. 172 . V^^ • V^^. QUADRATIC EQUATIONS 217. Define and illustrate the following kinds of equations : 1. Quadratic. 4. Incomplete quadratic. 2. Second degree. 6. Affected quadratic. 3. Pure quadratic. 6. Complete quadratic. PURE QUADRATIC EQUATIONS 218. Since pure quadratics contain only the second power of the unknown number, they may be reduced to the general form ax^ = c, in which a represents the coefficient of x^, and c the sum of the terms that do not involve x^, 219. Principle. — Every pure quadratic equation has tivo roots, numerically equal hut opposite in sign. It is proved in § 269 that every quadratic equation has two roots and only two roots. EXERCISES 220. 1. Find the roots of the equation 3 aj^ -f 15. = 0. Solution. 3 x'^ + 15 = 0. Transpose, 3 ic^ _ _ 15, Divide by 3, x'-^ = — 6. Extract the square root, Ax. 7, x = d= V— 5. Verification. — The given equation becomes = 0, and is therefore satisfied when either + V— 5 or — V— 5 is substituted for x. Solve, and verify each root : 2. 2aj2_4 = 4. 6. 3a;2 = 108. 10. (a; -f- 3)^ = 6 a;+6. 3. 3 ic2 + 2 a.'2 == 45. 7. 4 a;^ = J-g-. 11. (aj+ 5)^=10 a; +41. 4. 12-\-'6x'' = m. 8. \x^ = S. 12. (ic + 4y = 8 a;+24. 5. 12a;2 + 60 = 0. 9. |a;2-hl8=30. 13. 1 x''-2b = bx''^l^. Milne's sec course alg. — 11 161 162 QUADRATIC EQUATIONS Solve and verify : 14. 4. + x'' = 2(x + 12)-2x. 17. {x-5y-10 = 5(7-2x). 15. {x-\-2y = 2x(x-{-2)-{-12. 18. (x + 2y-4{x-\-2) -2 = 2, 16. {x-3y + 6(x-l) = -9. 19. (x-3y-i-10x=:x(4.+2x). 20. S{x^ + 4.)-{-5x==5{6-{-x). 21. (x -\-2y-^x + !)-{- 4. = 2S. 22. 4 x(x + 2)-5 = 12-{x- 4)^. 23. 2(3-2a!)+20=(aj-l)2-2a;. a? a?^ - 15 _ a; a; - 3 a^ + 3 __ h 7 24. T7. + -,: --. 27. ^^ + ---^_l3. i« _ 2 x + 2 40 a? x-' -15 X 12 T 5x 5 ^ + 3 , x-3 — '\ X — ■3 ' x-j-3 X — -2 a; + 2. -1 ' -1 " — . 25. ^^1-1^ -I- r:^ -z=L 28. 26. ^^^ ^ + ^^-1— = -1. 29. a^ + 2 2 - a; a^^ . x-^7 ^-7 __ a; -}- 1 a; — 1 ' ' a;^— 7 a; aj--f-7a; aj^ — 73 30. What negative number is equal to its reciprocal ? 31. If 25 is added to the square of a certain number, the sum is equal to the square of 13. What is the number ? 32. When 5 is taken from a certain number, and also added to it, the product of these results is 75. Find the number. 33. A certain number multiplied by \ of itself is equal to 16. Find the number. 34. The area of a sheet of mica is 48 square inches and its length is 1^ times its width. Find its length and its width. 35. At 75 cents per square yard, enough linoleum was pur- chased for $ 36 to cover a rectangular floor whose length was 3 times its breadth. Find the dimensions of the floor. 36. The sum of the squares of two numbers is 394, and the difference of their squares is 56. What are the numbers ? 37. A man had a rectangular field the width of which was I of its length. He built a fence across it so that one of the two parts formed a square containing 10 acres. Find the dimensions of the original field in rods. QUADRATIC EQUATIONS 163 AFFECTED QUADRATIC EQUATIONS 221. Since affected quadratic equations contain both the second and the lirst powers of the unknown number, they may always be reduced to the general form ax^ + 6a; + c = 0, in which a, h, and c may represent any numbers whatever, and x, the unknown number. The term c is called the absolute term. 222. Solution of affected quadratics by factoring. Reduce the equation to the form ax^ -f 6ic + c = 0, factor the first member, and equate each factor to zero, as in § 82, thus obtaining two simple equations together equivalent to the given quadratic, subject to the exceptions given in § 108 as to equivalence. EXERCISES 223. Solve by factoring, and verify results ; 1. 0^2 + Taj + 12 = 0. 15. 30 + r-?'2 = 0. 2. 2/2 -7 2/ + 12 = 0. 16. 5cc2 + 9a; = 2. 3. a;2 + 4 a; = 21. 17. 3 oj^ - 7 a; - 6 = 0. 4. 2;2 = ;3 + 72. 18. 6 a;^- 5 a; = - 1. 5. ^2 = 2/ + 110. 19. 2aj2 + 15 = 3(2a; + 5). 6. a:2 _p 2 a; = 120. 20. 6 (a;^ + 1) = 13 a;. 7. 2/' -20 2/ = 96. 21. 27.^^ - 3^/ - 14 = 0. 8. n2 + lln = -30. 22. 15^2-4 = - 17 s. 9. 36 = c2 + 16c. 23. 9a2 + 40 = 42a. 10. Z2 + 15Z-34 = 0. 24. 3(4a;2 + 2)+25a; = 8a:. 11. r2 = 6r + 135. 25. 2 (3a;2- 1)+ 7 a; = 18 a;. 12. a:2_24 = 4(a: + 2). 26. 3 - 13 a; = 6(aj2 _ 2). 13. 2a;2 + 3aj-2 = 0. 27. 4 i^(8i^ + 7) = 15. 14. 3a;2 + lla;-4 = 0. 28. 4a;(5 a; +4) = 7a; + 18. 164 * QUADRAXIC EQUATIONS 224. Solution of affected quadratics by completing the square. The general form of the perfect square of a binomial is x^ + 2 ax + a\ Consequently, an expression like x^ + 2 ax may be made a perfect square by adding the term a^, which it will be observed is the square of half the coefficient of x. This fact, as shown in the following solutions, is used to complete the square in one member of an affected 'quadratic, suitably prepared, so that it may be solved by extracting the square root of both members as was done in solving pure quadratics. EXERCISES 225. 1. Solve the equation a;^ - 3 a; - 10 = 0. Solution. x^ -3 x - 10 = 0. Transpose the absolute term, x^ — 3 x = 10. Complete the square in the first member by adding the square of half the coefficient of x, and add the same to the second member to preserve the equality, x2->3x + (1)2=10 + (1)2, or x2 - 3 X + I = -V-. Extract the square root of both members, whence, x = f + lorf-J; that is, X = 5 or — 2. Verification. — Either 5 or — 2 substituted for x in the given equa- tion reduces it to the identity = 0; that is, 5 and — 2 are roots of the equation. 2. Solve the general quadratic equation ax^ + hx + c=^ 0. Solution. ax^ + ?>x + c = 0. h c Transpose c and divide by a, x2 + -x = a a ^ Complete the square, etc., x2 + -x + a Extract the square root, whence, 2a • + 62 4a2 h _ 2a cr. — 52-4ac 4a2 x4 , v/?)2 _ 4 flfc ^ 2a ' _ 5 i V62 - 4 ac QUADRATIC EQUATIONS 165 Steps in the solution of an affected quadratic equation by the method of completing the square are : 1. Transpose so that the terms containing x^ and x are in one member and the knoivn terms in the other, 2. Make the coefficient of a? positive unity by dividing both members by the coefficient of x^, 3. Complete the square by adding to each member the square of half the coefficient of x, 4. Extract the square root of both members. 5. Solve the tivo simple equations thus obtained. Solve, and verify all results : 3. aj2-.2a;=143. 13. v'' + 15v = 54.. 4. x'' + 2x=zl68, 14. 'i;2 + 21 y = -54. 6. a;2~4a; = 117. 15. 2x'^ + Sx = 2T. 6. a;2-6a; = 160. 16. 3 a;^ + 16 a; = 12. 7. 8a; = a;2_i8o^ ^ 2 x''-}- 5x~l = 6. 8. a;2 + 2a;=120. 18. 4 a;^- 17 a; + 4 = 0. 9. 2/2 = 282/-187. 19. 6x^-5x-6 = 0. 10. a;2-12a;=189. 20. ,2 x'' -\- ,9 x = 3.5. 11. 2/' + 22?/ = -120. 21.2x^-\^-x = ^. 12. z^-180=:3z. 22. .03a;2_.07aj = .l. 226. Solution of quadratics by the quadratic formula. The general quadratic ax^ + bx-{-c = (1) has been solved in exercise 2, § 225. Its roots are ^^■5±V6-^-4ac^ (2) 2a ^ ^ Since (1) represents any quadratic equation, the student is now prepared to solve any quadratic equation whatever that contains one unknown number. The roots may be obtained by reducing it to the general form and employing (2) as a formula, known as the quadratic formula. 166 QUADRATIC EQUATIONS EXERCISES 227. 1. Solve the equation 6x^ = x-{-15. Solution. — Writing the equation in the general form 6 a;2 _ a; _ 15 = 0, we find that a = 6, 5 = — 1 , and c = — 16. ... by (2), § 226, X = 1 :i: V(^ 1)^ - 4 x 6(- 15) J \ J^ ^ ^ 2x6 ^l^i9^5__3 M 12 3 2 ^ 1 Solve by the quadratic formula : 2. 4:X^-x-S = 0. 13. l-3aj = 2a;2. 3. 2a!2 + 5aj + 2 = 0. 14. Sx^ = 5x-2. 4. 3x'' + llx + 6 = 0. 15. 4: = x(3x + 2). 5. 6x'^ + 2 = 7x. 16. x^-5x = -3. 6. 5x^-2x = 16. 17. 3x'-6x = -2, 7. 4ic2 + 4.T = 15. 18. 4aj2_3^_2 = 0. 8. 2 0^2 = 9-3 ir. 19. x^ + 10 = 6x. 9. a;(2a; + 3)=-l. 20. a;^ = - 4(a; + 3). 10. 13aj = 3i»2-10. 21. 4.{2x-5)=x\ 11. 7a;2 4-9a; = 10. 22. 5 a;^ 4. 18 = 6 a;. 12. 5aj2_18a; = 72. 23. a;(3 a; + 4) = - 2. 228. General directions for solving quadratic equations. 1. Reduce the equation to the general form ax'^ 4- ^^ -h c = 0. 2. If the factors are readily seen, solve by factoring. 3. If the factors are yiot readily seen, solve by completing the square or by the formula. 4. Verify all results, reject roots introduced in the process of reducing the equation to the general form, and account for roots that have been removed. Note. — In general, no root is introduced by clearing an equation of fractions, provided that : fractions having a common denominator are combined ; each fraction is expressed in its lowest terms ; and both members are then multiplied by the lowest common denominator. QUADRATIC EQUATIONS 167 ^^ MISCELLANEOUS EXERCISES 229. Solve according to the general directions just given : 1. 2x'^ — 5x = 0. ^a ^'^ 2 a; oq lb. — — - = Zo. 2. a;2-30=13a;. 4 3 1 18. ,.l^ + _3_=4. 3. r^ + 27r = -U0. 17. ^,_^^_^^ ^ 4. a;2-12a; = 0. 5. 18a;2 + 6a; = 0. '" '^^'+^ ' ^-^ 6. 6a;2-2a;-16 = 0. 19. ?iil^-^-=:i = i a; - 3 a; - 2 5 7. 8a;2_3 = _2x. 2 o 20. -Jl— = -^—-\-8. 8. 7a;2 + 2a; = 32. 2/ + 3 y + 3 9. 5a;'' = 4(a;-10). gl. ^-±1 + ^^-+1? = 7. V + 5 y + 6 10. a;2 - 4.3 a; = 27.3. a;-3 , a; + 2_23 11. x^ + .25x=.015. 22. __ + -_^__. 12. _J_ + _5_ = 12. 03 2a; + l 5^a;-8 a; + l x-1 3 ■ l-2a; 7 2 13. — JL_=r^. 24 2-fc±2)^ '■+^ % + l) 8 ^*- ^-^rrr r-3 14 a;' ar'-2a;^35 3a;- 1 , 2a; + l_2a;-4 9'^3a;-6 4* 15. ^^_'l^:z^=^±2. 26. Find roots to the nearest hundredth : 27. x^-2x-2 = 0, 30. 2/2 + 4 2/ + 2 = 0. 28. z''-\-2z-l=:0. 31. 2,92 + 4.9-7 = 0. 29. 7;2-f 4v + .4 = 0. 32. 2 a;2- 5 a; + 1.2 = 0. a;-l ' a;+ 1 a; — 2 5.V-2 15- '*' - 2/' + 2y 2/^ + 2y 168 QUADRATIC EQUATIONS LITERAL EQUATIONS 230. The methods of solution for literal quadratic equations are the same as for numerical quadratics. Results may be tested by substituting simple numerical values for the literal known numbers. EXERCISES 231. Solve for x by the method best adapted, and verify : 1. a;2-6 = 0. 2. 6aaj2_54a^ = 0. 3. x^ — cd = GX — dx. 4. aj2- 4 6a; -12 62 = 0. 5. x'^^2bx:=^b\ 6. cc2 + 3 aaj = 10 o?, 7. x^ — ax -\- bx -\- ex = 0. 8. {a-'Xy=(Zx'\-a){x—a). 9. abx^ + a^x — b^x — ab, 10. aj2_45a;_752^0. 11. ax^=(a-b)(a'^-¥)-bx\ 12. 5cx-2x''-2c' = 0. 13. 16x^+Sa^-16ax = 0, 14. (c^ 4- l)a; = cx^ + c. 15. a'^x'^-\-2ax^=(a'^-iy-x\ 16. 4:x'^-\-12ax-7 a'^ = 0. 17. 5x^-10bx-7b^=0. 18. 6ax^i-abx=2(6x + b). 19. a;^ — (6— a)c = ir(a — 5 + c). 20. (b — G)x'^ + (c — a)x=b — a, 21. a X _ab X a X 22. X ^— ^ _ A a -{- b X 23. aV b^_2ax ¥ c^~ c 24. 2x(b-x)_b 3b-2x 4 25. a 5 X fl?2 3"^ 4 3a~ ' 26. a:2 -f 1 1 X n^x — 2 71 2 — naj ri, 27. X , 2 ^ — 1 _ ^ ^^ a + l a; x{a-\-l) 28. ^ 1 ^ ^-0 a; — a X— b 29. 2x — a r.__ 4ta b ~ 2x-b 30. x + a X ~ a _ a^ -\-b^ X -\-b X — b x^ — b'^ 31. bx ,_a(i»4-2 6) a — X ' a-{-b 32. 1 111 X — c G d X— d QUADRATIC EQUATIONS 169 RADICAL EQUATIONS 232. The student has learned how to free radical equations of radicals, the cases in §§ 209, 210, being such as lead to simple equations. The radical equations given here lead to quadratic equations, but the methods of freeing them of radicals are the same as in the cases already considered. It was shown in § 211 that the processes of rationalizatioji and involution, used in freeing radical equations of radicals, are likely to introduce roots that do not verify in accordance with the convention adopted, and in § 228 certain precautions against introducing roots by clearing of fractions were given. It is important, therefore, to test the roots found in the solution of equations to see whether any are extraneous, as well as to examine the processes employed in reducing equa- tions to see whether any roots have been removed (§ 109). EXERCISES 233. Solve and verify, rejecting roots that do not satisfy the given equation, and accounting for roots that otherwise might be lost : 1. 2x- x-\- 3x- Find -3Va 2Vx = - -y/x- 7. 8 roots 9. 10. 11. 12. 13. ; = 2. 4. V25- -6a;+ V25 4-6a;= = 8. 2. 3. = 3Vx, 5. VI- F-3-l = 0. 6. V5- Vaj2 _ ^2 ^ v^ _|_ 5 y^ ^ 5 2a;- 2= VI -a;. X = V3 + X Va; — ^5. Va; 4- 3 + V4 a? + 1 - VlO bo the nearest hundredth : a; + 4 = 0. Vx^-\-9=^-s/2x-\-3 ■V2x- -3. V2 a; + 50 = Va? Vaj + 2. Va:2-f-3= V2a; + 1 V2a;- Va; + 6 = V3 a; - 2, -1. V2a;= Vaj + 2 + 1. 14. Va;2 + 5 = V2 a; + 3 Va; - 2. 170 QUADRATIC EQUATIONS Solve for x^ and verify as directed on page 169 : 15. V3a;-5+ Va;-9 = V4a;-4. 16 V^ + 2 g — -\Jx — 2 g _ x -^x-2a^ ^x + 2a 2g' 17. X -f Vic^ + m2 : ^ ^^ Vi»2^ m^ 18. a; + Vi^2 _ a2 = - v: X'' — g^ 19. 2^-h V4a^^-1 ^^^ 2 a; - V4 i»2 - 1 20. J^E« + J^±]2 = a^. ^a; + g ^a; — g 21. Vaj + «'- Va;-2g2= V2aj-5g2. 22. Vmn — a; — Va; Vmn — 1 = Vm^i Vl ■ X, Problems 234. 1. The product of two numbers is 14 and their sum is 9. Find the numbers. 2. Separate 16 into two parts whose product is 48. 3. Separate 24 into two parts whose product is 128. 4. Find two consecutive integers whose product is 156. 5. The sum of the squares of two consecutive integers is 2^h. What are the numbers ? 6. The difference between a certain number and its recipro- cal is ^f~. Find the number. 7. The sum of a certain number and its reciprocal is ^f. Find the number. 8. The sum of the reciprocals of two consecutive integers is Y^Y- Find the integers. 9. If g times the reciprocal of a number is added to the number, the result is g + 1. What is the number ? QUADRATIC EQUATIONS 171 10. The length of a sheet of paper is 14 inches more than its width and its area is 912 square inches. Find its length. 11. Find two consecutive even integers the sum of whose squares is 2(a^ -\-l), 12. A rectangular garden is 12 rods longer than it is wide and it contains 1 acre. What are its dimensions ? 13. The area of a car floor is 306 square feet.. If its length is 2 feet more than 4 times its width, what is its width ? 14. The area of a tablet is 2838 square incheSo If its length exceeds its width by 23 inches, what are its dimensions ? 15. An ice bill for a month was $4.80. If the number of cakes used was 4 less than the number of cents paid per cake, how many cakes were used ? 16. The height of a box is 5 feet less than its length and 2 inches more than its width. If the area of the bottom is 8| square feet, what are the dimensions of the box ? 17. A roll of parchment was worth $ 24. If the number of skins it contained was 20 more than the number of cents each skin cost, how many skins were there in the roll ? 18. The sum of the three dimensions of a block is 35 feet and its width and height are equal. The area of the top ex- ceeds that of the end by 50 square feet. Find its dimensions. 19. The sum of the three dimensions of a box is 58 inches and its length and width are equal. The area of the bottom exceeds that of one end by 176 square inches. Find its height. 20. A bale of cotton contains 21 cubic feet. Its length is 41 feet, and its width is -^^ of a foot less than its thickness. Find its width ; its thickness. 21. A man sold raisins for $480. If he had sold 2 tons more and had charged $ 20 less per ton, he would have re- ceived the same amount. How many tons of raisins did he sell? 172 QUADRATIC EQUATIONS 22. If a beet-sugar factory in Colorado sliced 200 tons less of beets per day, it would take 1 day longer to slice 6000 tons of them. How many beets are sliced per day ? 23. The senior class at a school had a banquet that cost $ 75. If there had been 5 persons less, the share of each would have been $ .50 more. How many persons were there in the class ? 75 Suggestion. — Let x=the number of persons. Then, — = the amount X each paid and , the amount each would have paid had there been 6 X— 5 persons less. Hence, = - • X — 5 5c 2 24. A party of people agreed to pay $ 12 for the use of a launch. As 2 of them failed to pay, the share of each of the others was 50 cents more. How many persons were there in the party ?. 25. A bricklayer and his helper in a certain day laid 1500 bricks. If they had laid 25 bricks more per hour and had worked 2 hours less time, they would have laid 1400 bricks. How many bricks did they lay per hour ? 26. A rectangular park, 60 rods long and 40 rods wide, is surrounded by a street of uniform width, containing 1344 square rods. How wide is the street ? 27. Two persons started at the same time and traveled toward a place 90 miles distant. A traveled 1 mile per hou;r faster than B, and reached the place 1 hour before him. At what rate did each travel ? 28. If the rate of a sailing vessel was 1^ knots more per hour, it would take \ of an hour less time to travel 150 knots. Find the rate of the vessel per hour. 29. A man rode 90 miles. If he had traveled ^ of a mile more per ho\ir, he would have made the journey in 10 minutes less time. How long did the journey last ? 30. A picture that is 18 inches by 12 inches has a frame of uniform width whose area is equal to that of the picture. Find the width of the frame. QUADRATIC EQUATIONS 173 31. A tank can be tilled by two pipes in 24| minutes. If it takes the smaller pipe 10 minutes longer to fill the tank than it does the larger pipe, in what time can the tank be filled by each pipe ? Suggestion. — Let x — the number of minutes required by the larger pipe and x -}- 10 = the number required by the smaller pipe. Then, - + ^^ — = — • ' X a; + 10 24i 32. A tank can be filled by two pipes in 35 minutes. If the larger pipe alone can fill it in 24 minutes less time than the smaller pipe, in what time can each fill the tank ? 33. A and B together can do a piece of work in 3 days. If it takes A working alone If days longer than it does B, in how many days can each do the work alone ? 34. A cistern can be emptied by two pipes in 3^ hours. The larger pipe alone can empty it in 1^ hours less time than the smaller pipe. In what time can each pipe empty the cistern ? 35. A farmer bought a horse for x dollars and sold it for $ 75, thus making a profit of o^ %. Find x, 36. A jeweler sold a clock for $ 24, thus gaining a per cent equal to the number of dollars the clock cost. How much did the clock cost ? 37. If a man puts $ 2000 at interest, compounded annually, and at the end of 2 years finds that it amounts to $ 2121.80, what rate of interest is he receiving ? 38. Find the price of eggs per dozen, when 2 less for 30 cents raises the price 2 cents per dozen. 39. By receiving two successive discounts, a dealer bought for $ 9 silverware that was listed at $ 20. What were the dis- counts in per cent, if the first was 5 times the second ? 40. Each page of a book of 400 pages was 10 inches by 6 inches. In later editions, the publishers saved 1550 square inches of paper by cutting down the margin equally on every side. By what width was the margin reduced ? 174 QUADRATIC EQUATIONS Formulae 235. Solve the formula : 1. ^^ = 4 7rr2, for r, 4. V= -^ 7rd% for d. nd^ V4,ah 4r2_ 3. i^= ^-^J^ ,for>S. / - R 7. h =r— V^^ — (^ ^t;)2, for i^. 8. The formula A = bh gives the area ^ of a parallelogram in terms of its base b and altitude h. It the area of a parallel- ogram is 96 square feet and its base is 4 feet more than twice its height, what is its height? its base? 9. The area JL of a trapezoid is ex- A V pressed by the formula A = ^h(a + b). / \ \ If the lower base a of a trapezoid is 5 feet / I \ longer than the upper base 6, the altitude Z ! A /i is 1 foot shorter than 6, and the area is 92 square feet, what are its dimensions ? 10. The square of the hypotenuse (K) of a right triangle is equal to the sum of the squares of the other two sides (a and 6). Write the formula for li, "%1. From the above formula and the figure, de- duce a formula for the diagonal (d) of a square whose side is s. 12. Find the diagonal of a square whose side is 8 feet. 13. If a baseball diamond is 90 feet square, what is the dis- tance, to the nearest tenth of a foot, from first base to third base? 14. Write the formula for the diagonal of a rectangle whose length is a and width is b. Solve for a. 15. The diagonal of a rectangle is 10 feet long. The rec- tangle is 2 feet longer than it is wide. Find its dimensions. QUADRATIC EQUATIONS 175 16. Express by an equality the area (A) of a square whose side is a; the area (A) of a hol- low square a units on the outside and b units on the inside. 17. The area of a hollow square is 40 square inches. If the out- side dimension is twice the inside dimension plus 1 inch, what is the inside dimension? 18. The area of a flat ring is the difference between the areas of two circles of radii R and r, respec- tively, or A = ir{R'^ — r'^). Solve for r. 19. When the area of a ring is 1320 square feet and E = r + 2, what is the value of r? (Use TT = 3i.) 20. The pressure P of the wind against a surface, in pounds per square foot, is computed from P= .005 F^ in which V is the velocity of the wind in miles per hour. Solve for F. 21. What is the velocity of the wind when it exerts a total pressure of 2.7 tons on a sign board 30 feet by 10 feet? 22. The formula for the volume of the frustum of a square pyramid is F IS when F = i /i(a2 -\-ab + ¥), Find a and h = 98, /i = 6, and a = 6 + 2. 23. If jL is the length of a pendulum that oscillates once in T seconds, and / the length of Ij T^ one that oscillates once in t seconds, then — = — . Solve for t. 24. If a pendulum 39.1 inches long oscillates once per second, how often does a pendulum 351.9 inches long oscillate? 25. A line is said to be divided in extreme and mean ratio when the longer part is a mean proportional between the whole line and the shorter part. Write the proportion for a line a whose longer part is b and shorter part c. Solve for b, 26. Divide a line 4^ feet long in extreme and mean ratio. 176 QUADRATIC EQUATIONS EQUATIONS IN THE QUADRATIC FORM 236. An equation that contains but two powers of an un- known number or expression, the exponent of one power being twice that of the other, as ax^"" -f hx^ + c = 0, in which n rep- resents any number, is in the quadratic form. EXERCISES 237. Solve the following equations : 1. aj4 + a;2-20 = 0. ' Solution x* + x2-20 = 0. (x2-4)(x2 4-5) =0. .•.a:2- 4 = or 0^24. 5 = 0, and x = ±2 or ± V— 6. Any one of these values substi- tuted for X in the given equation satisfies the equation, and is there- fore a root of it. 7. x^ - aji = 6. Solution x^ — o:^ = 6. .-. xi = 3or -r2; whence, x = 81 or 16. Since x = 16 does not verify, 16 is not a root and should be rejected. 8. x*-3iC* = -2. p9. a;*-f 3a;f-28 = 0. 10. a;-f3V^=4. 11. aj3-4a;3=12. 12. x^ = YI x^ -U. 2. o^-\-ll x?-^4. = 0, 3. 3a;4 + 5ic2_8^Q, 4. 5a;^ + 6aj2-ll = 0. 5. (a:-2)2+3(a;-2) = 10. 6. (aj2-fl)2+4(aj2 + l)=45. 13. a; — 4a;*-}-3x^ = 0. 11 1 Solution. —Factor, x^ (x^ _ 1) (x^ — 3) = ; that is, x^ = 0, X* — 1 = 0, or x^ - 3 = ; whence, x^ = 0, 1, or 3. Raise to the third power, x = 0, 1, or 27. Each of these values of x satisfies the given equation and is a root of it. 14. x^ —^x — bx^ = ^, 15. x—'dx^ +2 x^ = 0. 16. a; + 2aj* -3a;3 = 0. 17. 5 a; = x^x -|-.6 Vic. 18. 3a; = aj-^^-f 2^^2, 19. 2 X -\- -sjx = 15 x^x. Y QUADRATIC EQUATIONS 177 20. Solve x'-Sx -\- 2^x'- - 3a; + 6 = 18. Solution. — Adding 6 to both members, we have a:2 _ 3 X + 6 + 2Vx'^-Sx -f 6 = 24. (1) Put p for Vx'^ — Sx -\- 6 and p'^ for x^^ — Sx + 6. Then, p'^ -\-2p = 24. (2) Solving, we have p = 4 or — 6 ; (3) that is, \/x=2 - 3 X + 6 r^ 4, (4) or Vx2- 3x + 6 = - 6. (6) Square (4), a:^ _ 3 ^ + 6 = 16. (6) Solving (6), we have x = 5 or ~ 2. Since, in accordance with § 211, the radical in (6) cannot equal a nega- tive number, Vx:^ — 3 x + 6 = — 6 is an impossible equation. Hence, the only roots of the given equation are 5 and — 2. Solve and verify results : 21. x-2^x~^=7. 22. x'^-x-{- Wx^ - a; - 8 = 20. 23. Solve the equation a?^ — 9 a;^ 4- 8 = 0. Solution. a;6 - 9 x^ + 8 = 0. (1) Factor, (x^ ~ 1) (x^ - 8) = 0. (2) Therefore, x^ - 1 = 0, (3) or x3 - 8 = 0. (4) If the values of x are found by transposing the known terms in (3) and (4) and then extracting the cube root of each member, only one value of X will be obtained from each equation. But if the equations are factored, three values of x are obtained for each. Factor (3), (x - 1) (x2 -f- x + 1) = 0, (6) and (4), (a; - 2)(x2 + 2x + 4)= 0. (6) Writing each factor equal to zero, and solving, we have : From (5), x = 1, ^ (- 1 + V=:T), i (" ^ - ^^^)- (7) From (6), x = 2, - 1 + V^^, - 1 - V^^^. (8) Note. — Since the values of x in (7) are obtained by factoring x^ — 1 = 0, they may be regarded as the three cube roots of the number 1 . Also, the values of x in (8) may be regarded as the three cube roots of the number 8 (§ 166). milne's sec. course alg. — 12 178 QUADRATIC EQUATIONS Find the three cube roots of : 24. 27. 25. -27. 26. 64. 27. 125. 28. -64. Solve : 29. ic^-81 = 0. 30. a^-64 = 0. 31. a;^-f-4a^-8a; + 3 = 0. Solution a* + 4 a;3 - 8 X + 3 = 0. Factor, § 75, (x - l)(x + 3)(x2 + 2 a; - 1)= 0; whence, x = 1, — 3, — 1 ± \/2. 32. x'-j-2oc^-x = 30. 34. a;4 + 2if3 4.5a.-2 4-4a; = 60. 33. aj^-2a^-f i» = 132. S5. 0^-60^ + 15 x^ -IS x=: -S. 36. ^- + ^±1 = ?^. x + 1 x^ 12 Suggestion. — Since the second term is the reciprocal of the first, put p for the first term and - for the second. P 37. ?^±-%-^_ = 2. 38. 5l±l + _^ = ? 2 x^-\-x 4 a;2^1 2 MISCELLANEOUS EXERCISES 238. Solve the following equations : 1. -\/x + 3 V^ = 30. 4. a; = 11 - 3 V^TT^ 2. aa;2n ^ 5a;» + c = 0. 5. a;« -h 9 .^•3 + 8 = 0. 3. aj - 7 ic^ + 10 a;^ = 0. 6. a;^ - 5 x~i +4 = 0. 7. x'^-5x + 5Vx^ - 5 a; -f 1 = 49. 8. (a;2-aj)2-(aj2-a;)-132 = 0. 1 8 9. a;2 + a; + 1 — x^ + x-\-l 3' x^ —1 X 6 V^/\^/ QUADRATIC EQUATIONS 179 SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS 239. Two simultaneous quadratic equations in two unknown numbers generally lead to equations of the fourth degree, and they cannot be solved usually by quadratic methods, but some simultaneous equations involving quadratics are solvable by quadratic methods, as in the following cases. 240. When one equation is simple and the other of higher degree. Equations of this class may be solved by substitution. 241. 1. Solve the equations EXERCISES 'x-{-y: 5, a;2 + 2 2/2 = 17. Solution. — From (1), Substitute (3) in (2), Solving (4), we have Substitute 3 for x in (3), Substitute y- for x in (3) , y = b — x. x2 4-2(5-x)2=:17. a; =3 or-V. y = 2. 2/ = f Hence,x and y each have two corresponding values associated as follows x = 3;-V; 2/ = 2;f (1) (2) (3) (4) (5) (6) (7) Solve the following equations : 2. 1^ = ^^' [a:2 + i/2 = 40. ^ fa: + 2/ = 3, * [a;2 + 2a;?/ = 8. aj2 _ 2 2/2 = 7, 6. 4. 7. 3 2/2 - ;22 ^ 8, 2yz=2-z, (aj4-2/ = 3. j2 2/(0^-2)== 7, \2x = ^y. 242. An equation that is not affected by interchanging the unknown numbers involved is called a symmetrical equation. «2 -f xy -I- y2 = 7 and 3 a;2 + 3 y2 = 4 are symmetrical equations. 180 QUADRATIC EQUATIONS 243. When both equations are symmetrical. Though equations of this class may be solved by substitu- tion, it is better to solve first for x -{- y and x — y and then for X and y. EXERCISES 244. 1. Solve the equations | ^ + 2/ = H, (1) ^ [xy = 30, (2) Solution. — Square (1), x- + 2xy -\-y^ = 121. (3) Multiply (2) by 4, 4^xy = 120. (4) Subtract (4) from (3), x^ —^xy + y^ = l. (5) Extract the square root, x^y =± 1. (6) From (1) + (6), ic = 6or5. From (1) - (6), 2/ = -5 or 6. 2. Solve the equations ^ ^ "*" 2/ — ? K ) ^ \x + y==5, (2) Suggestion. — From the square of (2) subtract (1) ; then subtract this result from (1) and proceed as in exercise 1. 3. Solve the equations ^ ^ ' \ ^ \x + y=l. (2) Solution. — Raising (2) to the fourth power, we have x* 4- 4 x3?/ + 6 x:^y^ + 4 x?/^ + ?/* = 1. (3) Subtract (1) from (3), 4 x^i/ + 6 xhf -^4xy^ =- 96. (4) Divide (4) by 2, 2x^y + S x'Y + 2 xy^ = -48. (5) 2xy X square of (2), 2 x^y + 4 x^y^ + 2 xy^ = 2 xy, (6) Subtract (5) from (6), xV -2xy = 48. (7) Solve for xy, xy = — 6 or 8. (8) Equations (2) and (8) give two pairs of simultaneous equations, > + 2/ = 1 and 1 * + 2' = 1 , xy =— a [ xy = S Solve as. in exercise 1. The corresponding values of x and y are : fx = 8;-2; ^.(l+VUgi); i(l-Vi:31); \y^^2; 3; |(l_V-31); Kl+V-31). QUADRATIC EQUATIONS 181 Solve the following equations \xy = Q>, 9. 5. f-^y'=''^ t a?2/ = 4. p + 2/ = 4, 6. ^ 11. la;2 + a;t/4-i/2 = 13. 7. 12. lar'+^ = 117. 8. ^^ + a^ + 2/^ = 57, aj2 + 2/2 = 50. 13. |a;2+y2^26, = 21. |a^ + y2 = 13, 11. {a;y = 144. 19, 0^ + 2/^ = 17, = 21, = 7. 245. An equation all of whose terms are of the same degree with respect to the unknown numbers is called a homogeneous equation. 3 x2 + cc?/ = 1/2 and x^ — 2 ?/3 = are homogeneous equations. An equation like 2 x^ -h xy + ^/^ = 39 is said to be homoge- neous in the unknown terms. 246. When both equations are quadratic, one being homogeneous. In this case elimination may always be effected by substitu- tion, for by dividing the homogeneous equation through by ^-, it becomes a quadratic in '_ . The two values of - obtained y y from this equation give two simple equations in x and y, each of which may be combined with the remaining quadratic equa- tion as in §§ 240, 241. Thus, ax^ -f hxy + cy"^ = is the general form of the homo- geneous equation in which a, b, and c are known numbers. Dividing by y"^, we have a (^] + 6( ^ )-f- c = 0, a quadratic ^ \yj \yJ in - . y 182 QUADRATIC EQUATIONS EXERCISES 247. 1. Solve the equations ^ o . o ^ ) ^ \5x^-Jr4:xy-y^=0. (2) Solution. — Dividing (2) by y^ gives 5(-j +4f-j-l = 0, a quadratic in - which may be solved by factoring or by completing the square. y To avoid fractions, however, (2) may be factored at once ; thus, ,\ y =— X or 5 x. Substituting — x for y in (1), simplifying, etc., we have x2 -f-4a; = 5. Solving gives x = 1 or -- 5. (3) .•. y = — X =— 1 or 5. (4) Substituting 5 x for 2/ in (1), simplifying, etc., we have x2 - 2 X = 5. Solving gives x = 1 + \/6 or 1— V6. (6) .-. y = 6x = 6(1 4- V6) or 5(1 - \/6). (6) Hence, from (8), (4), (5), and (6) the roots of the given equations are x=l; -5; 1+V6; 1-V6; .2/=-l; 5; 5(1 +V6); 5(1-^6). Solve the following equations : {2x'^-3y-y^ = S, jSx'^ -7 xy - iOy^ = 0, I6a;2-5a^~ 62/2 = 0. [x'' - xy -12y^ = S. l5x'^-\-Sxy — 4:y^ = 0, ix'^-xy-y^ = 20, [xy + 2y'' = 60, '^' [sx^ ^13xy -}- 12y^ =0. 2x^-xy-y'^ = 0, \Sx'^ -7 xy -h 4:y'^ = 0, 4x'^ + 4:xy + y^=:S6. ' [5x^ - 7 xy -^Sy^ =:^ 4:, l6x'^-{-xy-12y^=:0, ix^ -{- y^ + x - y = 12, [x'^ + xy-y^l. ^' [3x'^-i-2xy-y^ = 0. QUADRATIC EQUATIONS 183 248. When both equations are quadratic and homogeneous in the unknown terms. In this case either : Substitute vy for x, solve for 2/^ in each equation, and com- pare the values of i/^ thus found, forming a quadratic in v. Or, eliminate the absolute term, forming a homogeneous equation ; then proceed as in § § 246, 247. EXERCISES 249. 1. Solve the equations | ^' " ^^ + 2/' = 21, (1) \y^-2xy = -lb. (2) First Solution. — Assume x = vy. (3) Substitute (3) in (1), v^ - '^V^ + V'^ = 21. (4) Substitute (3) in (2), y2^2vy^ = - 15. (6) Solve (4) for y^, 2/2 = 21 ^^^ v^ — t? + 1 Solve (6) for 2/2, y2 = ^^. (7) 2r — 1 15 _ 21 ,ov Compare the values of y^, gv- 1 "~ v^-v -\-l Clear, etc., 5 v'^ - 19 v + 12 = 0. (9) Factor, (i, _ 3)(5i, _ 4) = 0. (10) .'.v = S or f (11) Substitute 3 for v in (7) or in (6), y = ± VS, and since x = vy, x —± SVS. Substitute f for v in (7) or in (6), 2/ = i 5, 1 and since x= vy, x = ± 4. J When the double sign is used, as in (12) and in (13), it is understood that the roots shall be associated by taking the upper signs together and the lower signs together. Hence, |^ = 3V3; -3V3; 4; -4; [y=y/S; - V3 ; 5; -5. Suggestion for Second Solution. — Multiplying (1) by 5 and (2) by 7, and adding the results, we eliminate the absolute term and obtain the homogeneous equation x2 — 19 xy + 12 y2 = 0, which may be solved with either of the given equations, as in § 246. (12) (13) 184 • QUADRATIC EQUATIONS Solve the following equations : ' Xy'^ — xy = —1. ' [2 xy — y'^ = 16. (x' + xy = 2i, {x^^-xy-y^=20, ' \xy + 2y^ = 16. [ x'- - 3 xy -\-2 y'' = S, 4. {x(x-y)=6, i2x''-3xy-\-27/=100, U2 _ 3 2/2 = 3. ■' \x^-y^ = 75. ^ >2 + 2/2=13, ^ ix'-5xy+3y^ = S, xy -{-y^ = 15 ' [Sx'^ + xy -^y^=z 24. 250. Special devices. Many systems of equations belonging to the preceding classes and others not included in them may be solved readily by special devices, as illustrated in the following exercises. Though it is impossible to lay down any fixed line of procedure, the object often aimed at is to find values for a7iy two of the expressions x + y, x — y, and xy from which the values of x and y may be obtained. EXERCISES 251. 1. Solve the equations , ^ ' i ^y + .r = o- (1) (2) Solution Add (1) and (2), 5c2 + 2 x?/ 4- 2/2 - 25. (3) Extract the square root, x + i/ = + 5or — 6. (4) Subtract (2) from (1), x^ - 2/2 = 16. (5) Divide (5) by (4), X — 2/ = + 3 or - 3. (6) Add (4) and (6), etc., X = 4 or — 4. Subtract (6) from (4), etc. , 2/ = 1 or - 1. Note. — The first value of x ^ y corresponds only to the first value of X -h ?/, and the second value only to the second value. Consequently, there are only two pairs of values of x and y. Ix^-^y'^-^-x-^y^U, QUADRATIC EQUATIONS 185 2. Solve the equations , [xy = S. Suggestion. — Adding twice the second equation to the first, we have a;2 + 2 xy + 2/2 + X + 2/ = 20, or (a: + 2/)2 + (x + y) = 20, which may be solved for x + y and the results combined with xy = S, Symmetrical except as to sign. — Whether both equations are symmetrical, or one is symmetrical and the other would be so if some of its signs were changed, or both are of the latter type, the method of solution is the same as in § 243. f 0^2 + 2/2 = 53, (1) 3. Solve the equations [x-y = 5. (2) Suggestion. — Subtract the square of (2) from (1), obtaining 2 xy =28 ; add this equation to (1), and solve for x -\- y. 4. Solve the equations 'i + i = 74, 1-1 = 2. X y Suggestion. — Proceed as in exercise 3, solving for- + -, then for 11 ^ y - and - , and finally for x and y, X y Whether the equations are symmetrical or symmetrical except for the sign, it is often advantageous to substitute u + v for X and u^ v for y. r. . . . lx' + y' = S2, (1) 5. Solve the equations ^ \x-y = 2. (2) Solution. — Assume x= u -{- v, (3) and y = u — V. (4) Substitute these values in (1), + ?fc* - 4 uH + 6 u'^v^ - 4 uv^ + V* = 82, (6) and in (2), 2 1? = 2. (6) Divide (5) by 2, w* + 6 1*2^2 _^ 1,4 _ 41. (7) Divide (6) by 2, v = I. (8) Substitute 1 for v in (7) and solve, w = ± 2 or ± V— 10. (9) 186 QUADRATIC EQUATIONS Hence, substituting (8) and (9) in (3) and (4), we find the corre- sponding values of x and y to be (2) x=:3; -1; l+V-lO; I-V-IO; y=l\ -3; _1+V^^^^l0; _1-.V^^I0. Note. — The given system of equations may be solved also by the method of exercise 3, § 244. Division of one equation by the other. — The reduction of equations of higher degree to quadratics is often effected by dividing one of the given equations by the other, member by member, 6. Solve the equations I x^ — xy + 2/^ = 7. Solution. —Divide (1) by (2), x^ -\- xy -\- y'^ = 13. (3) Subtract (2) from (3), 2 xy = 6 ; whence, xy = 3. (4) Add (4) and (3), x^ -^2xy-\-y'^ = 16. (5) Subtract (4) from (2), x2 - 2 xy 4-2/^ = 4. (6) Extract the square root of (5), x + 2/ = 4or— 4. (7) Extract the square root of (6), x — y = 2 or — 2. (8) Solving these simultaneous equations in (7) and (8), we have x = 3; 1; -1; - 3; 2/ = !; 3; -3; -1. Note. — Since (7) and (8) have been derived independently, with the first value of x -}■ y we associate each value of x — y in succession, and with the second value of x + y each value of x — ?/ in succession, in the same order. Consequently^ there are fotir pairs of values of x and y. {^-f = ^^, (1) 7. Solve the equations ^ \x-y = 6. (2) Suggestion. — Divide (1) by (2) and solve the system made up of this result and (2). Elimination of similar terms. — When the equations are quad- ratic and each is homogeneous except for one teriUj if these excepted terms are similar in the tvro equations, they may be eliminated and the solution of the system be made to depend on the case of § 246. QUADRATIC EQUATIONS 187 Some equations belonging to this class, namely, those that are homo- geneous except for the absolute term, have been treated in § 248. 8. Solve the equations x^ + 2xy = ^y, [2 x'^ — xy -\- y^ = 2 y. Suggestion. — Eliminate the terms containing y and proceed as in § 246. Using the methods illustrated in exercises 1-8, solve : 9. 10. 11. 12. x^ + xy = 30, 13. . a^-\-y' = 17, a^ + y' = 6. x-y = l. pq = — 15. 36, 14. c2- cfZ + (^ = 3. y-z = l. 15. < x-y = ^. a^ y^ 1-1=1. la; y 16. x'^ -\-2xy = 7y, 2x^-xy + y^ = Sy MISCELLANEOUS EXERCISES 252. Solve the following systems of equations : 1. 2. 3. 5. 6. xy = — 4:, j3aj2-2 2/2=19, [2a'2-32/'= 6. I a;2 -1-2/2 = 52, [30;= 2y. \x'+y' = S2, x-{-y = 4:. x^ + xy=: 77, xy — y^ = 12. 2x-y=2, 2 a;2 ^ 2/^ = |. 7. 8. 10. 11. 12. lx' + 3xy = y' + 23, \x + 3y = 9. faj2-f 4a:-f 3 2/ = -l, \2x^ + 5xyi- 2 2/2 = 0. faj2+3a!2/-2/^ = 43, \x + 2y = 10. f2a;2-f-oa;2/-f 2/^ = 20, [5ic2 + 4?/2 = 41. {2xy^f = 12, [3 xy +5x^=104:. x^ -|- xy + 2/2 = 84, X — Vxy -{- yz=z6. 188 QUADRATIC EQUATIONS Solve the following systems of equations : ' [8 0^ + 2/^ = 65. " \x + y-\-xy = ll. |6a;2 + 6^/2 =13aJ2/, [x^ + y'^==^xy + ^, |a.'2-2/2 = 20. ^^' I 0^4 + 2/4 = 2. faj^-v^=175, \x^-lxy + 12y'^=:0, 15. ^ 20. ' u -r u ^ a;2 - 2/2 = 7. [xy + ^y =^2x + 21. ^^ U + 2/ = 10, ^^ j(a^ + 2/)(^' + ^')=65, I VaJ + V2/ = 4. * I (a; — 2/) (aJ^ — 2/^) = 5. ,^ faj3+2/' = 2252/, f ^2 ^2/ = ^ -2/' + 42, 23. i^ _ 2/2 = 75. [ aj2/ = 20. .T + 2/ + 2VSTi=24, a; — 2/ 4" 3 ■\Jx — y = 10. 24. 25. ^.2 4. 2/2 + 6 V.t2 -f- 2/' = 55, a;2 - 2/2 = 7. 6a;2/ + 9 2/2 + 2a;-62/~8 = 0, a;2 + 4 0^2/ + 42/2 - 4 a; — 8 2/ — 21 = 0. Suggestion. — The equations may be written in the quadratic form. Thus Ux-^Syy + 2(x^Sy)-S = 0, [ (x + 2yy-4:(x-{-2y)^21 =0. f a/'2 — xy = a^ -\- b^ ] 26. Solve for a; and y. [xy — y^ =:2ab J ^ , . lx-2y=:2(a + b) ]^ 27. Solve lor aj and y. \xy + 2y' = 2b{b-a)\ 28. Solve ^ for a and t. \^v= at J , s = 6 ^ + ^ a^2 1 29. Solve \ '^ tor i; and t. V = at QUADRATIC EQUATIONS 189 Problems 253. 1. The sum of two numbers is 16 and their product is 48. What are the numbers ? 2. The difference between two numbers is 4 and their product is 77. Find the numbers. 3. The product of two numbers is 108 and their quotient is 1^. Find the numbers. 4. The sum of two numbers is 8 and the sum of their squares is 40. Find the numbers. 5. The difference between two numbers is 2 and the dif- ference between their cubes is 26. Find the numbers. 6. The sum of two numbers is 82 and the sum of their square roots is 10. What are the numbers ? 7. The perimeter of a rectangle is 20 inches and its area is 24 square inches. Find its dimensions. 8. The product of two numbers is s^ and the difference between them is 8 times the smaller number. What are the numbers ? 9. The perimeter of a floor is 44 feet and its area is 120 square feet. Find its length and its width. 10. An electric sign is 10 feet longer than it is wide and its area is 6375 square feet. Find its dimensions. 11. The sum of the sides of two squares is 12. If the dif- ference between their areas is 3, what is the side of each ? 12. The area of a rectangular field is 3 acres and its length is 4 rods more than its width. Find its dimensions. 13. An Indian blanket has an area of 35 square feet. If its width were 1 foot less and its length 1 foot more, the former dimension would be ^ of the latter. Find its dimensions. 14. The product of two numbers is 18 less than 10 times the larger number and 8 less than 10 times the smaller number. Find the numbers. 190 QUADRATIC EQUATIONS 15. If a two-digit number is multiplied by its units' digit, the result is 24. If the sum of the digits is added to the num- ber, the result is 15. What is the number ? 16. The perimeter of a right triangle is 12 feet and its hypotenuse is 1 foot longer than its base. Find its base. 17. If a two-digit number is multiplied by the sum of its digits, the result is 198. If it is divided by the sum of its digits, the result is 5^. Find the number. 18. The denominator of a certain fraction exceeds its numer- ator by 1, and if the fraction is multiplied by the sum of its terms, the result is 3^. Find the fraction. 19. The base of a triangle was 7 inches longer than its alti- tude and its area was ^ of a square foot. Find the dimensions of the triangle. 20. The size of an oriental prayer carpet was 23 square feet. If the width was 10 inches more than ^ the length, what were the dimensions of the carpet ? 21. The difference between two numbers is 2 a and their product is h. Find the numbers. 22. A certain door mat has an area of 882 square inches. If its length had been 6 inches less and its width 5^ inches more, the mat would have been square. Find its dimensions. 23. I paid 75 ^ for ribbon. If it had cost 10 ^ less per yard, I should have received 2 yards more for the same money. How many yards did I buy, and what was the price per yard ? 24. A man expended $ 6.00 for canvas. Had it cost 4 cents less per yard, he would have received 5 yards more. How many yards did he buy, and at what price per yard ? 25. The central court of the New York State Capitol has an area of 12,604 square feet. What are the dimensions of the court, if the width is 2 feet more than twice the difference between the length and the width ? QUADRATIC EQUATIONS 191 26. The radius of one circle is | that of another circle. If the sum of the areas of the circles is 117 tt square feet, how long is the radius of each circle ? 27. A grocer sold carrots for $ 4.40. If the number of bunches had been 4 less and the price per bunch 1 ^ more, he would still have received $4.40. Find the price per bunch. 28. One machine sticks 720,000 pins into the papers per day. If the machine ran 2 hours longer daily and stuck into the papers 18,000 pins less hourly, the result would be the same. How long does the machine run per day ? 29. A merchant bought a piece of cloth for $ 147. He cut 12 yards that were damaged from the piece and then sold the remainder for $ 120.25 at a gain of 25 ^ per yard.' How many yards did he buy ? What was the cost per yard ? 30. A ship was loaded with 2000 tons of coal. If 50 tons more had been put on per hour, it would have taken 1 hour 20 minutes less time to load the whole amount. How long did it take to load the coal ? 31. A man packed 2000 pounds of cherries in boxes. If each box had contained 6 pounds more, he would have used 75 boxes less. How many boxes did he use and how many pounds of cherries did each contain ? 32. A farmer received 20 ^ less per bushel for oats than for rye, and sold 3 bushels more of oats than of rye. The receipts from the oats were $ 4.50 and from the rye $ 4.20. Find the number of bushels of each sold and the price per bushel. 33. Three men earned $ 87.36. If A had worked 3 days less he would have earned the same as B ; if 2^ times as long he would have earned the same as C. C earned $ 16.64 more than A and B together. Find the daily wages of each. 34. A boy has a large blotter, 4 inches longer than it is wide, and 480 square inches in area. He wishes to cut away enough to leave a square 256 square inches in area. How many inches must he cut from the length and from the width ? 192 QUADRATIC EQUATIONS 35. The total area of a rug whose length is 3 feet more than its width is 108 square feet. The area of the rug exclusive of the border is 54 square feet. Find the width of the border. 36. After a mowing machine had made the circuit of a 7-acre rectangular hay field 11 times, cutting a swath 6 feet wide each time, 4 acres of grass were still standing. Find the dimensions of the field in rods. 37. The amount of a sum of money for one year is $ 3990. If the rate were 1 % less and the principal were $ 200 more, the amount would be $ 4160. Find the principal and the rate. 38. My annual income from an investment is $ 60. If the principal were $ 500 less and the rate of interest 1 % more, my income would be the same. Find the principal and the rate. 39. A sum of money on interest for one year at a certain per cent amounted to $ 11,130. If the rate had been 1 % less and the principal $ 100 more, the amount would have been the same. Find the principal and the rate. 40. The fore wheel of a carriage makes 12 revolutions more than the hind wheel in going 240 yards. If the circumference of each wheel were 1 yard greater, the fore wheel would make 8 revolutions more than the hind wheel in going 240 yards. What is the circumference of each wheel ? 41. The town A is on a lake and 12 miles from B, which is 4 miles from the opposite shore. A man rows across the lake and walks to B in 3 hours. Returning, he walks at the same rate, but rows 2 miles an hour less than before. It takes him 5 hours to return. Find his rates of rowing and walking. 42. A, B, and C started at the same time to ride a certain distance. A and C rode the whole distance at uniform rates, A 2 miles an hour faster than C. B rode with C for 20 miles, and then by increasing his speed 2 miles an hour, reached his destination 40 minutes before C and 20 minutes after A. Find the distance and the rate at which each traveled. GRAPHIC SOLUTIONS QUADRATIC FUNCTIONS 254. Graphic solutions of quadratic equations in x. Let it be required to solve graphically, x^ — 6 a? + 5 = 0. To do this, we must construct the graph of f(x) =zx^ — 6x + 5, that is, of y = x^—6x + 5. The graph will represent all the corresponding real values of x and of x^ — 6x + 5, and among them will be the values of x that make x'^—6x + 5 equal to zero, that is, the roots of the equation x'^ — 6 x -{- 5 = 0. When the coefficient of a;^ ig ^ i^ as in this instance, it is convenient to take for the first value of a? a number equal to half the coefficient of x with its sign changed. Next, values of X differing from this value by equal amounts may be taken. Thus, first substituting x = o, it is found that i/ = — 4, locating the point ^=(3, —4). Next give vakies to x differing from 3 by equal amounts, as 2^ and 3^, 2 and 4, 1 and 5, and 6. It will be found that y has the same value for x = 3J as for x = 2J, for x = 4 as for x = 2, etc. The table below gives a record of the points and their coordinates. r n I ' E< 'E r\ i il •n 5ft D Imi ^ N j d'q \ 1 / r\ 1/ k J\^ J 3 2 B 1 1 _ X y Points 3 -4 A 2i, 3i -H B, B' 2,4 -3 C, C 1,5 D, i>' 0, 6 5 E, E' Plotting the points ^ ; B, B' ; (7, C ; etc., whose coordinates are given in the preceding table, and drawing a smooth curve tlirough them, we obtain the graph of y = x'^ — Qx -{- ^ 2iS shown in the figure. milne's sec. course alg. — 13 193 194 GKAPHIC SOLUTIONS Observe from the preceding graph and table that : " When x=3, x^— 6x-\-5=— A, which is represented by the negative^ OTdinsite PA. When x=2 and also when a? = 4, a;^ — 6 .t + 5 = — 3, which is represented by the equal negative ordinates MO and NO', When X = and also when x = 6, x"^ — 6x + 5 = 5, repre- sented by the equal positive ordinates OE and QE', The ordinates change sign as the curve crosses the a^axis. At D and at D', where the ordinates are equal to 0, the value oi x'^ — 6x -\-5 is 0, and the abscissas are x = 1 and x = 5. Hence, the roots of the given equation are 1 and 5. Note. — Half the coefficient of x with its sign changed, the number first substituted for x, is half the sum of the roots, or their mean value, when the coeflficient of x^ is +1. This will be shown in § 266. The curve obtained by plotting the graph of any quadratic function of the form ax"^ + &jr -|- c is a parabola. 255. Let it be required to solve each of the equations a,.2_8a; + 14 = 0, x^-Sx + 16 = 0, a;2-8a; + 18= 0. (1) (2) (3) The graphs corresponding to equa- tions (1), (2), and (3), found as in § 254, are marked I, II, and III, respectively. The roots of (1) are seen to be 0F= 2.6 and IT =5.4, approxi- mately. Since graph II has only one point, K, in common with the oj-axis, equation (2) appears to have only one root, 0K= 4. But it will be observed that if graph I, which represents two unequal real roots, V and OW, were moved upward two units, it would coincide with graph II. During this process the un- equal roots of (1), OF and OTF, would approach the value OK, which represents the roots of (2). 1 \ w l\l 1 1 ^ p ^' h 1 A ^ "^ /^ 7 v^^ ^ J] GRAPHIC SOLUTIONS 196 Consequently, the roots of (2) are regarded as two in number. They are real and equal, or coincident. The movement of the graph of ( 1 ) upward the distance JK^ or 2 units, corresponds to completing the square in (1) by adding 2 to each member. Since the roots of the resulting equation, x^ — 8x + 16 = 2, differ from those of (2) or from the mean value 0K= 4, by ± V2, or ± y/JK, it is evident that the roots of (1) are represented graphically by OK-^yJjK= 4 4- V2"= 5.414+, and 0^-\/J!^=4-V2 = 2.586-. Since graph III has no point on the aj-axis, there are no real values of x for which a;^ — 8 a; + 18 is equal to zero ; that is, (3) has no real roots. Consequently, the roots are imaginary. If graph III were moved downward 2 units, it would coincide with graph II. If the square in (3) were completed by subtracting 2 from each member, the roots of the resulting equation, x'^ — 8ic 4- 16 = — 2, would differ from the mean value by db V— 2, or i y/ LK. Hence, it is evident that the roots of (3) are represented graphically by OiT + VZiT^ 4 4- V^^, and OK-y/ZK=^--\/^^, The points J, K, and Z, whose ordinates are the least alge- braically that any points in the respective graphs can have, are called minimum points. 256. When the coeiticient of ar^ is + 1, it is evident from the preceding discussion that : Principles. — 1. The roots of a quadratic in x are equal to the abscissa of the minimum point, plus or minus the square root of the ordinate with its sign changed. 2. If the minimum p)oint lies on the x-a.xis, the roots are real and equal. 3. If the minimum point lies below the x-axis, the roots are real and unequal. 4. If the minimum point lies above the x-axis, the roots are imaginary. 196 GRAPHIC SOLUTIONS EXERCISES 257. Solve graphically, giving real roots to the nearest tenth . 1. x2 + a; - 2 = 0. 2. a;2 — a; + 6 = 0. 3. x^-3x-4. = 0, 4. x''-2x-15 = 0. 5. aj2 + 5 aj -h 14 = 0. 11. 6. a;2 4- 3 0^ - 10 = 0. 7. ar^ - 7 .-^ -h 18 = 0. 8. 0^2 + 4 aj -h 45 = 0. 9. a;2+6a;-27=:0. 10. aj2 14 i 51 := 0. 2a;2. 6 = 0. Suggestion. — Reduce the equation to the form x'^ -\- px + q = 0, in which the coefficient of x^ is -f- 1, and proceed as in the exercises above. 12. 2a;2-aj-15 = 0. 14. 6aj2_7.^^20. 13. 3a^2_^5aj-28 = 0. 15. 8 a;^ + 14 a; = 15. 258. Graphs of quadratic equations in x and y. EXERCISES 1. Construct the graph of the equation a;^ + ?/2 = 25. Solution. — Solve for y, y = ± V25 — x'^. Since any value numerically greater than 5 substituted for x will make the value of y imaginaiy, we substitute only values of x between and in- cluding — 5 and +5. The corresponding values of x and ?/, or ± V25 — x'-^, are recorded in the table below. It will be observed that each value substituted for x, except ± 5, gives two values of ?/, and that values of x numerically equal give the same values of y ; thus, when x=2, y= ±4.6, and also when x = — 2, ?/= ±4.6. X y ±5 ±1 ±4.9 ±2 ±4.6 ±3 ±4 ±4 ±3 ±5 1 >^'^ [^ r^ N K ■i^ / ^ ^ r \ / S =^ Vj l^ H r ^ The values given in the table serve to locate twenty points of the GRAPHIC SOLUTIONS 197 ^'raph of 0:2 + ?/'-2 = 25. Plotting these points and drawing a smooth curve through them, we see that the graph is apparently a circle. It may be proved by geometry that this graph is a circle whose radius is 5. The graph of any equation of the form x^ + y^ = r^ is a circle whose radius is r and whose center is at the origin. 2. Construct the graph of the equation x^ + ^^ = 49. 3. Construct the graph of the equation (^x-2f-\-{y— 3)2 = 9. Suggestion. — Solving for y^ we have y = Z ± VO — (ic — 'A)-. Since any value less than — 1 or greater than + 5 substituted for x makes the value of y imaginary, the graph lies between x = — 1 and + 5. The graph of any equation of the form (x — of -{- {y — by = r^ is a circle whose radius is r and center is at the point (a, b), 4. Construct the graph of the equation y'^ = S x -\- 9. Solution. — Solve for y, y =± VS x -\- 9. It will be observed that any value smaller than — 3 substituted for x will make y imaginary ; consequently, no point of the graph lies to the left of a: =:— 3. Beginning with x = — S, we substitute values for x and determine the corresponding values of y, as recorded in the table. X y -3 -2 ±1.7 -1 ±24 ±3 1 ±3.5 2 ±3.9 3 ±4.2 <'^ ^ =T^ -^ 1 p ^ r ^ / ?f / ^ ^ \ ^, ^ si s h" f ^ ^ Plotting these points and drawing a smooth curve through them, we find that the graph obtained is apparently a parabola. The graph of any equation of the form y^ = ax-\-c is a parabola. 5. Construct the graph of y'^ = 5 x -}- S. 6. Construct the graph of the equation 9x^-^25y'^=: 225. 198 GRAPHIC SOLUTIONS Solution. — Solve for y^ y = ± |\/25 — x^. Since any value numerically greater than 5 substituted for x will make the value of y imaginary, no point of the graph lies farther to the right or to the left of the origin than 5 units ; consequently, we substitute for x only values between and including — 5 and + 5. Corresponding values of x and y are given in the table. X y ±3 ± 1 ±2.9 ±2 ±2.7 ±3 ±2.4 ±4 ±1.8 ±5 Hx ^' f-? > ^ ^r< " fSL-^ /^i 1 r LL1»v \ s a. \y >^. fr^ ^ ^ 1 Plotting these twenty points and drawing a smooth curve through them, we have the graph of 9 x^ + 26y^ = 225, which is called an ellipse. The graph of any equation of the form b^x^ + aV = ^^^ is an ellipse. 7. Construct the graph of the equation 9 x^ -{-16y^ = 144. 8. Construct the graph of the equation 4 a;^ __ 9 ^2 _. 35^ Solution. — Solve for ?/, 2/ = ± | Vx^ — 9. Since any value numerically less than 3 substituted for x will make the value of y imaginary, no point of the graph lies between x = -\- S and x = — 3 ; consequently, we substitute for x only ± 3 and values numerically greater than 3. Corresponding values of x and y are given in the table. X y ±3 ±4 ±5 ±6 ±7 ±1.8 ±2.7 ±3.5 ±4.2 N y N k. / ki N A-i A ir f \ f \ ^ Va N \ ,< r N ^., > J N ^ _ Plotting these eighteen points, we find that half of them are on one GRAPHIC SOLUTIONS 199 side of the y-axis and half on the other side, and since there are no points of the curve between x = + 3 and x = — 3, the graph has two separate branches, that is, it is discontinuous. Drawing a smooth curve through each group of points, we see that the two branches thus constructed constitute the graph of the equation 4 x^ — 9y^ = 36, which is an hyperbola. The graph of any equation of the form b^x^ — aV = o^^ is an h3rperbola. An hyperbola has two branches and is called a discontinuous curve. 9. Construct the graph of the equation 9x^—16y'^ = 144. 10. Construct the graph of the equation xy = 10, Solution Substituting values for x and solving for y, we find the corresponding values of x and y as given in the table. X y X y 1 10 - 1 -10 2 5 -2 -5 3 H -3 -H 4 2i -4 -^ 5 2 -5 -2 6 If -6 -If 7 H -7 -H 8 n - 8 -li 9 H -9 -n 10 -10 -1 "~~ ■~" — — ~^ — — — — — — ' V- A, N f^ ^ ^» N s N ^ kn ^ »-t ^ ^ ^ ^c N h* V ^ i \ V ^ ^: Plotting these points and drawing a smooth curve through each group of points, we see that the two branches of the curve found constitute the graph of the equation xy = 10, which is an hyperbola. The graph of any equation of the form xy = c is an hyperbola. 11. Construct the graph of the equation xy = 12. 12. Construct the graph of the equation xy = — 12. 200 GRAPHIC SOLUTIONS 259. Graphic solutions of simultaneous equations involving quadratics. The graphic method of solving simultaneous equations that involve quadratics is precisely the same as for simultaneous linear equations (§§ 149-153), namely: Construct the graph of each equation, both being referred to the same axes, and determine the coordinates of the points where the graphs intersect. If they do not intersect, interpret this fact. 260. 1. Solve graphically EXERCISES 9aj2 + 252/2 = 225, 3 a; — 5?/= 15. Solution - U "S vg - ^' fe- ? / ^. \ . i \ ^ ^ ^ \ y^ / ^ —- - ^ y y ^ ^^ y^ ''^J >> 5i *" Constructing the graphs of these equations, we find the first to be an ellipse and the second a straight line. The straight line intersects the ellipse in two points, (5, 0) and (0, -3). Hence, there are two solutions, a; = 5, 2/ = ; and cc = 0, ?/ = — 3. Test. — The student may test the roots found by performing the numerical solution. 2. Solve graphically ■ a;2 + 2/2 = 25, J 4. Solution The graphs (a circle and a straight line) are found to intersect at the points, x = 3, y = 4;ic=:-3, 2/ = 4. Since the graphs have only these two points in common, their coordinates are the only values of x and y that satisfy both equations, and are the roots sought. The pairs of values found are real, and different, or unequal. V 6 V = 5 f^ ^ N V = 4 / s \ \ J /.< \ /^ ' V ■^^ X^ _,, _ GRAPHIC SOLUTIONS 201 3. Solve graphically Solution. — Imagine the straight line ?/ = 4 in the figure for exercise 2 to move upward until it coincides with the line y = b. The real unequal roots represented by the coordinates of the points of intersection ap- proach equality, and when the line becomes the tangent line y = ^, they coincide. Hence, the given system of equations has two real equal roots^ x = 0, 2/ =: 5, and x = 0, y — b. f a;2 + 2/2 = 25, 4. Find the nature of the roots of \ Solution. — Imagine the straight line ?/ = 4, in the figure for exercise 2 to move upward until it coincides with the line ?/ = 6. The graphs will cease to have any points in common, showing that the given equations have no common real values of x and y. It is shown by the numerical solution of the equations that there are two roots and that both are imaginary. A system of two independent simultaneous equations in x and y, one simple and the other quadraticy has two roots. The roots are real and unequal if the graphs intersect^ real and equal if the graphs are tangent to each other, and imaginary if the graphs have 7io points in common. x^ -\- y'^ = 25. Solution. — The graphs (the first an hyperbola and the second a circle) show that both of the given equations are satisfied hj four different pairs of real values of x and y : jx = 4.5; 4.5; -4.5; -4.5; I 2/ = 2 2; -2.2; -2.2; 2.2. Note. — The roots are estimated to the nearest tenth ; their accuracy may be tested by performing the numerical solution. 5. Solve graphically fh ^V -^/^ y^^i? \ ^^'? 2 ^?i^-:3^ k^ t^W^ \^.m ^ ^nr > I ^ J ;i \/\ '-.. ./ / \J X>^ x + c = is twice the other, what is the relation of 6^ to a and c? Solution Writing ax^ + 6x 4- c = in the form x2+^X + ^ = 0, (1) a a and representing the roots by r and 2 r, we have r + 2r = 3r = --, (2) a and r.2r = 2r2 = ^. (3) a On substituting the vakie of r obtained from (2) in (3) and reducing, 22. Obtain an equation expressing the condition that one root of 4 a:^ — 3 aoj + 6 = 3 is twice the other. 23. Find the condition that one root of ax^ + 6ic + c = shall be greater than the other by 3. 24. When one root of the general quadratic equation ax^ ~\-hx-\-c— is the reciprocal of the other, what is the rela- tion between a and c ? 25. If the roots of ax^ -\- hx -\- c = are i\ and rj, write an equation whose roots are — r^ and — r^. 208 PROPERTIES OF QUADRATIC EQUATIONS 26. Obtain the sum of the squares of the roots of 2 x^ — 12 X + 3 = y without solving the equation. Solution Sum of roots = n -\- Vo = 6, (1) Product of roots = rir2 — |. (2) Square (1), n^ + rg^ + 2 ri^s = 36. (3) (2) X 2, 2 nrs = 3. (4) (3)~(4), ri2 + r22 = 33. Find, without solving the equation: 27. The sum of the squares of the roots of xP — 5x — 6 = 28. The sum of the cubes of the roots of 2 aj^ — 3 a? + 1 = 0. 29. The difference between the roots of 12 o:^ + a; — 1 = 0. 30. The square root of the sum of the squares of the roots of a;2-7aj + 12 = 0. 31. The sum of the reciprocals of the roots of ax^-{-bx-\-c=0. Suggestion. - + - - *'^ '^ ^^ n ^2 ^1^2 32. The difference between the reciprocals of the roots of 8x^-10x + 3 = 0. 269. The number of roots of a quadratic equation. It has been seen (§ 266) that any quadratic equation may be reduced to the form oc^ -\- px + q = 0, which has two roots, as Ti and rg. To show that the equation cannot have more than two roots, write it in the form given in § 267, namely, (x-r^)(x-r2)=0. (1) If the equation has a third root, suppose it is 7-3. Substituting ^3 for x in (1), we have which is impossible, if r^ differs from both Vi and Vq. Hence, PRINCIPLE. — A quadratic equation has ttvo and only two roots. PROPERTIES OF QUADRATIC EQUATIONS 209 270. Factoring by completing the square. The method of factoring is useful in solving quadratic equa- tions when the factors are rational and readily seen. In more difficult cases we complete the square. This more powerful method is useful also in factoring quadratic expressions the factors of which are irrational or otherwise difficult to obtain. EXERCISES 271. 1. Factor 2x' -{-5x-3. Solution.— -Let 2x2 + 5x — 3=0. Divide by 2, etc., x"^ -h f x = |. Complete the square, x^ -}- ^x -\- f-f = f |. Solve, X = J or — 3. Forming an equation having these roots, § 267, we have (x-i)(x + 3)-0. Multiplying by 2 because we divided by 2, we have (2 X - 1) (x + 3) =r 2 x'-^ + 5 X - 3 = 0. Hence, the factors of 2 x^ + 5 x — 3 are 2 x — 1 and x + 3. Factor : 2. 5x^ + 3 X- 2. 5. 7 a;2 + 13 a; - 2. 3. 4 a;2 _ 4 a; - 3. 6, 15x^- 5.5 x-1, 4. Sx^-Ux + 3, 7. 24 aj2 - 10 aj - 25. 8. Factor x'^-\-2x — 4=. Solution. — Let x2 + 2 x — 4 = 0. I Complete the square, x^ -f- 2 x -f 1 = 5. Solve, X = — 1 4- V5 or — 1 — \/5. Hence, § 267, (x + 1- >/5)(x + 1 + V5) = x"^ -h 2 x - 4 = 0. That is, the factors of x'-^ -f 2 x — 4 are x + 1 — >/5 and x + 1 -f VS. 9. a;2 4- 4 a; — 6. 12. x^ -{- x -^ 1. 10. 2/^ -6 2/ + 3. 13. f^ + 3t + 7. 11. 2^2-5 21-1. 14. a2 + 3a~5. 15. Factor 2 - 3 a: - 2 a;2. Suggestion. — Since 2 — 3 x — 2x2= — 2(x2 -|-| x —1), factor x^ + f x — 1, in which the coefficient of x'^ is + 1, and multiply the result by — 2. milne's sec. course alg. — 14 210 PROPERTIES OF QUADRATIC EQUATIONS Factor : 16. 2x^ + 2 x-1, ' 19. 9a2-12a + 5. 17. dx^-Ax+l. 20. 16 '^(l - 'y) - 9. 18. 24a;-16i»2-3. 21. 16(3 + ti) + 3 nl 22. Factor 100 x'^ + 70 xy - 119 y\ Suggestion. — The coefficient of x^ being a perfect square, complete the square directly ; do not divide by 100. 23. 4 62 _ 48 6 + 143. 26. 16 p{p + 1) - 1517. 24. 9r2-12r + 437. 27. 256^ - 2 ^(5 e - 2 ^). 25. 4a2 + 12a-135. 28. 3h(4:k -3h) ^TkK 29. Factor a;* -f 4 0^3 _(_ 3 ^2^ 8 a; — 5. Solution. —Let a:* + 4 ic^ + 8 x^ + 8 x — 5 = 0. Complete the square, (x* + 4 x3 + 4 x2) + 4(x2 4- 2 x) + 4 = 9. Extract the square root, x^ + 2 x + 2 = 3 or — 3. ... x4 + 4x3 + 8x2 + 8x-6 = (x2 + 2x + 2-3)(x2-f 2x4-2 + 3) = (x2 + 2 X - l)(x2 + 2 X 4- 5). Factor the following polynomials : 30. a;^+6a^ + llaj2 4-6x-8. 31. x^ + 2x^ + 5x^ + Sx^+Sx'^-^^Sx + S. * 32. x^-4.x^ + 6x^-}-6a^-19x'^-}-10x + 9. 33. 4 a;6 4- 12 0^5 + 25 i»4 + 40 a^ + 40 a;2 4- 32 a; 4- 15. 34. Resolve x^ + 1 into factors of the second degree. Solution. x* + 1 = x* + 2 x2 4- 1 — 2x2 = (X2 4-1)2 _(x\/2)2 = (x2 4-xV2 4-l)(x2-x\/2 + l). Note. — Each of these quadratic factors may be resolved into two fac- tors of the first degree by completing the square. Resolve into quadratic factors : 35. ic*+16. 37. aj^ + 2aV-f 4a'». 36. a^ + b\ 38. 'y* - 4 ii^v^ — 2 n\ INTERPRETATION OF RESULTS 272. A number that has the same value throughout a dis- cussion is called a constant. Arithmetical numbers are constants. A literal number is constant in a discussion, if it keeps the same value throughout that discussion. 273. A number that under the conditions imposed upon it may have a series of different values is called a variable. The numbers .3, .33, .333, .3333, ... are successive values of a variable approaching in value the constant J. 274. When a variable takes a series of values that approach nearer and nearer a given constant without becoming equal to it, so that by taking a sufficient number of steps the difference between the variable and the constant can be made numerically less than any conceivable number however small, the constant is called the limit of the -variable, and the variable is said to approach its limit. This figure represents graphically a variable x ap- o x, x, x, x preaching its limit 0A'*=2. ' 1 * \ ' j ' i' ' ' The first value is 0X\ = 1 ; the second is OX2 = \\ ; the third is OX3 = If ; etc. At each step the difference between the variable and its limit is diminished by half of itself. Consequently, by taking a sufficient number of steps this difference may become less than any number, however small, that may be assigned. 275. A variable that may become numerically greater than any assignable number is said to be infinite. The symbol of an infinite number is 00 . 211 212 INTERPRETATION OF RESULTS 276. A variable that may become numerically less than any assignable number is said to be infinitesimal. An infinitesimal is a variable whose limit is zero. The character is used as a symbol for an infinitesimal num- ber as well as for absolute zero, which is the result obtained by subtracting a number from itself. 277. A number that cannot become either infinite or infini- tesimal is said to be finite. THE FORMS a x 0, ?, ? — 0' 0' 00 278. The results of algebraic processes may appear in the forms, a X 0, -, -, — , etc., which are arithmetically meaning- less; consequently, it becomes important to interpret the meaning of such forms. 279. Interpretation of a x 0. 1. Let represent absolute zero, defined by the identity, = n - n. (1) Multiplying a = a by (1), member by member, Ax. 3, we have a X = a(n'—n) = an — an by def . of zero, = 0. That is, Any finite number multiplied by zero is equal to zero. 2. Let represent an infinitesimal, as the variable whose successive values are 1, .1, .01, .001, •••. Then, the successive values of a X are (§ 20) a, .1 a, .01 a, .001 a, •••. Hence, a X is a variable whose li7nit is absolute zero. That is. Any finite number multiplied by an infinitesimal number is equal to an infinitesimal yxumber. INTERPRETATION OF RESULTS 213 280. Interpretation of ^. The successive values of the fractions, -, — , -x^, "aao' ^^^'^ are .5, 5, 50, 500, etc., and they continually increase as the denominators decrease. In general, if the numerator of the fraction - is constant X while the denominator decreases regularly until it becomes numerically less than any assignable number, the quotient will increase regularly and become numerically greater than any assignable number. .-. - = 00. That is. If a finite number is divided by an infinitesimal number, the quotient will be an infinite number, 281. Interpretation of -• Let represent absolute zero. Then, if a is any finite number, § 279, axO = 0; whence, . - = hx -\-o xy — ^ ah — ^ ay When is a fraction in its lowest terms ? 5 — 5 . 18. Show that = Give the principles accord- ing to which the signs of the terms of a fraction may be changed. [ 5 a; + 2/ = 22, 19. Solve by two different methods \ -^ I oj + 5 2/ = 14. 20. Represent the V8 by a line. / i\io 21. Find the fourth term oi 12 x + -) , when x = 5. 22. Define axiom ; elimination ; coordinate axes. 23. Construct the graph of 2 ?/ = 3 a; — 4. Tell how to determine where a graph crosses the a>axis ; the ^/-axis. 24. Show the difference in meaning between (a^y and a^ X a*". 25. Illustrate each of the following kinds of equations : numerical ; literal ; integral ; fractional ; identical ; condi- tional ; linear ; homogeneous ; symmetrical. 26. Solve the equation 2 x'^ — 5 x = 150 by three methods. Explain each. 27. Define evolution ; radical ; surd ; entire surd ; mixed surd ; binomial quadratic surd ; similar surds ; conjugate surds. 28. In the proportion a:b = b:G indicate the extremes ; the means ; the mean proportional ; the third proportional. 29. Illustrate how a root may be introduced in the solution of an equation ; how a root may be removed. What is meant by an extraneous root ? 30. Why is it specially important to test the values of the unknown number found in the solution of radical equations ? 31. Upon what axiom is the clearing of equations of fractions based ? What precautions should be taken to prevent intro- ducing roots? If roots are introduced, how may they be detected ? GENERAL REVIEW 233 32. Prove that a quadratic equation has two and only two roots. 33. Tell how to form a quadratic equation when its roots are given. Form the equation whose roots are | and i. 34. What is the meaning of " function of cc '' ? " infinite number '' ? Define variable. 35. Solve the equation ax^ -f 5a; + c = 0. Show the condi- tion under which the roots are real and unequal ; real and equal ; imaginary ; rational ; surds ; both positive ; both negative ; one positive and the other negative. 36. Derive the value of the sum of the roots of the equation x^ -\-px -\- q=0\ the value of the product of the roots. 37. In clearing a fractional equation of its denominators, why should we multiply by their lowest common multiple ? Illustrate by showing what happens when the equation 2x 10 ^ 7 X — 1 x^ — 1 x-\-l is multiplied by the product of all the denominators. 38. What powers of V— 1 are real ? imaginary? 39. Classify the following numbers as real or imaginary ; as rational or irrational : 2, Vi, V2, V5, V^, V^, VS^, ^a', ^f^^, a being a positive number. 40. Find the ratio of a? + W to a? — ah -^^ W. Indicate the antecedent and the consequent in the ratio found. 41. Write the inverse ratio of C6 to 6 ; the duplicate ratio. 42. For what values of x will x^ — x-{-l:x^ + x -{-1^=^:1 9 43. li a:h = c:d, show that 2a + 36:2a = 2c-+-3d:2c; ma : nh — mc : nd ; ma + nb : ma — nh = mc + nd : mc — nd. 44. Write the formula for the sum s of an arithmetical series. Find the sum of 10 terms of the series 1, 4, 7, •••. 46. Prove that in a finite geometrical progression s = ^ ~ ^ - 234 GENERAL REVIEW a-\-h a_ a-\-h 46. Multiply 2 o;^^ - 5 ?/ 2 by 2 a^^* + 5 2/ 47. Expand (a?" — y''){x'' + 2/'')(^^'' + ^^")- 48. Divide (a + 5) + aj by (a + &)* + a:*. 49. Factor 9.^2 -12 a;+4; 9a;2 + 9i« + 2; a^-3a;4-2; a*+l. 50. Show by the factor theorem that a; — a is a factor of a;" + 3 aa;""^ — 4 a'*. 51. Separate a^^ — 1 into six rational factors. 52. Factor 4(ac2 + 5c)2- (a^ - 6^ _ c^ + ^2)2. 53. Find the L.C.M. of x^ — 2/^ -^ + 2/> and xy — f, 54. Find the H.C.F. of 2a;4 _ Ta:^ + 4a;2 + 7a; - 6, 2 aj^ + a^ - 4a;2 + 7 a; - 15, and 2 a^ + a;3 _ ^ _ 12. 55. Expand (2a + 3 &)^; (VS + -v/^)^ (_1_V3)3. If a"* X a" = a"*"^" for all values of m and n, show that : 56. a-2 = l. 59. (a&)o = l. 57. a^ = V^ = (Va)l 60. (a6c)3 = ci^^V. 58. 2a-i=?:^^ ei. ^^ a Simplify, expressing results with positive exponents : 62. i^^x^i% i25-* ««• r^^ [a; ^2/ ' _ a;~y a+ 6 a — 6 67. 64. 50-^-32+ V256-8--. '• ^i_^,i ^i + ^i 2 3 68. Find a factor that will rationalize x^ -\- y^. 69 Interpret each of the following : - , — , - • 00 GENERAL REVIEW 235 r X 1-1 70. Simplify 71. Simplify Solve the following equations for x : 72. mx'^ — 7ix = mn. 74. (I + cc)^ +(1 — a;)^ = 242. 73. x^-\-S = 9x^, 75. x-{-x^-{-(l + x + x^y = 55. 76. 77. 78. H- X x. a ; a + b 1 1 . - X — - b a-[-h a ^ -\-x = 0. 1-i-x+Vl+x- : = a — l + x 1 — a; + Vl + a;2 Solve for x, y, and 2; : 79. 80. 81. 82. 83. ^A ^ ^2y2 + 2/4 = 21, x'^ — xy + y'^ = 7. aoj + 2/ + ^ = 2(^ + 1), x-\-ay + z = Sa + ly X -\- y -}- az = a^ + 3, x'^ + ocy = — 6, y'^ + xy=^W. ^x^-^3xy = 7y \xy + 4.y^ = lS. x^ ■i-x = 2& — y^ — y, xy = S. 84. 85. 86. 87. 88. 89. jVa??/ = 12, [x-\-y —Vx-\- y = 20, Ixy — Qcy^ = — 6, \x — xy^ = 9. {xy = x + y, \x' + f = S. x^y^ — 4:xy = 5, i»2 -f 4 ^2 ^ 29. 2a^-\-2f = 9xy, x + y = S. x^ + yi = 4, a;^ + 2/ = 16. 236 GENERAL REVIEW Problems 313. 1. The sum of two numbers is 72 and their quotient is 8. Find the numbers. 2. The sum of ^ and i of a number multiplied by 4 equals 88. Find the number. 3. Separate 54 into two parts such that yL of ^^^ differ- ence between them is ^. 4. Separate m into two parts such that - of the difference 1 ^ between them is - • r 5. A man sold his crop of raisins for $480, thus gaining ^ of the expense of raising them. What was this expense ? 6. A rare book sold for $ 15,000. If there was a gain of 87^ % , how much did the book cost ? 7. From a rose farm 400,000 plants are sent out yearly. How many plants are there in a carload, if 25 times the num- ber of cars is .001 of the number of plants in each ? 8. A man who had no room for 8 of his horses, built an ad- dition to his stable ^ its size. He then had room for 8 horses more than he had. How many horses had he ? 9. A woman on being asked how much she paid for eggs, replied, " Two dozen cost as many cents as I can buy eggs for 96 cents.'' What was the price per dozen ? 10. The denominator of a certain fraction exceeds the numerator by 3. If both terms are increased by 4, the frac- tion will be increased by |. Find the fraction. 11. An expert workman makes 36,000 beads in a certain time. If he worked 2 days longer and made 1000 beads less per day, the total number of beads would be 40,000. How many beads does he make per day? 12. A dealer sold a number of horses for $ 1320, receiving the same price for each. If he had sold 1 horse less, but had charged $10 apiece more, he would have received the same sum. Find the price of a horse. GENERAL REVIEW 237 13. Two numbers are in the ratio of 7 to 9, but if 14 is added to each they will be in the ratio of 5 to 6. Find the numbers. 14. The meshed wire in a bundle had an area of 400 square feet. If its width had been 2 feet more, it would have been I of its length. Find its dimensions. 15. The value of a fraction is |. If 4 is subtracted from its numerator and added to its denominator, the value of the resulting fraction is |. Find the fraction. 16. The greater of two numbers divided by the less gives a quotient of 7 and a remainder of 4 ; the less divided by the greater gives ^, Find the numbers. 17. The greater of two numbers divided by the less gives a quotient of r and a remainder of s ; the less divided by the greater gives t. Find the numbers. 18. There is a number whose three digits are the same. If 7 times the sum of the digits is subtracted from the number, the remainder is 180. What is the number ? 19. A certain fraction plus its reciprocal equals 2i. The numerator of the fraction minus the denominator equals 1. Find the fraction. 20. A firm finds that its monthly sales of toilet soap amount to $40 more if put up 3 cakes to a box and sold for 12^ a box, than if put up 6 cakes to a box and sold for 20 i^ a box. How many cakes does the firm sell per month ? 21. The perimeter of a rectangle is 8 m and its area is 2 m^. Find its dimensions. 22. The volumes of two cubes differ by 296 cubic inches and their edges differ by 2 inches. Find the edge of each. 23. The hypotenuse of a right triangle is 20. The sum of the other two sides is 28. Find the length of each side. 24. A sum of money at simple interest amounted in m years to a dollars and in n years to b dollars. Find the sum and the rate of interest. 238 GENERAL REVIEW 25. A farmer sold a wagon for $16 and lost as many per cent as the number of dollars in the cost of the wagon. How much did the wagon cost? 26. If a number of two digits is divided by the product of its digits, the quotient is 6. If 9 is added to the number, the sum equals the number obtained by interchanging the digits. What is the number ? 27. A piece of work can be done by A and B in 4 days, by A and C in 6 days, and by B and C in 12 days. Find the time it would take A to do it alone. 28. If the sides of an equilateral triangle are increased by 7 inches, 4 inches, and 1 inch, respectively, a right triangle is formed. Find the length of a side of the equilateral triangle. 29. A man planting peanuts with a machine used 30 pecks of shelled seed per day. If the number of acres planted per day was 1 more than the number of pecks of seed used per acre, how many acres of peanuts did he plant per day? 30. A jeweler has two silver cups. The first cup, with a cover on it valued at $ 1.50, is worth 1|^ times as much as the second cup, and the second cup with the cover on it is worth ii as much as the first cup. Find the value of each cup. 31. A merchant bought two lots of tea, paying for both $34. One lot was 20 pounds heavier than the other, and the number of cents paid per pound was in each case equal to the number of pounds bought. How many pounds of each did he buy ? 32. Three farms for raising black foxes once contained together 75 foxes. The number of foxes on the three farms form a series in arithmetical progression, the largest number being 30. How many foxes were on each of the other farms? 33. Find two numbers such that their sum, their product, and the difference of their squares are all equal. 34. A tank contains 400 cubic feet. Its height exceeds its width by 1 foot and its length is 5 times its width. Find its dimensions. GENERAL REVIEW 239 35. A takes 1^ hours longer than B to walk 15 miles, but if he doubles his rate he takes 1 hour less time than B. Find their rates of walking. ^ 36. The height (h) of an arch of ^^^^ width (w) is given by the formula, h = r-ir^-(lwy, in which r is the radius of the circle of which the arch is a segment. Solve for r, 37. The width of an arch for a culvert under a railroad em- bankment is 16 feet and its height is 6 feet. Find the radius of the arch. 38. How much does a teacher earn in 25 years, if she receives a salary of $ 720 the first year, and an increase of $ 80 each year for 14 years ? 39. The sum of all the even integers from 2 to a certain number inclusive is 702. Find the last of these integers. 40. A and B can together do a piece of work in 15 days. After working together for 6 days, A went away, and B finished it 24 days later. In what time would A alone do the whole ? 41. One machine makes 60 revolutions per minute more than another and in 5 minutes the former makes as many revolu- tions as the latter does in 8 minutes. Find the rate of each. 42. The area of the floor of a room is 120 square feet ; of one end wall, 80 square feet ; and of one side wall, 96 square feet. Find the dimensions of the room. 43. A company of soldiers attempted to form in a solid square, and 56 were left over. They attempted to form in a square with 3 more on each side, and there were 25 too few. How many soldiers were there in the company? 44. A tank can be filled by the larger of two faucets in 5 hours less time than by the smaller one. It is filled by them both together in 6 hours. How many hours will it take to fill the tank by each faucet separately ? 240 GENERAL REVIEW 45. How much pure alcohol must be added to a gallon of 92 % alcohol so that the mixture shall be 93 % alcohol ? 46. In a mass of copper, lead, and tin, the copper weighed 5 pounds less than ^ of the whole, and the lead and tin each 5 pounds more than ^ of the remainder. Find the weight of each. i 47. A new bronze was recently patented. It contained 5 ^ more copper than iron, twice as much nickel as aluminium, and 4 times the amount of aluminium was 3% less than the amount of copper. What per cent of each metal did the bronze contain ? 48. A needs 3 days more than B to do a certain piece of work, but working together the two men can do the work in 2 days. In how many days can B do the work ? 49. Find three numbers in geometrical progression, such that their product is 1000, and the sum of the second and third is 6 times the first. 50. A rectangular field is 119 yards long and 19 yards wide. How many yards must be added to its width and how many taken from its length, in order that its area may remain the same, while its perimeter is increased by 24 yards ? 51. It took a number of men as many days to pave a side- walk as there were men. Had there been 3 men more, the work would have been done in 4 days. How many men were there ? 52. By lowering the selling price of apples 2 cents a dozen, a man finds that he can sell 12 more than he used to sell for 60 cents. At what price per dozen did he sell them at first ? 53. If the distance traveled by a train in 63 hours had been 44 miles less and its rate per hour had been 4|^ miles more, the trip would have taken 50 hours. Find the total run. 54. In a quantity of gunpowder the niter composed 10 pounds more than | of the weight, the sulphur 3 pounds more than j^-g- of it, and the charcoal 3 pounds less than -^jj of the weight of the niter. What was the weight of the gunpowder ? GENERAL REVIEW 241 55. Two numbers whose product is 28,350, consist of three digits each. The hundreds' and units' digits of one are, respectively, 2 and 5, the corresponding digits of the other are 1 and 6, the tens' digit being the same in both numbers. Find the numbers. 56. If $ 820 is put at interest for a certain number of years at a certain rate, it amounts to $ 955.30. If the time were 1 year less and the rate ^ % more, the amount would be $ 918.40. Find the time and the rate. 57. A and B can do a piece of work in m days, B and C in 71 days, A and C in p days. In what time can all together do it ? How long will it take each alone to do it ? 58. At $2.50 per day, how many days did a man work to earn $ 24, if he forfeited $ 1.50 for every day he was idle, and worked 3 times as many days as he was idle ? 59. A train. A, starts to go from P to Q, two stations 240 miles apart, and travels uniformly. An hour later another train, B, starts from P, and after traveling for 2 hours comes to a point that A passed 45 minutes previously. The rate of B is now increased by 5 miles an hour and it overtakes A just on entering Q, Find the rates at which the trains started. 60. It takes A and B f of a day longer to tin and paint a roof than it does C and D, and the latter can do 50 square feet more a day than the former. If the roof contains 900 square feet, how much can A and B do in a day ? C and D ? 61. Find two numbers differing by 48, whose arithmetic mean exceeds the geometric mean by 18. 62. The formula for the weight of a hollow cylindrical column is I being expressed in feet, W and w in pounds, and D and d in inches. Find the weight of a hollow cylindrical cast iron column in which I = 10, iv = .2607 (pounds per cubic inch), the outside diameter Z> = 8, and the inside diameter c? = 4. Milne's sec. course alg. — 16 242 GENERAL REVIEW 63. A yacht goes 5 miles downstream in the same time that it goes 3 miles upstream ; but if its rate each way is diminished 4 miles an hour, its rate downstream will be twice its rate upstream. How fast does it go in each direction ? 64. On shipboard " eight bells '' is rung at midnight and every 4 hours thereafter. If 1 bell is rung at 12.30 a.m. and the number of bells increases by 1 every half hour up to " eight bells," how many bells are rung in the 24 hours ? 65. A man invested $ 2720 in railroad stock, a part at 95 yielding 2 % and the balance at 82 yielding 3 ^ . His income from both investments was $ 70. Find the amount invested in each kind of stock. 66. A rectangular piece of tin is 4 inches longer than it is wide. An open box containing 840 cubic inches is made from it by cutting a 6-inch square from each corner and turning up the ends and sides. What are the dimensions of the box ? 67. A projectile fired from a battleship was heard by the gunner to strike a mark 3360 feet away 4^ seconds after it was lired. An officer on another vessel 5600 feet from the first and 2240 feet from the mark heard the shot strike 1| seconds before the report reached him. Find the velocity of the sound and the average velocity of the projectile. 68. Find the common difference of the arithmetical progres- sion whose first term is 3 and whose second, fourth, and eighth terms are in geometrical progression. 69. If zinc weighs 437.5 pounds per cubic foot and copper 550 pounds, what per cent by volume is each of these metals in an alloy of them, 1 cubic foot of which weighs 532 pounds ? 70. The velocity acquired or lost by a body acted upon by gravity is given by the formula v = gt (take g = 32.16). A bullet is fired vertically upward with an initial velocity of 2010 feet per second. Find in how many seconds it will return to the earth (neglecting the friction of the air). Using the formula s = ^ gt^, find how high the bullet will rise. GENERAL REVIEW 243 71. The load on a wall column for an office building is 360,000 pounds, including the weight of the column itself, and is balanced, as shown in the figure, by a part of the load on an interior column. ISTeglecting the weight of the girder, find the load on the fulcrum. 72. A man bought some 50-dollar shares in one stock com- pany and I as many 100-dollar shares in another. At the end of the first quarter, dividends of 2 % and of 1^ %, respectively, were declared on these stocks, and the man received $ 120. How much money did he invest in each company ? 73. It took a passenger train, 175 feet long, 7|^ seconds to pass completely a freight train, 485 feet long, moving in the opposite direction. If the passenger train was going 3 times as fast as the freight train, what was the rate of each per hour ? 74. The distance a body 'will fall in t seconds, starting from rest, is given by the formula s— ^ gf. A man dropped a tor- pedo from a height and heard the report 5 seconds later. Tak- ing g = 32.16 and the velocity of sound 1125.6 feet per second, find, to the nearest tenth of a second, the time during which the torpedo was falling. 75. A mixture of graphite and clay, to be used as " lead " in pencils, was o % clay and weighed jj pounds. After the addition of clay to make the " lead " harder, the mixture was (c + 10)% clay and weighed 240 pounds. If graphite had been added, instead of clay, until the mixture weighed 250 pounds, the mixture would have ^een (c — 8) % clay. Solve for p and for c. 244 GENERAL REVIEW 314. The following examination was given recently by the College Entrance Board for Elementary Algebra Complete : 1. (a) Factor 2 mx -{-Qny — my — 12 rix ; 6 aj^ -|- 11 a; — 10 ; x"^ — a^x + 6a^ — a^h. - c4 -f cy + y^ ^ (c + yY & — if & — y'^ (6) Simplify l-{(7^.- 2. (a) Simplify and combine ^ _ i8Vi - iVlOS + 12* + 3^ ^- /o /T (&) Eationalize the denominator and simplify 2. ^^ V2+Vi 3. (a) Solve • ^ ' Associate properly the values of x and ?/. (^) ««!- V^ + 2V?^ = 3. 4. (a) Solve l + l = 4c2 + (^^ , . x^ y'^ Associate properly the \ values of x and y, ^^^ ~Jcd' (b) If b'.G = 5:3 in the equation x^ -}- bx + c^ = 0, are the roots of the equation real ? Give the reason for your answer. 5. At his usual rate a man can row 15 miles downstream in 5 hours less than it takes him to return. Could he double his rate, his time downstream would be only 1 hour less than his .time up. Find his rate in still water and the rate of the current. 6. The second term of an arithmetic progression is ^ of the 8th and the sum of 20 terms is 63. Find the progression. 7. (a) Graph 2/ = 1 + 3 iK2. (b) In the expansion of [3x ) find the term which, V SxV ^ when simplified, contains x^. SUPPLEMENTARY TOPICS CUBE ROOT Cube Root of Polynomials EXERCISES 315. 1. Find the process for extracting the cube root of PROCESS a^ + Sa'b + 3a¥ + ¥ \a -{- b ^ Trial divisor, Sa^ Complete divisor, 3 a'^ -\- 3 ab -^ ¥ 3a''b + 3ab^ + b^ 3a'b-\-3ab^-{-b^ Explanation. — Since a^ -\- S a^b -{- S ah'^ + b^ is the cube of (a + 5), we know that the cube root of a^ + 3 a^b + 3 ab^ -{- b^ is a -\- b. Since the first term of the root is a, it may be found by taking the cube root of a^, the first term of the power. On subtracting, there is a remainder of 3 a'^b + 3 ab^ + b^* The second term of the root is known to be 6, and that may be found by dividing the first term of the remainder by 3 times the square of the part of the root already found. This divisor is called a trial divisor. Since 3 a^b ■^Sab'^+ b^ is equal to b{Sa^ -^Sab + b^), the complete divisor, which multiplied by b produces the remainder 3 a^b 4- 3 ab^ + 6^, is Sa^ -\- S ab -{- b^; that is, the complete divisor is found by adding to the trial divisor 3 times the product of the first and second terms of the root and the square of the second term of the root. On multiplying the complete divisor by the second term of the root, and on subtracting, there is no remainder ; then, a -|- 5 is the required root. Since, in cubing a i- b -j- c, a + b may be expressed by x, the cube of the number will be cc^ + 3 x^c + 3xc^ + c^. Hence, it is obvious that the cube root of an expression whose root consists of more than two terms may be extracted in the same way as in exercise 1, by considering the terms already found as one terra. 246 6« 6« -3 6^+5 63-3 6-116^-6-1 36* 36*-36^+62 -3 6^ + 5 6^ -36= + 36*-63 3b*-Gb'+3b + l -36^ + 663-36-1 -36* + 66''-36-l 246 SUPPLEMENTARY TOPICS 2. Find the cube root of 5^ - 3 6^ + 5 5^ - 3 6 - 1. PROCESS Trial divisor, Complete divisor, Trial divisor, * Complete divisor, Explanation. — The first two terms are found in the same manner as in the previous exercise. In finding the next term, b^ — & is con- sidered as one term, which we square and multiply by 3 for a trial divisor. On dividing the remainder by this trial divisor, the next term of the root is found to be — 1. Adding to the trial divisor 3 times (b^ — b) multiplied by — 1, and the square of — 1, we obtain the com- plete divisor. On multiplying this by — 1, and on subtracting the product from — 3 6* + 6 ft*^ — 3 6 — 1, there is no remainder. Hence, the cube root of the polynomial is b^ — b — 1. EuLE. — Arrange the polynomial with reference to the consecu- tive powers of some letter. Extract the cube root of the first term, write the result as the first term of the root, and subtract its cube from the given polynomial. Divide the first term of the remainder by three times the square of the root already found, used as a trial divisor, and the quotient loill be the next term of the root. Add to this trial divisor three times the product of the first and second terms of the root, and the square of the second term. The result will be the complete divisor. Multiply the complete divisor by the last term of the root found, and subtract this product from the dividend. Find the next term of the root by dividing the first term of the remainder by the first term of the trial divisor. Form the complete divisor as before, considering the part of the root already found as the first term, and continue in this manner until all the terms of the root are found. SUPPLEMENTARY TOPICS 247 Find the cube root of : 4. a^ — Sx'^y-i-Sxy'^ — f. 5. m3-9m2 + 27m-27. 6. cM + 12 aV + 48 ax^ + 64. 7. 8 m^ — 60 m^Ti -|- 150 m7i^ — 125 n'. 8. 27a^-189i«2^4-441a;?/2-34t2/^ 9. 125 a^ + 675 a2a7 + 1215 aa;2 + 729 a^. 10. 64 aV - 240 a'^b^c + 300 abc'' - 125 c^. 11. a:6-6a;^ + 15a;4-20a;3^15^^2__5a;-f 1. 12. m6 + 6m^ + 15m^ + 20m3-}- 15m2 + 6m + l. 13. x' + 12aj5 + 63a;4 + 184a^ -f- 315a:2 + 300a; + 125. 14. x^-{-6x'-lHx'- 1000 + 180a;2 _ 112a;3 ^ qqq^ 15. 8c« - 60c^ + 198c^ - 365c3 + 396c2 - 240c + 64. 16. c^-3c'd-3 g'(P 4- 11 c^^^ + 6 c^d^ - 12 cd' - 8 d^ 17. .T3_i2a:2 + 54a;-112 + ^-^ + ^. a; x^ x^ ^^ aWx^ &x^ , 3aca;^ 3a^6i:c^ lo. 1 • c^* 6^ b c 19. a;6 + 15a;2+l^ + 20-h-4--+6a;l x^ x^ x^ 20. l-A + ^^§ + ?^^6.^ + 8a:3. ' a^ 2x'' 4.x 8 2 21. 71^ _ 1^5 _|. 9 ^4 _ _y_^^3 4. 9 ^2 _ 1^^ ^ 1^ 22. lx^+ ^x^y -\- xY — ^f — I ^Y -^ i^^ — ttV^' 248 SUPPLEMENTARY TOPICS Cube Root of Arithmetical Numbers 316. Compare the number of digits in the cube root of each number with the number of digits in the number itself : Dumber 1 EOOT 1 Number I'OOO Root 10 Number I'OOO'OOO EoOT 100 27 3 27'000 30 27'000'000 300 729 9 970'299 99 997'002'999 999 Observe that : Principle. — If gl number is separated into periods of three digits each, beginning at units, its cube root will have as many digits as the number has periods. The left-hand period may be incomplete, consisting of only one or two digits. 317. If the number of units expressed by the tens' digit is represented by t, and the number of units expressed by the units' digit is represented by u, any number consisting of tens and units may be represented by ^ + u, and its cube by (t + u)^, or ^3 _j_ 3 f-2^ ^ 3 ^^^2 _^ yz^ Thus, 25 = 2 tens + 5 units, or (20 + 5) units, and 253 = 20^ + 3(202 x 5)+ 3(20 x 5^) +53= 15,625. EXERCISES 318. 1. Extract the cube root of 12,167. FIRST PROCESS 12167 120 + 3 f= 8 000 Trial divisor, 3 ^^ ^ 1200 ^tu= 180 Complete divisor, = 1389 4167 4167 Explanation. — On separating 12,167 into periods of three figures each (§316), there are found to be two digits in the root, that is, the root is composed of tens and units. Since the cube of tens is thousands, and the thousands of the power are less than 27, or 33, and more than 8, or 23, the tens' figure of the root is 2. 2 tens, or 20, cubed is 8000, and 8000 sub- SUPPLEMENTARY TOPICS 249 tracted from 12,167 leaves 4167, which is equal to 3 times the tens 2 x the units + 3 times the tens x the units 2 -f the units ^. Since 3 times the tens 2 x the units is much greater than 3 times the tens X the units 2 + the units 3, #4167 is only a little more than 3 times the tens 2 X the units. If, then, 4167 is divided by 3 times the tens 2, or by 1200, the trial divisor, the quotient vsrill be approximately equal to the units, that is, 3 will be the units of the root, provided proper allowance has been made for the additions necessary to obtain the complete divisor. Since the complete divisor is found by adding to 3 times the tens 2 the sum of 3 times the tens x the units and the units 2, the complete divisor is 1200 + 180 -f 9, or 1389. This multiplied by 3, the units, gives 4167, which, subtracted from 4167, leaves no remainder. Therefore, the cube root of 12,167 is 20 -f 3, or 23. Explanation. — In practice it is usual to place figures of the same order in the same column, and to disregard the ciphers on the right of the products. Since a root expressed by any number of figures may be regarded as composed of tens and units, the processes of exercise 1 have a gen- eral application. Thus, 120 = 12 tens + units ; 1203 = 120 tens + 3 units. 2. Extract the cube root of 1,740,992,427. SECOND PROCESS 12167 123 t^= 8 3<2=1200 3tu= 180 u'= 9 4 167 1389 4 167 Solution 1'740'992'427 11203 1 1^ s 'E 5' 3^2 = 3(10)2 Stu =3(10 X 2) ^2 = 22 = 300 = 60 = 4 740 3^2 = 3(120)2 364 = 4^ 728 J200 12 992 • a > 5^ 3^2-3(1200)2 3 ^w = 3(1200 X 3) t|2 = 32 = 432( = 1( )000 )800 9 12 992 427 433( )809 12 992 427 Since the third figure of the root is 0, it is not necessary to form the complete divisor, inasmuch as the product to be subtracted will be 0. 250 SUPPLEMENTARY TOPICS Rule. — ■ Separate the number into periods of three figures each, beginning at units. Find the greatest cube in the left-hand period, and write its root for the first digit of the required root. Cube this root, subtract the result from the left-hand period, and, annex to the remainder the next period for a nev) dividend. Take three times the square of the root already found, annex two ciphers for a trial divisor, and by the result divide the divi- dend. The quotient, or the quotient diminished, will be the second figure of the root. To this trial divisor add three times the product of the first part of the root ivith a cipHer annexed, multiplied by the second part, and also the square of the second part Their sum tvill be the complete divisor. Multiply the complete divisor by the second part of the root, and subtract the product from the dividend. Continue thus until all the figures of the root have been found. 1. When there is a remainder after subtracting the last product, annex decimal ciphers, and continue the process. 2. Decimals are pointed off from the decimal point toward the right. 3. The cube root of a common fraction may be found by extracting the cube root of the numerator and the denominator separately or by reducing the fraction to a decimal and then extracting its root. Extract the cube root of : 3. 29,791. 9. 2,406,104. 15. .000024389. 4. 54,872. 10. 69,426,531. 16. .001906624. 5. 110,592. 11. 28,372,625. 17. .000912673. 6. 300,763. 12. 48.228544. 18. .259694072. 7. 681,472. 13. 17,173.512. 19. 926.859375. 8. 941,192. 14. 95.443993. 20. 514,500.058197. Extract the cube root to three decimal places : 21. 2. 23. .8. 25. ^V 27. f 22. 5. 24. .16. 26. |. 28. yV SUPPLEMENTARY TOPICS 251 VARIATION 319. One quantity or number is said to vary directly as another, or simply to vary as another, when the two depend upon each other in such a manner that if one is changed the other is changed in the same ratio^ Thus, if a man earns a certain sum per day, the amount of wages he earns varies as the number of days he works. 320. Tlie sign of variation is oc. It is read ' varies as J Thus, X X 2/, read ^ x varies as j/,' is a brief way of writing the proportion x:x' = y :t/', in which x' is the vakie to which x is changed when y is changed to y'. 321. The expression xccymedms that if y is doubled, a; is doubled, or if y is divided by a number, x is divided by the same number, etc. ; that is, that the ratio of a? to 2/ is always the same, or constant. If the constant ratio is represented by kj then when X(xy, - = J€, or x = ky. Hence, y If X varies as y, x is equal to y multiplied by a constant, 322. One quantity or number varies inversely as another when it varies as the reciprocal of the other. Thus, the time required to do a certain piece of work varies inversely as the number of men employed. For, if it takes 10 men 4 days to do a piece of work, it will take 5 men 8 days, or 1 man 40 days, to do it. In a? Qc - , if the constant ratio of a; to - is A;, ^ = k, or xy = k, y y 1 ^ y Hence, If X varies inversely as y, their product is a constant, 323. One quantity or number varies jointly as two others when it varies as their product. Thus, the amount of money a man earns varies jointly as the number of days he works and the sum he receives per day. For, if he should work three times as many days, and receive twice as many dollars per day, he would receive six times as much money. 252 SUPPLEMENTARY TOPICS In 03 QC yz, if the constant ratio of x to yz is k, — = Ic, 01' X = kyz. Hence, yz If X varies jointly as y and z, x is equal to their product multi- plied by a constant. 324. One quantity or number varies directly as a second and inversely as a third when it YSbvies jointly as the second and the reciprocal of the third. Thus, the time required to dig a ditch varies directly as the length of the ditch and inversely as the number of men employed. For, if the ditch were 10 times as long and 5 times as many men were employed, it would take twice as long to dig it. 1 y . . Ill xocy * -, or xcc '-^, if k is the constant ratio, z z x-r'- = k, or X = k-. Hence, z z . Ifx varies directly as y and inversely as z, x is equal to - 'mul- tiplied by a constant, 325. If X varies as y when z is constant, and x varies as z when y is constant, then x varies as yz when both y and z are variable. Thus, the area of a triangle varies as the base when the altitude is con- stant ; as the altitude when the base is constant ; and as the product of the base and the altitude when both vary. Proof. — Since the variation of x depends upon the variations of y and z, suppose the latter variations to take place in succession, each in turn producing a corresponding variation in x. While z remains constant, let y change to ?/i, thus causing x to change to x'. Then, ^=y.. (1) Now while y keeps the value 2/1, let ;? change to zi^ thus causing x' to change to xi. Then, ^ = 1, (2) Xi Zi SUPPLEMENTARY TOPICS 253 Multiply (1) by (2), ^ = -^. (3) xi yizi x = ^^.yz. (4) yizi Since, if both changes are made, xi, ?/i, and zi are constants, ^i- is a constant, which may be represented by k. ^^^^ Then, (4) becomes x = kyz. Hence, x oc yz. Similarly, if x varies as each of three or more numbers, y, z, V, '" when the others are constant, when all vary x varies as their product. That is, X oc yzv — . Thus, the volume of a rectangular solid varies as the length, if the width and thickness are constant ; as the width, if the length and thickness are constant ; as the thickness, if the length and width are constant ; as the product of any two dimensions, if the other dimension is constant ; and as the product of the three dimensions, if all vary. EXERCISES 326. 1. If a; varies inversely as y, and x = 6 when y = 8, what is the value of x when y = 12? Solution. — Since x cc - , let A: be the constant ratio of x to - • y y Then, § 322, xy = k. (1) Hence, when x = Q and y = 8, A; = 6 x 8, or 48. (2) Since k is constant. A; = 48 when y = 12, and (1) becomes 12 x = 48. Therefore, when y = 12, x = 4. 2. li xcc^ , and if a; = 2 when v = 12 and z = 2, what is the z ^ ' value of X when 2/ = 84 and 2; = 7 ? 3. If ajQC^, and if x = 60 whei z t^lie value of y when a; = 20 and z = 7? 3. If a; oc ^, and if x = 60 when y = 24 and z = 2, what is z 254 SUPPLEMENTARY TOPICS 4. If X varies jointly as y and z and inversely as the square of w, and if a:^ = 30 when y=SjZ = 5, and iv = 4, what is the value of x expressed in terms of y, z, and w ? 5. If xacy and y ccz, prove that xocz. Proof. — Since xccy and y cc z, let m represent the constant ratio of X to y, and n the constant ratio of y to z. Then, § 321, x - my, (1) and y = nz. (2) Substitute nz for y in (1), x = mn;s. (3) I 1 1 6. If a; oc - and y oc ~, prove that xccz. y z 7. If xocy and zocy, prove that (x ± z) cc y. 8. The volume of a cone varies jointly as its altitude and the square of the diameter of its base. When the altitude is 15 and the diameter of the base is 10, the volume is 392.7. What is the volume when the altitude is 5 and the diameter of the base is 20 ? Solution. — Let F, jGT, and D denote the volume, altitude, and diam- eter of the base, respectively, of any cone, and V the volume of a cone whose altitude is 5 and the diameter of whose base is 20. Since Foe HD^, or V= kHD^ and F=392.7 whenif=15andD = 10, 392.7= A; X 15x100. (1) Also, since V becomes V when H=b and D = 20, F/ = A;x5x400. (2) Dividing (2) by (1), Ax. 4, ^^-\^,-\- (3) .-. v 9. The circumference of a circle varies as its diameter. If the circumference of a circle whose diameter is 1 foot is 3.1416 feet, what is the circumference of a circle 100 feet in diameter ? 10. The area of a circle varies as the square of its diameter. If the area of a circle whose diameter is 10 feet is 78.54 square feet, what is the area of a circle whose diameter is 20 feet ? SUPPLEMENTARY TOPICS 255 11. The distance a body falls from rest varies as the square of the time of falling. If a stone falls 64.32 feet in 2 seconds, how far will it fall in 5 seconds ? 12. The volume of a sphere varies as the cube of its diameter. If the ratio of the sun's diameter to the earth's is 109.3, how many times the volume of the earth is the volume of the sun ? 13. If 10 men can do a piece of work in 20 days, how long will it take 25 men to do it ? 14. If a men can do a piece of work in b days, how many men will be required to do it in c days ? 15. The illumination from a source of light varies inversely as the square of the distance. How far must a screen that is 10 feet from a lamp be moved so as to receive i as much light ? 16. The number of times a pendulum oscillates in a given time varies inversely as the square root of its length. If a pendulum 39.1 inches long oscillates once a second, what is the length of a pendulum that oscillates twice a second? 17. Three spheres of lead whose radii are 6 inches, 8 inches, and 10 inches, respectively, are united into one. What is the radius of the resulting sphere, if the volume of a sphere varies as the cube of its radius ? 18. A wrought-iron bar 1 square inch in cross section and 1 yard long weighs 10 pounds. If the weight of a uniform bar of given material varies jointly as its length and the area of its cross section, what is the weight of a wrought-iron bar 36 feet long, 4 inches wide, and 4 inches thick ? 19. The distances, from the fulcrum of a lever, of two weights that balance each other vary inversely as the weights. If two boys weighing 80 pounds and 90 pounds, respectively, are balanced on the ends of a board 8^ feet long, how much of the board has each on his side of the fulcrum ? 20. The weight of wire of given material varies jointly as the length and the square of the diameter. If 3 miles of wire .08 of an inch in diameter weigh 288 pounds, what is the weight of I of a mile of wire .16 of an inch in diameter? 256 SUPPLEMENTARY TOPICS LOGARITHMS 327. The exponent of the power to which a fixed number, called the base, must be raised in order to produce a given num- ber is called the logarithm of the given number. When 2 is the base, the logarithm of 8 is 3, for 8 = 2^. 328. When a is the base, x the exponent, and m the given number, that is, when a^ = m, x is the logarithm of the num- ber m to the base a, written log„ m = x. When the base is 10, it is not indicated. Thus, the logarithm of 100 to the base 10 is 2, and of 1000, 3 ; written, log 100 = 2 ; log 1000 := 3. 329. Logarithms may be computed with any arithmetical number except 1 as a base, but the base of the common, or Briggs, system of logarithms is 10. Since lO'^ = 1, the logarithm of 1 is 0. Since 10^ = 10, the logarithm of 10 is 1. Since 102 ^ iqo, the logarithm of 100 is 2. Since 10" i = J^^, the logarithm of .1 is — 1. Since 10"^ = y^^^, the logarithm of .01 is — 2. 330. Then, the logarithm of any number between 1 and 10 is greater than and less than 1, and that of any number between 10 and 100 is greater than 1 and less than 2. For example, the logarithm of 4 is approximately 0.6021, and of 50, approximately 1.6990. Most logarithms are endless decimals. 331. The integral part of a logarithm is called the character- istic ; the fractional or decimal part, the mantissa. In log 50 = 1.6990, the characteristic is 1 and the mantissa is .6990. 332. The following illustrate characteristics and mantissas: log 4580 = 3.6609 ; that is, 4580 = los.md, log 458.0 = 2.6609 ; that is, 458.0 = 102-6609. log 45.80 = 1.6609 ; that is, 45.80 = 10i-6609. log 4.580 = 0.6609 ; that is, 4.580 = looeeoe. log .4680 = 1.6609 ; that is, .4580 = lo-i+.66()9. log .0458 = 2^609 ; that is, .0458 = 10-2+-6r,09. log .00458 = 3.6609 ; that is, .00458 = 10-3+6609. SUPPLEMENTARY TOPICS 257 333. From the preceding examples it is evident that : Principles. — 1. The characteristic of the logarithm of a num- ber greater than 1 is either positive or zero and 1 less than the number of digits in the integral part of the number. 2. The characteristic of the logarithm of a decimal is negative and numerically 1 greater than the number of ciphers immediately following the decimal point, 334. To avoid writing a negative characteristic before a positive mantissa, it is customary to add 10 or some multiple of 10 to the negative characteristic, and to indicate that the number added is to be subtracted from the whole logarithm. Thus, 1.6609 is written 9.6609 - 10 ; 2.3010 is written 8.3010 - 10 or sometimes 18.3010 - 20 ; 28.3010 - 30 ; etc. 335. It is evident, also, from the examples in § 332, that in the logarithms of numbers expressed by the same figures in the same order, the decimal parts, or mantissas, are the same, and the logarithms differ only in their characteristics. Hence, tables of logarithms contain only the mantissas. 336. The table of logarithms on the two following pages gives the mantissas, to the nearest fourth place, of the common logarithms of all numbers from 1 to 1000. 337. To find the logarithm of a number. EXERCISES 1. Find the logarithm of 765. Solution. — In the following table, the letter N designates a vertical column of numbers from 10 to 99 inclusive, and also a horizontal row of figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first two figures of 765 appear as the number 76 in the vertical column marked N on page 259, and the third figure 5 in the horizontal row marked N. In the same horizontal row as 76 are found the mantissas of the logarithms of the numbers 760, 761, 762, 763, 764, 765, etc. The mantissa of the logarithm of 765 is found in this row under 6, the third figure of 765. It is 8837 and means .8837. By Prin. 1, the characteristic of the logarithm of 765 is 2. Hence, the logarithm of 765 is 2.8887. milne's sec. course alg. — 17 258 SUPPLEMENTARY TOPICS Table of Common Logarithms N lO 1 2 3 4 5 6 7 8 9 0000 0043 0086 0128 0170 02I2 0253 0294 0334 0374 II 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1^73 1206 1239 1271 1303 1335 ^3^7 1399 1430 14 1 46 1 1492 1523 1553 1584 I614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2i75f 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 243^ 2455^' 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 25 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 43H 4330 4346 4362 4378 4393 4409 4425 4440 4456 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 501 1 5024 5038 32 5051 5065 5079 5092 5105 5"9 5132 5H5 5159 5172 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 37 5682 5694 5705 5717 5729 5740 5752. 5763 5775 5786 38 5798 5809 5821 5832 5843 5855 5866 ^Hl 5888 5899 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6oio 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 ^r^ 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 691 1 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 53 7243 7251 7259 7267 7275 7284 7292 7300 73?^ 7316 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 N 1 2 3 4 5 6 7 8 9 SUPPLEMENTARY TOPICS 259 Table of Common Logarithms N 1 2 3 4 5 6 7 8 9 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 63 7993 8cx)o 8007 8014 8021 8028 8035 8041 8048 8055 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 82C4 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 68 8325 8331 8338 8344 8351 8357 8363 8370 ^Z1(> 8382 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 N 1 2 3 4 1 5 6 7 8 9 260 SUPPLEMENTARY TOPICS 2. Find the logarithm of 4. Solution. — Although the numbers in the table appear to begin with 100, the table really includes all numbers from 1 to 1000, since numbers expressed by less than three figures may be expressed by three figures by adding decimal ciphers. Since 4 = 4.00, and since, § 335, the mantissa of the logarithm of 4.00 is the same as that of 400, which is .6021, the mantissa of the logarithm of 4 is .6021. By Prin. 1, the characteristic of the logarithm of 4 is 0. Therefore, the logarithm of 4 is 0.6021. Verify the following from the table : 3. log 10 = 1.0000. 9. log .2 = 9.3010 - 10. 4. log 100 = 2.0000. 10. log 542 = 2.7340. 5. log 110 = 2.0414. 11. log 345 = 2.5378. 6. log 2 =0.3010. 12. log 6.07 = 0.7050. 7. log 20 =1.3010. 13. log 78.5 = 1.8949. 8. log 200 = 2.3010. 14. log .981 = 9.9917 -10.. 15. Find the logarithm of 6253. Solution. — Since the table contains the mantissas not only of the logarithms of numbers expressed by three figures, but also of logarithms expressed by four figures when the last figure is 0, the mantissa of the logarithm of 625 is first found, since, § 336, it is the same as the mantissa of the logarithm of 6250. It is found to be .7959. The next greater mantissa is .7966, the mantissa of the logarithm of 6260. Since the numbers 6250 and 6260 differ by 10, and the mantissas of their logarithms differ by 7 ten-thousandths, it may be assumed as sufficiently accurate that each increase of 1 unit, as 6250 increases to 6260, produces a corresponding increase of .1 of 7 ten-thousandths in the mantissa of the logarithm. Consequently, 3 added to 6250 will add .3 of 7 ten-thousandths, or 2 ten-thousandths, to the mantissa of the loga- rithm of 6250 for the mantissa of the logarithm of 6253. Hence, the mantissa of the logarithm of 6253 is .7959 + .0002, or .7961. Since 6253 is an integer of 4 digits, the characteristic is 3 (Prin. 1). Therefore, the logarithm of 6253 is 3.7961. Note. — The difference between two successive mantissas iix the table is called the tabular difference. SUPPLEMENTARY TOPICS 261 Find the logarithm of : 16. 1054. 20. 21.09. 24. .09095. 17. 1272. 21. 3.060. 25. .10125. 18. .0165. 22. 441.1. 26. 54.675. 19. 1906. 23. .7854. 27. .09885. 338. To find a number whose logarithm is given. • The number that corresponds to a given logarithm is called its antilogarithm. Thus, since the logarithm of 62 is 1.7924, the antilogarithm of 1.7924 is 62. EXERCISES 339. 1. Find the number whose logarithm is 0.9472. Solution. — The two mantissas adjacent to the given mantissa are .9469 and .9474, corresponding to the numbers 8.85 and 8.86, since the given characteristic is 0. The given mantissa is 3 ten-thousandths greater than the mantissa of the logarithm of 8.85, and the mantissa of the logarithm of 8.86 is 5 ten-thousandths greater than that of the logarithm of 8.85. Since the numbers 8.85 and 8.86 differ by 1 one-hundredth, and the mantissas of their logarithms differ by 5 ten-thousandths, it may be assumed .as sufficiently accurate that each increase of 1 ten-thousandth in the mantissa is produced by an increase of J of 1 one-hundredth in the number. Consequently, an increase of 3 ten-thousandths in the man- tissa is produced by an increase of f of 1 one-hundredth, or .006, in the number. Hence, the number v^hose logarithm is 0.9472 is 8.856. 2. Find the antilogarithm of 9.4180 - 10. Solution. — Given mantissa, .4180 Mantissa next less, .4166 ; figures corresponding, 261. Difference, 14 Tabular difference, 17)14(.8 Hence, the figures corresponding to the given mantissa are 2618. Since the characteristic is 9— 10, or — 1, the number is a decimal with no ciphers immediately following the decimal point (Prin. 2). Hence, the antilogarithm of 9.4180 - 10 is .2618. 262 SUPPLEMENTARY TOPICS Find the antilogarithm of : 3. 0.3010. 8. 3.9546. 13. 9.3685-10. 4. 1.6021. 9. 0.8794. 14. 8.9932-10. 5. 2.9031. 10. 2.9371. 15. 8.9535-10. 6. 1.6669. 11. 0.8294. 16. 7.7168-10. 7. 2.7971, 12. 1.9039. 17. 6.7016 - 10. 340. Multiplication by logarithms. Since logarithms are the exponents of the powers to which a constant number is to be raised, it follows that : 341. Principle. — The logarithm of the product of two or more numbers is equal to the sum of their logarithms ; that is, To any base, log (mn) = log m -f- log n. For, let logatn = x and log^n = y, a being any base. It is to be proved 'that logo (mw) = x-\- y. § 327, a' = m, and ay = n. Multiplying, we have a*+w = mn. Hence, § 328, logo (inn) = x+y = logom + log«w. EXERCISES 342. 1. Multiply .0381 by 77. Solution Prin., § 341, log (.0381 x 77) = log .0381 + log 77. log .0381 = 8.5809 - 10 log 77 = 1.8865 Sum of logs = 10.4674 - 10 = 0.4674. 0.4674 = log 2.934. .-. .0381 X 77 = 2.934. SUPPLEMENTARY TOPICS 263 Note. — Three figures of a number corresponding to a logarithm may be found from this table with absolute accuracy, and in most cases the fourth will be correct. In finding logarithms or antilogarithms, allow- ance should be made for the figures after the fourth, whenever they express .5 or more than .5 of a unit in the fourth place. Multiply : 2. 3.8 by 56. 6. 2.26 by 85. 10. 289 by .7854. 3. 72 by 39. 7. 7.25 by 240. 11. 42.37 by .236. 4. 8.5 by 6.2. 8. 3272 by 75. 12. 2912 by .7281. 5. 1.64 by 35. 9. .892 by .805. 13. 1.414 by 2.829. 343. Division by logarithms. Since the logarithms of two numbers to a common base represent exponents of the same number, it follows that : 344. Principle. — The logarithm of the quotient of two num- bers is equal to the logarithm of the dividend minus the logarithm of the divisor ; that is, To any base, log (m -f- n) = log m — log n. For, let loga m = x and log„ n = y, a being any base. It is to be proved that logo (m ^ n) = x — y. § 327, a- = m, and ay = n. Dividing, we have a^-V'= m -^ n. Hence, § 328, log^ {m^n) =x -y = logaW-log„n. EXERCISES 345. 1. Divide .00468 by 73.4. Solution Prin., § 344, log (.00468 - 73.4) = log .00468 - log 73.4. log .00468 = 7.6702 - 10 log 73.4 = 1.8657 Difference of logs = 6.8045 - 10 5.8045 - 10 = log .00006376. .-. .00468 -- 73.4 = .00006376. 264 SUPPLEMENTARY TOPICS 2. Divide 12.4 by 16. Solution Prin., § 344, log (12.4 -h 16) = log 12.4 - log 16. log 12.4 = 1.0934 =11.0934-10 log 16 = 1.2041 Difference of logs = 9.8893 - 10 9.8893 - 10 = log .775. .-. 12.4 - 16 = .775. Suggestion. — The positive part of the logarithm of the dividend may be made to exceed that of the divisor, if necessary to avoid subtracting a larger number from a smaller one as in the above solution, by adding 10 - 10 or 20 - 20, etc. Divide: 3. 3025 by 55. 8. 10 by 3.14. 13. 1 by 40. 4. 4090 by 32. 9. .6911 by .7854. 14. 1 by 75. 6. 3250 by 57. 10. 2.816 by 22.5. 15. 200 by .5236. 6. .2601 by .68. 11. 4 by .00521. 16. 300 by 17.32. 7. 3950 by .250. 12. 26 by .06771. 17. .220 by .3183. 346. Extended operations in multiplication and division. Though negative numbers have no common logarithms, opera- tions involving negative numbers may be performed by con- sidering only their absolute values and then giving to the result the proper sign without regard to the logarithmic work. Since dividing by a number is equivalent to multiplying by its reciprocal, for every operation of division an operation of multiplication may be substituted. In extended operations in multiplication and division with the aid of logarithms, the latter method of dividing is the more convenient. 347. The logarithm of the reciprocal of a number is called the cologarithm of the number. The cologarithm of 100 is the logarithm of yJo' which is — 2. It is written, colog 100 =—2. SUPPLEMENTARY TOPICS 265 348. Since the logarithm of 1 is and the logarithm of a quotient is obtained by subtracting the logarithm of the divisor from that of the dividend, it is evident that the cologarithm of a number is minus the logarithm of the number, or the logarithm of the number with the sign of the logarithm changed ; that is, if log„ m = x, then, colog„ m = — x. Since subtracting a number is equivalent to adding it with its sign changed, it follows that : 349. Principle. — Instead of subtracting the logarithm of the divisor from that of the dividend^ the cologarithm of the divisor may he added to the logarithm of the dividend ; that is, To any base, log (m -r- n) = log m + colog n. EXERCISES 350. 1. Find the value of -^^^ ^f^-^ ^^^^"^ by logarithms. 458 X 15.6 X .029 ^ ^ Solution •Q^^x^^-^x^^^ = .063 X 58.5 X 799 X J- X J- X J- . 458 X 15.6 X .029 458 15.6 .029 log .063= 8.7993-10 / log 58.5= 1.7672 ^^ ^A/^ 6%^, ' log 799= 2.9025 ^"^ SioU /Vwi<^ , colog 458= 7.3391-10 U^ tlA^^i colog 15.6= 8.8069-10 -^ ^C;^^^ colog .029= 1.5376 ^^"^ ^=U^ log of result = 31.1526 - 30 ' " = 1.1526. '' .-. result = 14.21. Find the value of : ^ 110 X 3.1 X .653 ^ 15 X .37 x 26.16 ,33x7.854x1.7 - ' 11 x 8 x .18 x 6.67 266 SUPPLEMENTARY TOPICS (_ 3.04) X .2608 .4051 X (-12.45) ■ 2.046 X .06219 ' ' (- 8.988) x .01442* , . 600 X 5 X 29 „ 78 X 52 X 1605 0» TTZ ^ ^, ^ - . O. .7854 X 25000 x 81.7 ' " 338 x 767 x 431 ' 3.516 X 485 x 65 « -^ X -315 x 428 3.33 X 17 X 18 X 73 .317 x .973 x 43.7 351. Involution by logarithms. Since logarithms are simply exponents, it follows that : 352. Principle. — TJie logarithm of a power of a number is equal to the logarithm of the number multiplied by the index of the poiver; that is, To any base, log m^ = n log m. For, let logo m = x, and let n be any number, a being any base. It is to be proved that logo wi" = nx. §327, a^ = m. Raise each member to the nth power. Ax. 6, <*« a. ^ ^^^ h,^^ Hence, § 328, logo m^ = nx = n logo w. EXERCISES 353. 1. Find the value of .25\ Solution Prin,, § 362, log .25^ = 2 log .25. log .25 = 9.3979 - 10. 2 log .25 = 18.7958 - 20 = 8.7958-10. 8.7958 - 10 = log .06249. .-. .252 = .06249. Note. —By actual multiplication it is found that .252 = .0625, whereas the result obtained by the use of the table is .06249. Also, by multiplication, 18^ = 324, whereas by the use of the table it is found to be 324.1. Such inaccuracies must be expected when a four-place table is used. i SUPPLEMENTARY TOPICS 267 Find by logarithms the value of : 2. 72. 7. .782. 12. 4.071 17. (^)2. 3. 112. 8. 8.052. 13. .5433. 18. (1)1 4. (-47)2. 9. 8.332. 14^ (-T)\ 19. (tWs)'- 5. 4.92. 10. 6.611 15. 1.02^ 20. (^Vt)'- 6. 5.22. 11^ 7142, iQ 17383^ 21. (yis)-^. 354. Evolution by logarithms. Since logarithms are simply exponents, it follows that : 355. Principle. — The logarithm of a root of a number is equal to the logarithm of the number divided by the iyidex of the required root ; that is, To any base, log vm = • For, let loga m = x and let n be any number, a being any base. It is to be proved that loga Vm = x -4- n. § 327, , a" = m. Take the wth root of each member, Ax. 7, Qxn = Vm. Hence, §328, logo\/m = x ^ n ^lOga^ n EXERCISES 356. 1. Find the square root of .1296 by logarithms. Solution Prin., § 356, logV.1296 = I log .1296. log .1296 =9.1126 -10. 2 )19.1126- 20 9.6663 - 10 9.5663 - 10 = log .360. .-. v. 1296 = .36. 268 SUPPLEMENTARY TOPICS Find by logarithms the value of : 2. 225i 8. (-1331)1 14. V2. 20. V-2. 3. 12.25*. 9. 1024*. 15. V3. 21. ^.027. 4. .2023*. 10. .6724^. 16. V5. 22. V30|. 5. 326*. 11. 5.929*. 17. V6. 23. VM. 6. .512* 12. .4624*. 18. /^T ^^ ^/ — |- Divide : 22. 3byl-V^^. 25. a^ + ¥ hj a - bV^^. 23. 2 by 1 + V^^. 26. a - ?>i by ai H- 6. 24. 4+V4by 2-V'^^. 27. (1 + i)^ by 1 - /. 28. Find by inspection the square root of 3 + 2 V— 10. Solution 3 + 2V- 10 = (5-2)+2V5. -2 = 6 + 2V5. -2 + (-2). .-. V3 + 2V- 10=V6 + 2\/5. -2 + (-2) =V5 + v'-2. Find by inspection the square root of : 29. 4 + 2V^21. 33. 4V-3-1. 30. 1+2V^^. 34. 12V^^-5. 31. 6-2V^, 35. 24V^^-7. 32. 9-f2V-22. 36. b'' + 2ab^-l-a\ 37. Verify that — 1 4-V— 1 and — 1 —V— 1 are roots of the equation x^ -{- 2 x -\- 2 = 0. 362. TJie sum arid the product of two conjugate complex num- bers are both real. For, let a + hV^^X and a — 6 V— 1 be conjugate complex numbers. Their sum is 2 a. Since (V— 1)2 =— 1, their product is, a2 - {hy/^^Y = a^ - (~ h'^) = a2 + &2. 272 SUPPLEMENTARY TOPICS 363. If two complex numbers are equal, their real parts are equal and also their imaginary parts. For, let a -f bV^ = x 4- yV^^. Then, a — x = (y — b)V— 1, which, § 358, is impossible unless a = x and y = b. ' 364. If a + ^V— 1 = 0, a and b being real, then a = and 6 = 0. For, if then, and, squaring, we have whence, a + bV- 1 =0, 6V— 1 =— a, - b-' = aP' ; oP' 4- ?>■- = 0. Now, a2 and h^ are both positive ; hence, their sum cannot be unless each is separately ; that is, a = and h — ^. 365. Relation between the units +1, —1, V— 1, and — V — 1. The use of rectangular axes is a device for representing simultaneous values of two varying numbers. In the preced- ing discussions only real numbers have been represented. But by confining real numbers to the a^-axis, it is possible, in har- mony v^ith established algebraic laws, to devote the ?/-axis to the representation of imaginary numbers. V In the accompanying figure the length of any radius of the circle rep- resents the arithmetical unit 1. The line drawn ^ — X from to A, called the line OA, represents the positive unit -f- 1, and the line 0^" represents the negative unit — 1. Every real number lies some- where on the line X'X, which is supposed to extend indefi- nitely in both directions from 0. X^X is called the axis of real numbers. j!. / A'\ \}\ ? *u SUPPLEMENTARY TOPICS 273 The direction of any line drawn from 0, as OB, that is, the quality or direction sign of the number represented by that line,. is determined with reference to the fixed line OA by find- ing what part of a revolution is required to swing the line from the position OA to the required position. By common con- sent revolution of the line OA is performed in a direction op- posite to that of the hands of a clock, as shown by the arrows. OB is reached after i of a revolution, OA^ after ^ of a revolu- tion, 0^1" after i of a revolution, etc. Since 0^", or — 1, represents ^ of a revolution of OA, the square of 0A'\ or (— 1)^, represents 1 revolution of OA, which produces OA, or -|- 1. Hence, OA'^, or — 1, represents the square root of -|- 1, or (4-1)2. Similarly, since OA^ represents ^ of. i of a revolution of OA, and OA^^ represents ^ of a revolution of OA, OA' repre- sents the square root of OA'', or of — 1 ; that is, OA' = V— 1. If OA" is swung i of a revolution from the position 0^" to the position OA'", OA" will be multiplied by V— 1 just as OA is multiplied by V— 1 to produce OA', Hence, the result OA'" = -1 . V^=T. = - V^=^. -\- 1, represented by OA, and — 1, by OA", are the units for real numbers, that is, are real units. Just as the real number + a is represented by a line a units long extending from toward X, and the real number — a by a line a units long ex- tending from in the opposite direction, so the imaginary number -faV— 1, or (+V— l)xa, is represented by a line a units long extending from toward Y, and the imaginary number — a V— 1, or (— V— 1) X a, by a line a units long ex- tending from in the opposite direction, toward Y'. Hence, -f V— 1 and — V— 1 are the units for imaginary numbers, that is, they are imaginary units; + aV— 1 is called a posi- tive imaginary number and — aA/'— 1 a negative imaginary number. Hence, it is seen that imaginary numbers have as much reality as real numbers. Imaginary numbers were named ])efore their nature was understood. milne's sec. course alg. — 18 274 SUPPLEMENTARY TOPICS 366. Graphical representation of a complex number. The sum of 3 positive real units and 2 positive imaginary units is found by counting 3 units along OX in the positive direction from and from ^a t point, Z), measuring 2 WBlbs upward at right angles to OX in the direction of the axis of imaginary numbers. The line OP represents by its ^ length and direction the com- bined effect or sum of the directed lines OD and DP, that is, the complex number The same result may be obtained by counting 2 units along Y upward from and from the end of the second division measuring 3 units toward the right at right angles to OF in the direction of the axis of real numbers. Hence, the line OP represents either 3-f2V— lor2V- -f 3. Similarly, the line OF represents either 2^ — V — 1 -V^l+21 and the line OP" either 3 ^/ZTx _ i p r ^ p X— ■■ • 'o .""^ l i+lV-1 -1 or or EXERCISES 367. Represent the following numbers graphically : 4. i_3V^^. 5. 2-W^. 6. 3V^=T:-5. 2 + V- 7. 8. 9. 1. 3+4V^^. 2. 4 + 2V^^. 3. 3-2\/^=^. 10. Represent graphically — 2 + V— 13. Suggestion. — The given expression may be written The approximate value of Vl3 is 3.6. Represent graphically : 11. 2+V^^. 13. -4-V 12. 3-V^^. 14. -2-2V-1. 2 2 ^ ■^* _ .3 I 3 V— 1 2 r 9 V J-. ■ 2+VT3 V- 1. 10. 2_V-13. 15. 16. 5+V' 2-V~- 17. 20. SSARY Abscissa. A distance measured along or parallel to the x-axis. Absolute term. A term that does not contain an unknown number. Absolute value. The value of a number without regard to its sign. Addends. Numbers to be added. Addition. The process of finding a single expression for the algebraic sum of two or more numbers. Affected quadratic. A quadratic that contains both the second and first powers of one unknown number. Algebra. That branch of mathematics which treats of general num- bers and the nature and use of equations. It is an extension of arithmetic and it uses both figures and letters to express numbers. Algebraic expression. A number represented by algebraic symbols. Algebraic numbers. Positive and negative numbers, whether integers or fractions. Algebraic sum. The result of adding two or more algebraic numbers. Alternation. When the antecedents of a proportion are to each other as the consequents, the numbers are said to be in proportion by alterna- tion. Antecedent. The first term of a ratio. Antilogarithm. The number that corresponds to a given logarithm. Arithmetical progression. Same as arithmetical series. Arithmetical series. A series, each term of which after the first is derived from the preceding by the addition of a constant number. Arrangement. When a polynomial is arranged so that in passing from left to right the several powers of some letter are successively higher or lower, the polynomial is said to be arranged according to the ascending or descending powers, respectively, of that letter. Axes of reference. Two straight lines that intersect, usually at right angles, used to locate a point or points in a plane. Axiom. A principle so simple as to be self-evident. Base of a logarithm. See , logarithm. Binomial. An algebraic expression of two terms. Binomial formula. The formula by means of which any irdicated power of a binomial may be expanded. Binomial quadratic surd. A binomial surd whose surd or surds are of the second order. Binomial surd. A binomial, one or both of whose terms are surds. 275 276 GLOSSARY Binomial theorem. The principle by means of which any indicated power of a binomial may be expanded. Biquadratic surd. A surd of the fourth order. Briggs logarithms. Same as common logarithms. Characteristic. The integral part of a logarithm. Clearing an equation of fractions. The process of changing an equa- tion containing fractions to an equation without fractions. Coefficient. When one of the two factors into which a number can be resolved is a known number, it usually is written first and called the coefficient of the other factor. In a broader sense, either one of the two factors into which a number can be resolved may be considered the coefficient of the other. Co-factor. Same as coefficient. Cologarithm. The logarithm of the reciprocal of a number is called the cologarithm of the number. Common denominator. Two or more fractions that have the same denominator are said to have a common denominator. Common difference. The constant number that is added to any term of an arithmetical progression to produce the next. Common factor. A factor of each of two or more numbers. Common logarithms. The system of logarithms whose base is 10. Common multiple. An expression that exactly contains each of two or more given expressions. Complete quadratic. Same as affected quadratic. Complex fraction. A fraction one or both of whose terms contains a fraction. Complex number. The algebraic sum of a real number and an imagi- nary number. Composition. When the sum of the terms of one ratio of a proportion is to one of the terms as the sum of the terms of the other ratio is to its corresponding term, the numbers are said to be in proportion by compo- sition. Composition and division. When the sum of the terms of one ratio of a proportion is to their difference as the sum of the terms of the other ratio is to their difference, the numbers are said to be in proportion by composition and division. Compound expression. Same as polynomial. Conditional equation. An equation that is true for only certain values of its letters. Conjugate complex numbers. Two complex numbers that differ only in the signs of their imaginary terms. ' Conjugate surds. Two binomial quadratic surds that differ only in the sign of one of the terms. Consequent. The second term of a ratio. Consistent equations. Same as simultaneous equations. Constant. A number that has the same value throughout a discussion. GLOSSARY 277 Continued fraction. A complex fraction of the form . Continued proportion. A multiple proportion in which each conse quent is repeated as the antecedent of the following ratio. Coordinate axes. Sanne SiS axes of reference. Coordinates. See rectangular coordinates. Couplet. The two terms of a ratio. Cube. Same as third power. Cube root. One of the three equal factors of a number. Cubic surd. A surd of the third order. Degree of a term. The sum of the exponents of the literal factors of a rational integral term determines the degree of the term. Degree of an expression. The term of highest degree in any rational integral expression determines the degree of the expression. Denominator. The divisor in an algebraic fraction. Dependent equations. Two or more equations that express the same relation between the unknown numbers involved are often called depend- ent equations, for each may be derived from any one of the others. Derived equations. Same as dependent equations. Difference. The result of subtracting one number from another. That is, the difference is the algebraic number that added to the subtra- hend gives the minuend. Discriminant. The expression 6^ _ 4 ^j^, which appears in the roots of the general quadratic equation ax'^ -]- bx + c =0, Dissimilar monomials. Monomials that contain different letters or the same letters with different exponents. Dissimilar terms. Terms that contain different letters or the same letters with different exponents. Dividend. In division, the number that is divided. Division. The process of finding one of two factors when their product and one of the factors is given. Division in proportion. When the difference of the terms of one ratio of a proportion is to one of the terms as the difference of the terms of the other ratio is to its corresponding term, the numbers are said to be in proportion by division. Divisor. In division, the number by which the dividend is divided. Duplicate ratio. The ratio of the squares of two numbers is called their duplicate ratio. Elimination. The process of deriving from a system of simultaneous equations another system involving fewer unknown numbers. Entire surd. A surd that has no rational coefficient except unity. Equation. A statement of the equality of two numbers or expressions. Equation of a problem. The statement in algebraic language of the conditions of the problem. Equation of condition. Same as conditional equation. 278 GLOSSARY Equation of the first degree. Same as simple equation. Equation of the second degree. Same as quadratic equation. Equivalent equations. Two or more equations that have the same roots, each equation having all the roots of the other. Equivalent fractions. Fractions that are of the same value, though they may differ in form. Even root. A root whose index is an even number. Evolution. The process of finding any required root of a number. Exponent. A small figure or letter placed at the right and a little above a number to indicate how many times the number is to be used as a factor. Extraneous root. A value found for the unknown number, in the solution of an equation, that does not satisfy the equation. Extremes of a proportion. The first and fourth terms. Extremes of a series. The first and last terms. Factor. Each of two or more numbers whose product is a given number. Factorial n. The product of the successive integers from 1 to 7t or from n to 1, n being any integer. Factoring. The process of separating a number into its factors. Finite number. A number that cannot become either infinite or infinitesimal. Finite series. A series consisting of a limited number of terms. First degree equation. Same as simple equation. Formula. An expression of a principle, a rule, or a law in symbols. Fourth proportional. The fourth number of four different numbers that form a proportion. Fourth root. One of the four equal factors of a number. Fraction. In algebra, an indicated division ; in arithmetic, one or more of the equal parts of a unit. Fractional equation. An equation that involves an unknown number in any denominator. Fractional expression. An expression, any term of which is a fraction. Fulcrum. The point or edge upon which a lever rests. Function. An expression involving one or more letters is called a function of those letters. General number. A literal number to which any value may be as- signed. Geometrical progression. Same as geometrical series. Geometrical series. A series, each term of which after the first is derived by multiplying the preceding term by some constant mul- tiplier. Graph. A picture (line or lines) every point of which exhibits a pair of corresponding values of two related quantities. Graph of an equation. The line or lines containing all the points, and only those, whose coordinates satisfy a given equation. GLOSSARY 279 Greater than. One number is said to be greater than another when the remainder obtained by subtracting the second from the first is positive. Higher equation. An equation that contains a higher power of the unknown number than the second. Highest common factor. The common factor of two or more expres- sions that has the largest numerical coefficient and is of the highest degree. It is equal to the product of all the common factors of the expressions. Homogeneous equation. An equation all of whose terms are of the same degree with respect to the unknown numbers. Homogeneous expression. An expression all of whose terms are of the same degree. Identical equation. An equation whose members are identical, or such that they may be reduced to the same form. Identity. Same as identical equation. Imaginary number. A number that involves an indicated even root of a negative number. Incomplete quadratic. Same as pure quadratic. Inconsistent equations. Two or more equations that are not satisfied in common by any set of values of the unknown numbers. Independent equations. Two or more equations that express different relations between the unknown numbers involved, and so cannot be re- duced to the same equation. Indeterminate equation. An equation that is satisfied by an unlimited number of sets of values of its unknown numbers. Index of a power. Same as exponent. Index of a root. A small figure or letter written in the opening of a radical sign to indicate what root of a number is sought. Inequality. An algebraic expression indicating that one number is greater than or less than another. Infinite number. A variable that may become numerically greater than any assignable number. Infinite series. A series consisting of an unlimited number of terms. Infinitesimal number, A variable that may become numerically less than any assignable number. Infinity. The same as infinite number. Integer. Same as whole number. Integral equation. An equation that does not involve an unknown number in any denominator. Integral expression. An expression that contains no fraction. Inverse ratio. Same as reciprocal ratio. Inversion. When the terms of each ratio of a proportion are written in inverse order, the numbers are said to be in proportion by inversion. Involution. The process of finding any required power of an expres- sion. 280 GLOSSARY Irrational equation. An equation involving an irrational root of an unknown number. Irrational expression. An expression containing an irrational number. Irrational number. A number that cannot be expressed as an integer or as a fraction with integral terms. Known number. A general number or a number Tthose value is known. Less than. One number is said to be less than another when the re- mainder obtained by subtracting the second from the first is negative. Lever. Any sort of a bar resting on a fixed point or edge. Like degree. The same degree. Like terms. Same as similar terms. Limit of a variable. A constant which the value of the variable con- tinually approaches but never reaches. Linear equation. Same as simple equation. Linear function. A function of the first degree in the variable or variables involved. Literal coefficient. A coefficient composed of letters. Literal equation. An equation one or more of whose known numbers is expressed by letters. Literal numbers. Letters that are used for numbers. Logarithm. The exponent of the power to which a fixed number, called the base, must be raised in order to produce a given number is called the logarithm of the given number. Lowest common denominator. The denominator of lowest degree, having the least numerical coefficient, to which two or more fractions can be reduced. It is equal to the lowest common multiple of the given denominators. Lowest common multiple. The expression having the smallest numer- ical coefficient and of lowest degree that will exactly contain each of two or more given expressions. Lowest terms. When the terms of a fraction have no common factor, the fraction is said to be in its lowest terms. Mantissa. The fractional or decimal part of a logarithm. Mean proportional. A number that serves as both means of a propor- tion. Means of a proportion. The second and third terms. Means of a series. All of the terms except the first and the last. Members of an equation. In an equation, the number on the left of the sign of equality is called the first member of the equation, and the number on the right is called the second member. Minuend. In subtraction, the number from which the subtraction is made. Mixed coefficient. A coefficient composed of both figures and letters. Mixed expression. An expression some of whose terms are integral and some fractional. GLOSSARY 281 Mixed number. Same as mixed expression. Mixed surd. A surd that has a rational coefficient. Monomial. An algebraic expression of one term only. Multiple proportion. A proportion that consists of three or more equal ratios. Multiplicand. In multiplication, the number multiplied. Multiplication. When the multiplier is a positive integer, the process of taking the multiplicand as many times as there are units in the multiplier. In general, the process of finding a number that is obtained from the multiplicand just as the multiplier is obtained from unity. Multiplier. In multiplication, the number by which the multiplicand is multiplied. Natural numbers. The numbers 1, 2, 3, 4, and so on. Negative number. A number less than zero. Negative term. A term preceded by the sign — . Numerator. The dividend in an algebraic fraction. Numerical coefficient. A coefficient composed of figures. Numerical equation. An equation all of whose known numbers are expressed by figures. Odd root. A root whose index is odd. Order of a radical or of a surd is indicated by the index of the root or by the denominator of the fractional exponent. Ordinate. A distance measured along or parallel to the ?/-axis. Origin. The intersection of the axes of reference. Parentheses. One of the signs of aggregation ( ). Pascal's triangle. The triangular array of coefficients of the ex- pansion of successive powers of a binomial, beginning with the zero power. Perfect square. An expression that may be separated into two equal factors. Polynomial. An algebraic expression of more than one term. Positive number. A number greater than zero. Positive term. A term preceded by + , expressed or understood. Power of a number. The product obtained when the number is used a certain number of times as a factor. Prime factor. A factor that is a prime number. Prime number. A number that has no factors except itself and 1. Prime to each other. Expressions that have no common prime factor except 1 are said to be prime to each other. Principal root. A real root of a number that has the same sign as the number itself. Problem. A question that can be answered only after a course in reasoning. Product. The result of multiplying one number by another. Proportion. An equality of ratios. 282 GLOSSARY Pure quadratic. An equation that, when simplified, contains only the second power of the unknown number. Quadratic equation. An equation that, when simplified, contains the square of the unknown number, but no higher power. Quadratic form. An expression that contains but two powers of an unknown number or expression, the exponent of one power being twice that of the other. Quadratic formula. A formula that expresses the roots of the general quadratic equation ax^ -f 6a: + c = 0. Quadratic function. A function of the second degree in the variable or variables involved. Quadratic surd. A surd of the second order. Quotient. The result of dividing one number by another. Radical. An indicated root of a number. Radical equation. Same as irrational equation. Radical expression. An expression that involves a radical in any way. Radicand. A number whose root is required. Ratio. The relation of two numbers that is expressed by the quotient of the first divided by the second. Ratio of a geometrical series. The number by which any term of the series is multiplied to produce the next. Ratio of equality. A ratio whose antecedent is equal to its consequent. Ratio of greater inequality. A ratio whose antecedent is greater than its consequent. Ratio of less inequality. A ratio whose antecedent is less than its consequent. Rational expression. An expression that contains no irrational number. Rational factor of a surd. A factor whose radicand is a perfect power of a degree corresponding to the order of the surd. Rational number. A number that is, or may be, expressed as an integer or as a fraction with integral terms. Rationalization. The process of multiplying an expression containing a surd by any number that will make the product rational. Rationalizing factor. The factor by which a surd expression is mul- tiplied to render the product rational. Rationalizing the denominator. The process of reducing a fraction having an irrational denominator to an equal fraction having a rational denominator. Real number. A number that does not involve the even root of a negative number. Reciprocal of a number is 1 divided by the number. Reciprocal of a fraction is the fraction inverted or 1 divided by the fraction. Reciprocal ratio. The ratio of the reciprocals of two numbers is called the reciprocal ratio of the numbers. GLOSSARY 283 Rectangular coordinates. The abscissa and ordinate of a point re- ferred to two perpendicular axes are called the rectangular coordinates of the point. Reduction. The process of changing the form of an expression with- out changing its value. Remainder in subtraction. Same as difference. Root of an equation. Any number that satisfies the equation. Root of a number. When the factors of a number are all equal, one of the factors is called a root of the number. Satisfied. When an equation is reduced to an identity by the substi- tution of certain known numbers for the unknown numbers, the equation is said to be satisfied. Second degree equation. Same as quadratic equation. Second power. When a number is used twice as a factor, the product is called the second power of the number. Second root. Same as square root. Series. A succession of numbers, each of which aftei the first is derived from the preceding number or numbers according to some fixed law. Sign of a fraction. The sign written before the dividing fine of a fraction. Similar monomials. Monomials that contain the same letters with the same exponents. Similar radicals. Radicals that in their simplest form are of the same order and have the same radicand. Similar surds. Surds that in their simplest form are of the same order and have the same radicand. Similar terms. Terms that contain the same letters with the same exponents. Simple equation. An integral equation that involves only the first power of one unknown number in any term when similar terms have been united. Simple expression. Same as monomial. Simplest form of a radical. A radical is in its simplest form when the index of the root is as small as possible, and when the radicand is integral and contains no factor that is a perfect power of a degree corre- sponding to the index of the root. Simultaneous equations. Two or more equations that are satisfied by the same set or sets of values of the unknown numbers form a system of simultaneous equations. Solution of an equation. The process of finding the roots of an equation. Square. Same as second power. Square root. One of the two equal factors of a number. Substitution. When a particular number takes the place of a letter, or general number, the process is called substitution. 284 GLOSSARY Subtraction. The process of finding one of two numbers when their sum and the other number are given. Subtraction is the inverse of addition. Subtrahend. In subtraction, the number that is subtracted. • Sum. See algebraic sum. Surd. The indicated root of a rational number that cannot be ob- tained exactly. Symmetrical equation. An equation that is not affected by inter- changing the unknown numbers involved. Term. An algebraic expression whose parts are not separated by the signs 4- or — . Terms of a fraction. The numerator and denominator of a fraction. Terms of a series. The successive numbers that form the series. Third power. When a number is used three times as a factor, the product is called the third power of the number. Third proportional. The consequent of the second ratio when the means of a proportion are identical. Third root. Same as cube root. Transposition. The process of removing a term from one member of an equation to the other. Trinomial. An algebraic expression of three terms. Trinomial square. A trinomial that is a perfect square. Triplicate ratio. The ratio of the cubes of two numbers is called their triplicate ratio. Unknown number. A number whose value is to be found. Unlike terms. Same as dissimilar terms. Variable. A number that under the conditions imposed upon it may have a series of different values. Vary. Same as vary directly. Vary directly. One quantity or number is said to vary directly as another, when the two depend upon each other in such a manner that if one is changed the other is changed in the same ratio. Vary inversely. One quantity or number varies inversely as another when it varies as the reciprocal of the other. Vary jointly. One quantity or number varies jointly as two others when it varies as their product. Whole number. A unit or an aggregate of units. X-axis. The horizontal axis of reference is usually called the x-axis. Y-axis. The vertical axis of reference is usually called the y-axis. INDEX (The numbers refer to pages.) Abscissa, 110, 275. Absolute term, 45, 163, 275. Absolute value, 13, 275. Absolute zero, 212. Addends, 275. Addition, 14-15, 65-66, 142-143, 158-159, 269-270, 275. of complex numbers, 269-270. of fractions, 65-66. of imaginary numbers, 158-159. of monomials, 14. of polynomials, 15. of radicals, 142-143. Affected quadratics, 161, 163-167, 193- 196, 275. solved by completing the square, 164- 165. solved by factoring, 163. solved by formula, 165-166. solved by graphs, 193-196. Algebra, 275. Algebraic expression, 9, 275. Algebraic fraction, 61, 278. Algebraic numbers, 13, 275. Algebraic signs, 8. Algebraic sum, 13, 275. Antecedent, 87, 275. Antilogarithms, 261-262, 275. Arithmetical means, 220-221. Arithmetical progressions, 215-223, 275. Arithmetical series, 215, 275. Arrangement of polynomial, 24, 275. Associative law, for addition, 14. for multiplication, 21. Axes of reference, 110, 275. Axioms, 33, 275. Base of a logarithm, 256, 275. Binomial, 9, 275. Binomial formula, 120-121, 275. Binomial quadratic surd, 147, 275, Binomial surd, 147, 275. Binomial theorem, 118-121, 276. Biquadratic surd, 138, 276. Braces, 8. Brackets, 8. Briggs system of logarithms, 256, 276. Characteristic, 256, 276. Circle, 197. Clearing equations of fractions, 74-77, 276. Coefficient, 9, 276. Co-factor, 276. Collecting coefficients, 20. Cologarithm, 264, 276. Common denominator, 64, 276. Common difference, 215, 276. Common factor, 41, 276. Common multiple, 41, 276. Common system of logarithms, 256, 276. Commutative law, for addition, 14. for multiplication, 21. Complete quadratic, 161, 276. Complex fractions, 70-71, 276. Complex numbers, 269-274, 276. Compound expression, 276. Conditional equation, 276. Conjugate complex numbers, 269, 276. Conjugate surds, 147, 276. Consequent, 87, 276. Consistent equations, 95, 276. Constant, 107, 211, 276. Continued fractions, 71, 277. Continued proportion, 277. Coordinate axes, 277. Coordinates, 110, 277. Couplet, 87, 277. Cube, 10, 277. Cube root, 122, 245-250, 277. of arithmetical numbers, 248-250. of polynomials, 245-247. Cubic surd, 138, 277. Degree, of expression, 41, 277. of term, 41, 277. Denominator, 61, 277. Dependent equations, 95, 277. Derived equations, 277. Detached coefficients, 24, 30. Difference, 277. Discontinuous curve, 199. Discriminant, 203, 277. Dissimilar monomials, 10, 277. Dissimilar terms, 277. Distributive law, for division, 27. for evolution, 122, 129, 130. for involution, 117, 129, 130. for multiplication, 22. Dividend, 277. Division, 27-32, 69-72, 145, 160, 263-264, 264-266, 271, 277. by detached coefficients, 30. by logarithms, 263-264, 264-266. by monomials, 27-28. by polynomials, 28-31. of complex numbers, 271. of fractions, 69-71. of imaginary numbers, 160. of radicals, 145. Divisor, 277. Duplicate ratio, 87, 277. Elimination, 95-101, 277. by addition, 95-96. by substitution, 96-97. by subtraction, 95-96. Ellipse, 198. Entire surd, 138, 277. Equation of a problem, 36, 277. Equation of condition, 73, 277. Equations, 33-40, 58, 73-86, 95-106, 112- 115, 137, 152-156, 161-192, 193-196, 200-202, 203-210, 235-243, 277. in the quadratic form, 176-178. 285 286 INDEX Equations, of condition, 73, 277. of the first degree, 278. of the second degree, 161, 278. solved by factoring, 58, 163. Equivalent equations, 73, 278. Equivalent fractions, 63, 278. Even root, 122, 278. Evolution, 117, 122-128, 146-149, 245- 250, 267-268, 271, 278. by logarithms, 267-268. of arithmetical numbers, 125-128, 248- 250. of complex numbers, 271. of monomials, 123. of polynomials, 123-125, 245-247. of radicals, 146-149. Examination questions, 244. Expansion of (a + x)'^, 118. Exponential equations, 137. Exponents, 10, 129-137, 278. Exponents and radicals, 129-156. Extraneous roots, 74, 169, 278. Extremes, of a proportion, 90, 278. of a series, 215, 278. Factor, 41, 278. Factor theorem, 52. Factorial n, 8, 120, 278. Factoring, 41-57, 209-210, 278. binomials, 42-43. by completing the square, 209-210. polynomials, 50-54. trinomials, 44-49. Factors and multiples, 41-60. Finite number, 212, 278. Finite series, 225, 278. First degree equation, 278. First member of an equation, 33, 280. Formation of quadratic equations, 206-208. Formula, 12, 278. Formula for the rth term of the expansion of (a + x)«, 120. Formulae, 84-86, 174-175, 268. Fourth proportional, 90, 278. Fourth root, 122, 278. Fractional equation, 73, 278. Fractional exponents, 132-137. Fractional expression, 278. Fractions, 61-72, 157, 278. Fractions indeterminate in form, 214 . Fulcrum, 94, 278. Function, 107, 278. Fundamental property of imaginaries, 157. General directions, 36, 55, 152, 166. for factoring, 55. for solving problems, 36. for solving quadratics, 166. for solving radical equations, 152. General number, 278. General review, 231-244. Geometrical means, 228. Geometrical progressions, 223-230, 278. Geometrical series, 223, 278. Graph, 108, 278. Graph of an equation, 278. Graphic solutions, 107-116, 193-202. of quadratics in x, 193-196. of simultaneous equations involving quadratics, 200-202. Graphic solutions, of simultaneous linear equations, 112-115. Graphical representation, 108-110, 115- 116, 138, 274. of complex numbers, 274. of a radical of the second order, 138. Graphs of quadratic equations in x and y, 196-199. Grouping terms in parentheses, 19. Higher equation, 279, Highest common factor, 41, 59, 279. Homogeneous equation, 181, 279. Homogeneous expression, 279. Homogeneous in unknown terms, 181. Hyperbola, 199. Identical equation, 73, 279. Identity, 34, 73, 279. Imaginary numbers, 122, 157-160, 203, 279. Impossible equation, 155. Incomplete quadratic, 161, 279. Inconsistent equations, 95, 113, 279. Independent equations, 95, 279, Indeterminate equations, 95, 113, 279. Index, of power, 279. of root, 122, 279, Inequality, 87, 279. Infinite number, 211, 279. Infinite series, 225, 279. Infinitesimal, 212, 279. Infinity, 279. Integer, 279. Integral equation, 73, 279. Integral expression, 41, 62, 279. Interpretation, of forms a XO, ^, ^, -^,212-214. 0°^ of results, 211-214, Introducing roots, 74, 155-156, 166, 169. Introductory review, 9-40. Inverse ratio, 87, 279, Involution, 117-121, 146, 158, 266-267, 279, by binomial theorem, 118-119. by logarithms, 266-267, of imaginaries, 158, of monomials, 117-118, of polynomials, 119. of radicals, 146. Involution and evolution, 117-128. Irrational equation, 152, 280. Irrational expression, 138, 280. Irrational number, 138, 203, 280. Known number, 34, 280. Last term, of arithmetical series, 216-217. of geometrical series, 223-225. Law of a series, 215. Law of coefficients, for division, 27, for multiplication, 21, Law of exponents, for division, 27, 32, 129. for evolution, 122, 129, 130. for involution, 117, 129, 130. for multiplication, 21, 129. Law of grouping, for addition, 14. for multiplication, 21. Law of order, for addition, 14. for multiplication, 21. INDEX 287 Law of signs, for division, 27, 32. for evolution, 122. for involution, 117. for multiplication, 21. Lever, 94, 280. Like degree, 19, 280. Like terms, 10, 280. Limit of variable, 211, 280. Linear equation, 112, 280. Linear functions, 107-116, 280. Literal coefficient, 9, 280. Literal equations, 35, 73, 77, 84-86, 99, 101, 106, 154-155, 168, 170, 174-175, 188, 280. Literal numbers, 280. Logarithm of a number, 257-261. . Logarithms, 256-268, 280. Lowest common denominator, 04, 280. Lowest common multiple, 41, 60, 280. Lowest terms, 63, 280. Mantissa, 256, 280. Mean proportional, 90, 280. Means, of a proportion, 90, 280. of a series, 215, 280. Members of an equation, 33, 280. Minimum points, 195. Minuend, 280. Mixed coefficient, 9, 280. Mixed expression, 280. Mixed number, 62, 281. Mixed surd, 138, 281. Monomial, 9, 281. Monomial factors, 41. Multiple proportion, 281. Multiplicand, 281. Multiplication, 21-26, 67-68, 143-144, 159-160, 262-263, 264-266, 270, 281. by detached coefficients, 24. by logarithms, 262-263, 264-266. of complex numbers, 270. of fractions, 67-68. of imaginary numbers, 159-160. of monomials, 21. of polynomials, 23-24. of polynomials by monomials, 22. of radicals, 143-144. Multiplier, 281. Natural numbers, 157, 281. Nature of roots of a quadratic equation, 203-205. Negative exponents, 131, 133-137. Negative numbers, 13, 157, 281. Negative term, 281. Notation and definitions, 9-10. Number of roots of a quadratic equation, 208. Numerator, 61, 281. Numerical coefficient, 9, 281. Numerical equation, 73, 281. Numerical substitution, 11-12. Odd root, 122, 281. Order, of operations, 11. of radical, 281. of surd, 138, 281. Ordinate, 110, 281. Origin, 111, 281. Parabola, 194, 197. Parentheses, 10, 18-20, 281. Pascal's triangle, 118, 281. Perfect square, 44, 281. Plotting points and constructing graphs, 111-112. Polynomial, 9, 281. Positive and negative numbers, 13. Positive numbers, 13, 281. Positive term, 281. Powers, 10, 117-119, 146, 158, 266-267, 281. by binomial theorem, 118-119. by logarithms, 266-267. of V-1, 158. of monomials, 117-118. of polynomials, 119. of radicals, 146. Prime factor, 41, 281. Prime number, 281. Prime to each other, 41, 281. Principal root, 122, 281. Problems, 36-40, 78-86, 92-94, 101-106, 128, 162, 170-175, 189-192, 218, 221- 223, 224-225, 229-230, 236-243, 254- 255, 281. Product, 281. Progressions, 215-230. Properties, of complex numbers, 271-272. of proportions, 90-91. of quadratic equations, 203-210. of quadratic surds, 147-148. of ratios, 88. Proportion, 90-94, 281. by alternation, 91, 275. by composition, 91, 276. by composition and division, 91, 276. by division, 91, 277. by inversion, 91, 279. Pure quadratics, 161-162, 282. Quadratic equations, 161-192, 193-202, 203-210, 282. solved by completing the square, 164- 165. solved by factoring, 163. solved by formula, 165-166. solved by graphs, 193-196, 200-202. Quadratic form, 176, 282. Quadratic formula, 165, 282. Quadratic functions, 193-202, 282. Quadratic surd, 138, 282. Quotient, 282. Radical equations, 152-156, 169-170, 282. Radical expression, 138, 282. Radical sign, 8. Radicals, 138-156, 282. Radicand, 138, 282. Ratio, 87-89, 282. of equality, 87, 282. of geometrical series, 223, 282. of greater inequality, 87, 282. of less inequality, 87, 282. Ratio and proportion, 87-94. Rational expression, 41, 138, 282. Rational factor of a surd, 138, 282. Rational number, 138, 203, 282. Rationalization, 149-151, 282. Rationalizing factor, 149, 282. 288 INDEX Rationalizing the denominator, 149, 282. Real number, 122, 157, 203, 282. Reciprocal, 69, 282. Reciprocal ratio, 87, 282. Rectangular coordinates, 110, 283. Reduction, 62, 283. Reduction of fractions, 62-64. to integers or mixed numbers, 62. to lowest common denominator, 64. , to lowest terms, 63. Reduction of mixed expressions to frac- tions, 65. Reduction of mixed surd to entire surd, 141. Reduction of radicals, 139-142. to same order, 141-142. to simplest form, 139-140. Relation, between +1, — 1, +a/— 1, -V^, 272-273. of roots and coefficients in a quadratic equation, 206. Remainder, 283. Removal of parentheses, 18-19. Removing roots, 74, 169. Root, of an equation, 73, 283. of a number, 10, 117, 283. Roots, 122-128, 146-149, 245-250, 267- 268 271 by logarithms, 267-268. of arithmetical numbers, 125-128, 248- 250. of complex numbers, 271. of monomials, 123. of polynomials, 123-125, 245-247. of radicals, 146-149. Satisfied, 73, 283. Satisfying an equation, 34. Scale of algebraic numbers, 13. Second degree equation, 161, 283. Second member of an equation, 33, 280. Second power, 283. Second root, 283. Series, 215, 283. Signs, 8. of aggregation, 8, 18. of fractions, 61, 283. of roots of a quadratic, 204. Similar monomials, 10, 283. Similar radicals, 138, 283. Similar surds, 283. Similar terms, 283. Simple equations, 33-40, 73-86, 95-100, 111-115, 152-156, 283. Simple expression, 283. Simplest form of a radical, 139, 283. Simultaneous equations, 95-106, 112-115, 179-192, 200-202, 283. Simultaneous equations involving quad- ratics, 179-192, 200-202. both quadratic and homogeneous in un- known terms, 183-184. both quadratic, one homogeneous, 181- 182. both symmetrical, 180-181. division of one by the other, 186. elimination of similar terms, 186. one simple the other higher, 179, solved by graphs, 200-202. solved by special devices, 184-187. symmetrical except as to sign, 185. Simultaneous simple equations, 95-106, 112-115. solved by graphs, 112-115. Solution, of a problem, 33. of an equation, 33, 73, 283. Special cases, in division, 32. in multiplication, 25-26. Square, 10, 283. Square root, 122-128, 147-149, 271, 283. of arithmetical numbers, 125-128. of binomial quadratic surds, 147-149. of complex numbers, 271. of monomials, 123. of polynomials, 123-125. Substitution, 283. Subtraction, 16-17, 65-66, 142-143, 158- 159, 270, 284. of complex numbers, 270. of fractions, 65-66. of imaginary numbers, 158-159. of monomials, 16. of polynomials, 17. of radicals, 142-143. Subtrahend, 284. Sum, 284. of arithmetical series, 217-218. of finite geometrical series, 225-226. of infinite geometrical series, 226-227. Summary of factoring, 55-57. Supplementary topics, 245-274. Surd, 138, 157, 284. Symbols, 8. Symmetrical equation, 179, 284. Synthetic division, 30-31. Table of logarithms, 258-259. Tabular difference, 260. Term, 9, 284. Terms, of a fraction, 61, 284. of a series, 215, 284. Theory of exponents, 129-137. Third power, 284. Third root, 284. Third proportional, 90, 284. Transposition in equations, 34-35, 284. Trinomial, 9, 284. Trinomial square, 44, 284. Triplicate ratio, 87, 284. Unknown number, 34, 284. Unlike terms, 10, 284. Variable, 107, 211, 284. Variation, 251-255. Varv, 251, 284. Vary directly, 251, 284. Vary inversely, 251, 284. Vary jointly, 251, 284. Vertical bar, 8. Vinculum, 8. Whole number, 284. X-axis, 110, 284. Y-axis, 110, 284. Zero exponents, 131, 133-137. Sts^ SyJi:i^<, cW^^ ms^^ ^^.. <>y H(£X^' i^ ^V«^ oj ■^"St^^^"^\JL£-<&^uL^ tl'.ZVstJ^''^ °^ 25 CENTS THIS BOOK In THE D^^/^'^""^ ™ R^rURN W.LI. INCREASE ;^^3^^J_^^UE. "^"E PENAL^ DAY AND TO.„.oo ON T^^J^HE --OURTH OVERDUE. '^ "*^^ SEVENTH DAY 893304 THE UNIVERSITY OF CALIFORNIAJLIBB^WI^' 4 "'"■ iiiiill 'iiiiiiii WmM , mi^ 'ifeii "'