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 Complete Course 
 
 ALGEBRA 
 
 ACADEMIES AJ^D HIGH'* SCHOOLS, 
 
 BY 
 
 WEBSTER WELLS, S.B., 
 
 ASSOCIATE PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS 
 INSTITUTE OF TECHNOLOGY. 
 
 LEACH, SHEWELL, & SANBORN. 
 
 BOSTON AND NEW YORK. 
 
<^ 
 
 
 ^^-^^^^til^^ 
 
 Copyright, 1885, 
 By WEBSTER WELLS. 
 
 J. S. Gushing & Co., Pkinters, Boston. 
 
PEEFAOE. 
 
 rr^HE present work cootains a full and complete treat- 
 -^ ment of the topics usuall}' included in an Elementary 
 Algebra. The author has endeavored to prepare a course 
 sufficiently advanced for the best High Schools and Acad- 
 emies, and at the same time adapted to the requirements 
 of those who are preparing for admission to college. 
 
 Particular attention has been given to the selection of 
 examples and problems, a sufficient number of which have 
 been given to afford ample practice in the ordinar}- processes 
 of Algebra, especially in such as are most likely to be met 
 with in the higher branches of mathematics. Problems of 
 a character too difficult for the average student have been 
 purposely excluded, and great care has been taken to obtain 
 accuracy in the answers. 
 
 The author acknowledges his obligations to the elemen- 
 tary text-books of Todhunter and Hamblin Smith, from 
 which much material and many of the examples and prob- 
 lems have been derived. He also desires to express his 
 thanks for the assistance which he has received from expe- 
 rienced teachers, in the way of suggestions of practical 
 value. 
 
 WEBSTER WELLS. 
 Boston, 1885. 
 
 797968 
 
CONTENTS. 
 
 PAGE 
 
 I. Definitions and Notation 1 
 
 Symbols of Quantity 1 
 
 Symbols of Operation 2 
 
 Symbols of llelation 4 
 
 Symbols of Abbreviation 4 
 
 Algebraic Expressions 5 
 
 Axioms 10 
 
 Solution of Problems by Algebraic Methods ... 10 
 
 Negative Quantities 13 
 
 n. Addition 15 
 
 Addition of Similar Terms 16 
 
 Addition of Polynomials 18 
 
 ni. Subtraction 20 
 
 Subtraction of Polynomials 21 
 
 rV. Use of Parentheses 24 
 
 V. Multiplication 27 
 
 Multiplication of Monomials 29 
 
 Multiplication of Polynomials by Monomials . . 30 
 
 Multiplication of Polynomials by Polynomials . . 31 
 
 VI. Division 36 
 
 Division of Monomials 37 
 
 Division of Polynomials by Monomials 38 
 
 Division of Polynomials by Polynomials .... 39 
 
 VII. FORMUL/E 44 
 
 VIII. Factoring 51 
 
vi CONTENTS. 
 
 PAGE 
 
 IX. Highest Common Factor 65 
 
 X. Lowest Common Multiple 75 
 
 XL Fractions 80 
 
 General Principles 80 
 
 To Reduce a Fraction to its Lowest Terms ... 82 
 To Reduce a Fraction to an Entire or Mixed Quan- 
 tity 86 
 
 To Reduce a Mixed Quantity to a Fractional Form 88 
 To Reduce Fractions to their Lowest Common 
 
 Denominator 89 
 
 Addition and Subtraction of Fractions 91 
 
 Multiplication of Fractions 97 
 
 Division of Fractions 99 
 
 Complex Fractions 101 
 
 XIL Simple Equations 106 
 
 Transposition 107 
 
 Solution of Simple Equations 108 
 
 Solution of Equations containing Fractions . . . Ill 
 
 Solution of Literal Equations 115 
 
 Solution of Equations involving Decimals . . . . 117 
 
 XIII. Problems leading to Simple Equations con- 
 
 taining One Unknown Quantity .... 119 
 
 XIV. Simple Equations containing Two Unknown 
 
 Quantities 132 
 
 Elimination by Addition or Subtraction . . . . 133 
 
 Elimination by Substitution 135 
 
 Elimination by Comparison 136 
 
 XV. Simple Equations containing more than Two 
 
 Unknown Quantities 144 
 
 XVI. Problems leading to Simple Equations con- 
 taining more than One Unknown Quan- ^ 
 tity 148 
 
CONTENTS. vii 
 
 PAGE 
 
 XVir. Involution 158 
 
 Involution of Monomials 158 
 
 Square of a Polynomial 159 
 
 Cube of a Binomial 161 
 
 Any Power of a Binomial 162 
 
 XVIII. Evolution 165 
 
 Evolution of Monomials 165 
 
 Square Root of Polynomials 167 
 
 Square Root of Numbers 170 
 
 Cube Root of Polynomials 173 
 
 Cube Root of Numbers 176 
 
 XIX. The Theory of Exponents 180 
 
 XX. Radicals 190 
 
 To Reduce a Radical to its Simplest Form . . 190 
 
 Addition and Subtraction of Radicals . . . . 193 
 To Reduce Radicals of Different Degrees to 
 
 Equivalent Radicals of the Same Degree . . 195 
 
 Multiplication of Radicals 196 
 
 Division of Radicals 198 
 
 Involution and Evolution of Radicals . . . . 199 
 To Reduce a Fraction having an Irrational De- 
 nominator to an Equivalent Fraction whose 
 
 Denominator is Rational 200 
 
 Imaginary Quantities 203 
 
 Multiplication of Imaginary Quantities . . . 204 
 
 Properties of Quadratic Surds 206 
 
 Square Root of a Binomial Surd 207 
 
 Solution of Equations containing Radicals . . 208 
 
 XXI. Quadratic Equations 211 
 
 Pure Quadratic Equations 211 
 
 Affected Quadratic Equations 212 
 
 First Method of Completing the Square . . . 213 
 
 Second Method of Completing the Square . . 216 
 
 Solution of Quadratic Equations by a Formula . 222 
 
 XXII. Problems involving Quadratic Equations. 223 
 
viii CONTENTS. 
 
 PAGE 
 
 XXIII. Equations in the Quadratic Form .... 228 
 
 XXIV. Simultaneous Equations involving Quad- 
 
 ratics 233 
 
 XXY. Theory of Quadratic Equations .... 246 
 
 Factoring 250 
 
 Discussion of the General Equation 253 
 
 XXVL Ratio and Proportion 255 
 
 Properties of Proportions 256 
 
 XXVII. Arithmetical Progression 264 
 
 XXVIII. Geometrical Progression 273 
 
 XXIX- Binomial Theorem 282 
 
 Proof of the Theorem for a Positive Integral 
 
 Exponent 282 
 
 XXX. Logarithms 287 
 
 Properties of Logarithms 289 
 
 Use of the Table 296 
 
 Applications 299 
 
 Arithmetical Complement 301 
 
 Exponential Equations 304 
 
 Answers 307 
 
ALGEBRA, 
 
 I. DEFINITIONS AND NOTATION. 
 
 1. Algebra is that branch of mathematics in which the 
 rehitions of numbers are investigated, and the reasoning 
 abridged and generaHzed by means of symbols. 
 
 Note. Writers on Algebra employ the word "quantity " as synony- 
 mous with " number " ; this definition of the word will be understood 
 throughout the present work. 
 
 2. The Symbols of Algebra arc of four kinds : 
 
 1. Symbols of Quantity. 
 
 2. Symbols of Operation. 
 
 3. Symbols of Relation. 
 
 4. Symbols of Abbreviation. 
 
 SYMBOLS OF QUANTITY. 
 
 3. The symbols of quantity generally used are the Jigures 
 of Arithmetic, and the letters of the alphabet. 
 
 Figures are used to represent known quantities and deter- 
 mined values ; while letters may represent any quantities 
 whatever, known or unknown. 
 
 4. Known Quantities, or those whose values are given, 
 when not expressed b}- figures, are usuall}- represented by 
 the first letters of the alphabet, as a, 6, c. 
 
 5. Unknown Quantities, or those whose values are to be 
 determined, are usually represented by the last letters of the 
 alphabet, as x, y, z. 
 
2 ALGEBRA. 
 
 6. Quantities occupying similar relations in the same 
 problem, are often represented by the same letter, distin- 
 guished by different accents; as a', a", a'", read " a prime," 
 " a second," " a third," etc. 
 
 They may also be distinguished by different subscript 
 figures; as ai, as, ag, read "a one," ''a two," "a three," 
 etc. 
 
 7. Zero, or the absence of quantity, is represented by the 
 symbol 0. 
 
 SYMBOLS OF OPERATION. 
 
 8. The Sign of Addition, +, is called ''plus.'' 
 
 Thus, a-\-b, read " a plus 6," indicates that the quantity 
 b is to be added to the quantity a. 
 
 9. The Sign of Subtraction, — , is called " minus." 
 Thus, a — b, read "a minus &," indicates that the quan- 
 tity b is to be subtracted from the quantity a. 
 
 Note. The sign '^ indicates the difference of two quantities ; thus, 
 a '^ h denotes that the difference of the quantities a and h is to be 
 found. 
 
 10. The Sign of Multiplication, x, is read ''times" 
 
 " into" or " multipUed by." 
 
 Thus, a X 5 indicates that the quantity a is to be multi- 
 plied by the quantity b. 
 
 The sign of multiplication is usually omitted in Algebra, 
 except between arithmetical figures ; the multiplication of 
 quantities is therefore indicated by the absence of any sign 
 between them. Thus, 2a6 indicates the same as 2 x a X 6. 
 
 A point is sometimes used in place of the sign X between 
 two or more figures ; thus, 2 • 3 • 4 denotes 2x3x4. 
 
 11. Quantities multiplied together are called factors^ and 
 the result of the multiplication is called the product. 
 
 Thus, 2, a, and b are the factors of the product 2db. 
 
DEFINITIONS AND NOTATION. S 
 
 12. A Coefficient is a number prefixed to a quantity to 
 indicate how many times the quantity is to be taken. 
 
 Thus, in Aax^ 4 is the coefficient of ax, and indicates that 
 ax is to be taken 4 times; that is, 4: ax is equivalent to 
 ax -\- ax -{- ax -{- ax. 
 
 When no coefficient is expressed, 1 is understood to be 
 the coefficient. Thus, a is the same as 1 a. 
 
 When any number of factors are multiplied together, the 
 product of any of them may be regarded as the coefficient 
 of the product of the others. Thus, in abed, ah is the 
 coefficient of cd ; 6 of acd\ abd of c ; etc. 
 
 13. An Exponent is a figure or letter written at the right 
 of, and above a quantity, to indicate the number of times 
 the quantity is taken as a factor. 
 
 Thus, in re*, the * indicates tliat x is taken three times as a 
 factor ; that is, x^ is equivalent to xxx. 
 
 14. The product obtained by taking a factor two or more 
 times is called a power. A single letter is also often called 
 the Jirst power of that letter. Thus, 
 
 a^ is read '' a to the second power," or " a square,'' and 
 indicates aa ; 
 
 a* is read " a to the third power," or " a cube,'' and indi- 
 cates aaa; 
 
 a* is read " rt to the fourth power," or ^^ a fourth,'' and 
 indicates aaaa ; etc. 
 
 When no exponent is written, the Jirst power is under- 
 stood ; thus, a is the same as «\ 
 
 15. The Sign of Division, h-, is read " dimded by." 
 Tims, a-i-b denotes that the quantity a is to be divided 
 
 by the quantity b. 
 
 Division is also indicated by writing the dividend above, 
 
 and the divisor below, a horizontal line. Thus, - indicates 
 
 a ^ 
 
 the same as a-i-b. When thus written, - is often read " a 
 
 7 » b 
 
 over b. 
 
ALGEBRA. 
 
 SYMBOLS OF RELATION. 
 
 16. The symbols of relation are signs used to indicate 
 the relative magnitudes of quantities. 
 
 17. The Sig^ of Equality, =, read " egz/aZs," or " zs 
 equal to" indicates that the quantities between which it is 
 placed are equal. 
 
 Thus, x = y indicates that the quantities x and y are equal. 
 
 A statement that two quantities are equal is called an 
 equatio7i. 
 
 Thus, ic+ 4= 2a;— 1 is an equation, and is read " x plus 
 4 equals 2 a; minus 1." 
 
 18. The Sign of Inequality, > or <, read " zs greater 
 than" and " is less than" respectively, when placed between 
 two quantities, indicates that the quantity toward which the 
 opening of the sign turns is the greater. 
 
 Thus, ic > ?/ is read " a? is greater than y" ; x — 6 <y is 
 read " x minus 6 is less than y," 
 
 SYMBOLS OF ABBREVIATION. 
 
 19. Th» Sig^ of Deduction, .*., stands for therefore or 
 hence. 
 
 20. The Signs of Aggregation, the parenthesis ( ) , the 
 
 brackets [ ] , the braces { ] , and the vinculum , indicate 
 
 that the quantities enclosed by them are to be taken col- 
 lectiveh'. Thus, 
 
 (a-{-b)x, [a4-6]£c, {a-{-b\x^ a-\~bxx, 
 
 all indicate that the quantity x)btained by adding a and b is 
 to be multiplied by x. 
 
DEFINITIONS AND NOTATION. 5 
 
 21. The Sign of Continuation, ..., stands for ^^ and so 
 on " or " continued by the same law." Thus, 
 
 a, a-f-6, a + 26, a-\-Sb, ... 
 
 reads "a, a plus 6, a plus 26, a plus 3b, and so on." 
 
 ALGEBRAIC EXPRESSIONS. 
 
 22. An Algebraic Expression is any combination of alge- 
 braic symbols ; as 2 ar^ — 3 ah -f c^. 
 
 23. A Term is an algebraic expression whose parts are not 
 separated b}' the signs + or — ; as 2 or, — 3a5, or -f- c^. 
 
 2ar^, — 3a6, and c^ are called the terms of the expression 
 2x^-Sab + (^. 
 
 24. Positive Terms are those j)receded by a plus sign ; as 
 
 + 2a2, or 4-c3. 
 
 For this reason, the sign + is often called the positive 
 sign. If no sign is expressed, the term is understood to be 
 positive ; thus, a is the same as + a. 
 
 25. Negative Terms are those preceded by a minus sign ; 
 as 
 
 — 3a6, or — bc^. 
 
 For this reason, the sign — is often called the negaiive 
 sign; it can never be omitted before a negative term. 
 
 Note. In a negative term, the numerical coefficient indicates how 
 many times the quantity is to be taken suhtractivehj . (Compare Art. 
 12.) 
 
 Thus, — 3a6 is equivalent to —ah — ah — ah. 
 
 26. In Arithmetic, if the same number be both added to 
 and subtracted from another, the value of the latter will not 
 be changed. Thus, 
 
 5 + 3-3 = 5. 
 
6 ALGEBRA. 
 
 Similarly, in Algebra, if any quantity b be both added to 
 and subtracted from another quantity a, the result will be 
 equal to a. That is, 
 
 a -\- b — b = a. 
 
 Consequently, equal terms affected by utiUke signs, in an 
 expression, neutralize each other, or cavicel. 
 
 27. A Monomial is an algebraic expression consisting of 
 only one term ; as 5a, 7o6, or —Sb^c. 
 
 A monomial is sometimes called a simple quantit3\ 
 
 28. A Polynomial is an algebraic expression consisting of 
 more than one term ; as a + &, or 3a^ + 6 — 55^. 
 
 A polynomial is sometimes called a compomid quantity. 
 
 29. A Binomial is a polynomial of two terms ; as a — b, 
 or 2a + bK 
 
 30. A Trinomial is a polynomial of three terms ; as 
 ab^2c^-b\ 
 
 31. Similar or Like Terms are those which differ only in 
 their numerical coefficients. Thus, 
 
 2xy^ and — 7xy^ are similar terms. 
 
 32. Dissimilar or Unlike Terms are those which are not 
 similar. Thus, 
 
 boiy^y and bxy^ are dissimilar terms. 
 
 33. The Degree of a term is the number of literal factors 
 which it contains. Thus, 
 
 2 a is of the Jlrst degree, since it contains but 07ie literal 
 factor ; 
 
 ab is of the second degree, since it contains two literal 
 factors ; 
 
 Sa^b^ is of the Jifth degree, since it contains Jive literal 
 factors. 
 
DEFINITIONS AND NOTATION. 7 
 
 The degree of any term is determined by adding the expo- 
 nents of its several letters. Thus, 
 
 a6V is of the sixth degree. 
 
 34. Homogeneous Terms are those of the same degree. 
 ' a^, 36c, and — 4a^ are homogeneous terms. 
 
 35. A polynomial is called homogeneous when all its terms 
 are homogeneous ; as a^ -|- 2 ahc — 3 b^. 
 
 36. A polynomial is said to be arranged according to the 
 ascending powers of any letter, when the term containing 
 the lowest exponent of that letter is placed first, that having 
 the next higher immediately after, and so on. Thus, 
 
 b* + 3ab^ - 2a^b^ -i- 3a^b - 4:a'' 
 
 is arranged according to the ascending powers of a. 
 
 Note. The term 6*, which does not involve a at all, is regarded as 
 containing the lowest exponent of a in the above expression. 
 
 37. A polynomial is said to be arranged according to the 
 descending powers of any letter, when the term containing 
 the highest exponent of that letter is placed first, that having 
 the next lower immediately after, and so on. Thus, 
 
 64 4. 3a63 _ 2a26^ -^Sa^'b -4a* 
 is arranged according to the descending powers of b. 
 
 38. The Reciprocal of a quantity is 1 divided by that 
 quantity. Thus, the reciprocal of a is _ ; and of x-\-y is 
 
 x-\-y 
 
 39. The Numerical Value of an expression is the result 
 obtained by jendering it into an arithmetical quantity, by 
 means of the numerical values assigned to its letters. 
 
8 ALGEBRA. 
 
 Thus, the numerical vahie of 
 
 when a = 4, 6 = 3, c = 5, and d = 2, is 
 
 4 X 4 + 3 X 3 X 5 - 2« = 16 + 45 - 8 = 53. 
 
 EXERCISES. 
 
 40. Find the numerical value of the following expressions, 
 when a = 2, 6 = 3, c = 1, and d = 4 : 
 
 6« 
 
 1. 
 
 a' + 2ab-c-\-d. 
 
 6. 
 
 a'' 
 
 2. 
 
 ^a^-2a'b + (f. 
 
 7. 
 
 cd ah 
 6^ "^ c^ ' • 
 
 3. 
 
 6a^b + 4.ab^'- 27 cd. 
 
 8. 
 
 b^ - a}'b\ 
 
 4. 
 
 2a2 + 36c--. 
 cd 
 
 9. 
 
 3^2 2a 
 5ac 36^ 
 
 5. 
 
 7 + - + J 
 6 c c^ 
 
 10. 
 
 a2 62 c^ 
 62"^c^"*"d2' 
 
 If the expression involves parentheses, the operations indi- 
 cated within the parentheses must be performed ^?'s^. 
 
 Thus, to find the numerical value, when a = 3, 6 = 2, and 
 
 C=l,Of ^^2, 7,2 
 
 W 
 
 we have, 
 
 2a-3c=6-3= 3 
 a2 + c2 = 9 + l = 10 
 a2+62 = 9 + 4 = 13 
 a2-62=9-4= 5 
 
 Hence the numerical value of the expression is 
 
 2x3xlO-M = 60-i| = 2|?. 
 5 5 5 
 
DEFINITIONS AND NOTATION. 9 
 
 Find the numerical values of the following, when a = 4, 
 6 = 2, c = 3, and d=l : 
 
 11. a\a-{-b)-2abc. iq ^ r ^ 
 
 3a — 3c 3 
 
 12. 7a' + (a-b)(a-c). 
 
 1- 25 a — 30 c — d 
 
 13. 15a-7{b + c)-d. ^^' ^:j7^ 
 
 14. c(a^- + a^-^), ^^ a^+6^ c^-6^ 
 16. 25a2-7(62 + c2)4.(Z2. * a^-ft^ c^ + ft^ 
 
 Find the values of the following, when a = -, b = -, ^ == 7' 
 
 2 o O 
 
 and a; = 2 : 
 
 19. (2a + 36 + 5c) (8a + 36 -5c) (2a -36 + 15c). 
 
 21. iB*-(2a + 36)ar^ + (3a-26).x--ca; + 6c. 
 
 2 b 
 or 
 
 «« 2 a; 
 
 8 6c — a a + 6 + c 
 
 41. Put the following into the form of algebraic expres- 
 sions : 
 
 1. Five times a, added to twice 6. 
 
 2. Two times a;, minus y to the second power. 
 
 3. The product of (/, 6, c square, and d cube. 
 
 4. Three times the cube of a, minus twice the product of 
 f/, square and 6, plus the cube of c. 
 
 6. The product of a; + ?/ and a. 
 
 6. The product of x-{-y and a — 6. 
 
 7. a square, divided b^' the product of 6 and c. 
 
 8. a square, divided by 6 — c. 
 
10 ALGEBRA. 
 
 9. X divided by 3, plus 2, equals three times y minus 11. 
 
 10. The product of m and a + 6 is less than the recipro- 
 cal of a; cube. 
 
 AXIOMS. 
 
 42. An Axiom is a truth assumed as self-evident. 
 
 Algebraic operations are based upon definitions, and the 
 following axioms : 
 
 1. If equal quantities be added to equal quantities, the 
 sums will be equal. 
 
 2. If equal quantities be subtracted from equal quanti- 
 ties, the remainders will be equal. 
 
 3. If equal quantities be multiplied by equal quantities, 
 the products will be equal. 
 
 4. If equal quantities be divided by equal quantities, the 
 quotients will be equal. 
 
 5. If the same quantity be both added to and subtracted 
 from another, the value of the latter will not be changed. 
 
 6. If a quantity be both multiplied and divided by 
 another, the value of the former will not be changed. 
 
 7. Quantities which are equal to the same quantity are 
 equal to each other. 
 
 SOLUTION OF PROBLEMS BY ALGEBRAIC METHODS. 
 
 43. The following examples will illustrate the application 
 of the notation of Algebra in the solution of problems. 
 
 1. The sum of two numbers is 30, and the greater is 4 
 times the less. What are the numbers? 
 
 We will first solve the problem by the method of Arith- 
 metic, and afterwards by Algebra. The marginal letters 
 refer to the corresponding steps of the two methods ; that is, 
 the operation (a) in the algebraic solution is equivalent to 
 the operation (a) in the arithmetical ; and so on. In this 
 way the student can compare the two processes step by step. 
 
DEFINITIONS AND NOTATION. H 
 
 Solution by Arithmetic. 
 
 The less number, plus the greater number, equals 30. 
 (a) Hence the less number, plus 4 times the less number, equals 30. 
 {b) Therefore 5 times the less number equals 30. 
 
 (c) Hence the less number is one-fifth of 30, or C. 
 
 (d) Then the greater number is 4 times 6, or 24. 
 
 Solution by Algebha. 
 
 Let X = the less number. 
 
 Then 4 a: = the greater number, 
 
 (a) By the conditions, x + 4j: = 30. 
 (h) Or, 5x = 30. 
 
 (c) Dividing by 6, x — G, the less number. 
 
 ((/) Whence, 4t = 24, the greater number. 
 
 2. A, B, and C together have $66. A has one-half as 
 much as B, and C has as much as A and B together. How 
 much has each ? 
 
 Let X = the number of dollars A has. 
 
 Then 2 ar = the number of dollars B has, 
 and ^ a:-f2T, or3a; = the number of dollars C has. 
 
 By the conditions, x+2z-f-8a: = 60. 
 
 Or, 6x = 6Q. 
 
 Whence, ^ x = 11, the number of dollars A has. 
 
 Therefore, 2x = 22, the number of dollars B has, 
 
 and 3 a? = 33, the number of dollars C has, 
 
 3. The sum of the ages of A and B is 109 years, and A is 
 13 years younger than B. What are their ages? 
 
 Let X = the number of years in A's age. 
 
 Then a: + 13 = the number of years in B's age. 
 
 By the conditions, ar + a: + 13 = 109. 
 
 Or, 2a:+13=109. 
 
 AViience, 2 a: = 96. 
 
 And, X = 48, the number of years in A's age. 
 
 Therefore, ar + 13 = 61, the number of years in B's age. 
 
12 ALGEBRA. 
 
 PROBLEMS. 
 
 4. The greater of two numbers is 5 times the less, and 
 their sum is 42. What are the numbers? 
 
 5. The sum of the ages of A and B is 68 years, and B is 
 6 years older than A. What are their ages ? 
 
 6. Divide $1200 between A and B, so that A may receive 
 $128 less than B. 
 
 7. A man had $3.72 ; after spending a certain sum, he 
 found that he had left 3 times as much as he h^d spent. 
 How much had he spent? 
 
 8. Divide $260 between A, B, and C, so that B may 
 receive 3 times as much as A, and C 3 times as much as B. 
 
 9. Divide the number 125 into two parts, one of which is 
 21 less than the other. 
 
 10. The sum of three numbers is 98 ; the second is 3 
 times the first, and the third exceeds the second by 7. 
 What are the numbers? 
 
 11. A, B, and C together have $127; C has twice as 
 much as A, and $13 more than B. How much has each? 
 
 12. My horse, carriage, and harness together are worth 
 $400. The horse is worth 11 times as much as the harness, 
 and the carriage is worth $175 less than the Horse. What 
 is the value of each? 
 
 13. The sum of three numbers is 108. The first is one- 
 third of the second, and 33 less than the third. What are 
 the numbers? 
 
 14. Divide the number 210 into three parts, such that the 
 first is one-half of the second, and one-third of the third. 
 
 15. A man bought a cow, a sheep, and a hog, for $75 ; 
 the price of the sheep was $27 less than the price of the 
 cow, and $6 more than the price of the hog. What was the 
 price of each? 
 
DEFINITIONS AND NOTATION. 13 
 
 NEGATIVE quantities; 
 
 44. The signs + and — , besides indicating the opera- 
 tions of addition and subtraction, are also used, in Algebra, 
 to distinguish between quantities which are the exact reverse 
 of each other in quality or condition. 
 
 Thus, in the thermometer, we may speak of a temperature 
 above zero as +, and of one below as — . For example, 
 + 25° means 25° above zero, and —10° means 10° below 
 zero. 
 
 In navigation, north latitude is considered +, and south 
 latitude — ; west longitude is considered +, and east lon- 
 gitude — . For example, a place in latitude —30°, longi- 
 tude + 95°, would be in latitude 30° south of the equator, 
 and in longitude 95° west of Greenwich. 
 
 Again, in financial transactions, we may consider assets 
 as +, and debts or liabilities as — . For example, the 
 statement that a man's property is — $100, means that he 
 owes or is in debt Si 00. 
 
 And in general, when we have to consider quantities the 
 exact reverse of each other in quality or condition, we may 
 regard quantities of either quality or condition as positive, 
 and those of the opposite quality or condition as negative. 
 
 45. The thermometer affords an excellent illustration of 
 
 the relation between positive and negative quan- 
 
 4- 6 Let OA represent the scale for temperatures 
 ^3 above zero, and OB the scale for temperatures 
 + 2 below zero ; and let us consider the following 
 
 1 At 7" A.M. tbe temperature is — 6° ; at noon it is 
 o 11° warmer, and at 6 p.m. it is 9° colder than at 
 4 noon. Required the temperatures at noon and at 
 
 -5 
 
 G P.M. 
 
 -6 
 
 — 7 Beginning at the scale-mark —6, and counting 
 
14 ALGEBRA. 
 
 11 degree-spaces upwards, we reach the scale-mark -|-5; 
 and counting from the latter 9 degree-spaces downwards, 
 we reach the scale-mark — 4. Hence, the temperature at 
 noon is + 5°, and at 6 p.m. — 4°. 
 
 EXERCISES. 
 
 46. 1. At 7 A.M. the temperature is — 8° ; at noon it is 
 7° warmer, and at 6 p.m. it is 3° colder than at noon. 
 Required the temperatures at noon and at 6 p.m. 
 
 2. A certain city was founded in the year 151 B.C., and 
 was destroyed 203 years later. In what year was it de- 
 stroyed ? 
 
 3. At 7 a.m. the temperature is + 4° ; at noon it is 10° 
 colder, and at 6 p.m. it is 6° warmer than at noon. Re- 
 quired the temperatures at noon and at 6 p.m. 
 
 4. What is the difference in latitude between two places 
 whose latitudes are +56° and —31°? 
 
 5. A man has bills receivable to the amount of $2000. 
 and bills payable to the amount of $3000. How much is he 
 worth ? 
 
 6. At 7 A.M. the temperature is — 3°, and at noon it is 
 -}-ll°. How many degrees warmer is it at noon than at 7 
 
 A.M.? 
 
 7. What is the difference in longitude between two places 
 whose longitudes are + 25*^ and — 90° ? 
 
 8. The temperature at 6 a.m. is — 7°, and during the 
 morning it grows warmer at the rate of 3° an hour. Re- 
 quired the temperatures at 8 a.m., at 9 a.m., and at noon. 
 
 47. The absolute value of a quantity is the number repre- 
 sented by the quantity, taken independently of the sign 
 affecting it. 
 
 Thus, the absolute value of — 5 is 5- 
 
ADDITION. 15 
 
 II. ADDITION. 
 
 48. Addition, in Algebra, is tlie process of collecting two 
 or more quantities into one equivalent expression, called the 
 sum. 
 
 Thus, the sum of a and b is a -\-b (Art. 8) . 
 
 49. If either quantit}^ is negative, or a polynomial, it 
 should be enclosed in a parenthesis (Art. 20) ; thus, 
 
 The sum of a and — 6 is indicated by a -h ( — ft) . 
 The sum of a — b and c — c? is indicated by 
 
 (rt — &)-+- (c — d) . 
 
 50. Required tlie siun of a and — b. 
 
 Using the interpretation of negative quantities as ex- 
 plained in Art. 44, if a man incurs a debt of $100, we ma}- 
 regard the transaction either as adding — $100 to his prop- 
 erty, or as subtracting $100 from it That is, 
 
 Adding a negative quantity is equivalent to subtracting a 
 positive quantity of the same absolute value (Art. 47). 
 
 Thus, the sum of a and — 6 is obtained by subtracting b 
 
 from a ; or, 
 
 a+( — ^)=a — 6. 
 
 51. It follows from Arts. 48 and 50 that the addition of 
 monomials is effected by uniting the quantities with their 
 respective signs. 
 
 Thus, the sum of a, —b^c, — d, and -— e, is 
 
 a — b + c — d — e. 
 
 It is immaterial in what order the terms are united, pro- 
 vided each has its proper sign. Thus, the above result ma}^ 
 also be expressed 
 
 c -h a—e — d — b, 
 
 — cZ — &-f-c — e-j-a, etc. 
 
16 ALGEBRA. 
 
 ADDITION OF SIMILAR TERMS. 
 
 52. 1. Required the sum of 5 a and 3 a. 
 
 5 a signifies a taken 5 times (Art. 12), and 3 a signifies a 
 taken 3 times. We have, therefore, a taken in all 8 times, 
 or 8 a. That is, 
 
 5a -|-3a = 8a. 
 
 2. Required the sum of — 5 a and — 3 a. 
 
 — 5a signifies a taken 5 times subtractively (Art. 25), and 
 — 3a signifies a taken 3 times subtractively. We have, 
 therefore, a taken in all 8 times subtractively, or —8 a. 
 That is, 
 
 — 5a — 3a = — 8a. 
 
 Therefore, 
 
 To add two similar (Art. 31) terms of like sigri, add the 
 coefficients, affix to the result the common symbols, and prefix 
 the common sign. 
 
 53. 1. Required the sum of 8 a and — 5 a. 
 
 Since 8 a is the sum of 3 a and 5 a (Art. 52, 1), the sum 
 of 8a and —5a is equal to the sum of 3a, 5a, and —5a, 
 which is 
 
 3a 4- 5a -5a. (Art. 51.) 
 
 But, by Art. 26, 5a and —5a cancel each other, leaving 
 the result 3 a. 
 
 Hence, 8a+( — 5a)=3a. 
 
 2. Required the sum of —8a and 5a. 
 
 Since —8a is the sum of —3a and —5a (Art. 52, 2), the 
 sum of —8a and 5a is equal to the sum of —3a, —5a, 
 and 5 a, which is 
 
 — 3a — 5a + 5a, or —3a. 
 
 Hence, ( — 8 a) +5a = — 3 a. 
 
ADDITION. 17 
 
 Therefore, 
 
 To add two similar terms of unlike sign, subtrax^t the less 
 coefficient from the greater^ affix to the result the common sym- 
 bols^ and prefix the sign of the greater coefficient. 
 
 Note. A clear understanding of the nature of the processes in Arts. 
 52 and 53 may be obtained by comparing them with the following, the 
 negative quantities being interpreted as explained in Art. 44. 
 
 1. If a man owes §5, and incurs a debt of §3, he will be in debt to 
 the amount of .$8. That is, the sum of — $5 and — $3 is — §8. 
 
 2. If a man's assets amount to $8, and his liabilities to §5, he is 
 worth $3. That is, the sum of $8 and - $6 is $3. 
 
 3. If a man's liabilities amount to §8, and his assets to $6, he is in 
 debt to the amount of §3. That is, the sum of — $8 and $6 is — $3. 
 
 EXAMPLES. 
 54. Add the following : 
 1. 11 and —5. 7. —llmand— 8m. 
 
 '2. - 13 and 3. 8. be and 166c. 
 
 3. 12 and — 1. 9. — 2aic and lax. 
 
 4. _ 4 and - 7. 10.-3 a^}^ and — d'bK 
 6. —2a and 7a. 11. 12mn^ and — 19?)m^ 
 6. b and - 3 6. 12. - 13a6c and 22a6c. 
 
 13. Required the sum of 2 a, — a, 3 a, — 12 a, and 6 a. 
 Since the order of the terms is immaterial (Art. 51), we 
 
 may add the positive terms first, and then the negative, and 
 finally combine these results by the rule of Art. 53. 
 
 The sum of 2a, 3a, and 6 a is 11 a. 
 
 The sum of — a and — 12 a is — 13 a. 
 Hence, the required sum is 11a — 13a, or — 2a. Ans. 
 
 Add the following : 
 
 14. 7a, — a, and — 3a. 15. — 6m, m, — 11m, and 5m. 
 
 16. 13a6, -7a6, — 8a6, and — 6a6. 
 
18 ALGEBRA. 
 
 17. 7n\ -n\ -^n\ lln\ and -10n\ 
 
 18. ISoic^, — aa^, — 20aaT^, 6aic^, and —basi?. 
 
 If the terms are not all similar, we may combine the simi- 
 lar terms, and unite the others with their respective signs. 
 
 19. Required the sum of 12 a, —5a:, —3?/, —5a, 8 a;, 
 
 and — 3aj. 
 
 The sum of 12 a and — 5a is 7a. 
 The sum of — 5ic, Sx^ and — 3 a; is 0. 
 
 Hence, the required sum is 7a — Sy. Ans. 
 Add the following : 
 
 20. 5aa;, —116, —ax, and 6&. 
 
 21. 2a, 56, -3c, -86, and 9c. 
 
 22. 5 m, — 2n^, n, — 2m, — 6^, and 3n^. 
 
 23. 3a;, —y, —a;, 6, —8?/, —2a;, 4?/, and —5. 
 
 ADDITION OF POLYNOMIALS. 
 
 55. A polynomial may be regarded as the sum of its 
 monomial terms (Art. 51). Thus, 2a — 36 + 4c is the sum 
 of the terms 2 a, —3 6, and 4 c. 
 
 Hence, the addition of two or more polynomials is effected 
 by uniting their terms with their respective signs. 
 Thus, the sum of a — 6 and c — cZ is a — 6 + c — dJ. 
 
 56. Required the sum of 6a — 7a;, Sx — 2a-^Sy, and 
 2 a; — a — mn. 
 
 It is convenient in practice to set the expressions down 
 one underneath the other, similar terms being in the same 
 vertical column. Thus, 
 
 6a— 7x 
 
 — 2a-{-3x-\-Sy 
 
 — a 4- 2 a; — mn 
 
 3a — 2a; + 3?/ — mn, Ans. 
 
ADDITIOX. 19 
 
 From the above principles we derive the following rule : 
 
 To add two or more expressions^ set them down one under- 
 neath the other, similar terms being in the same vertical column. 
 Find the sum of the terms in each column, and unite the results 
 with their respective signs. 
 
 
 EXAMPLES. 
 
 
 Add the following : 
 
 
 1. 
 
 2a-7x 
 
 — a + 4a; 
 a-f X 
 
 2. 
 
 -Sab-{-2cd 
 
 — lab-hScd 
 
 4a6 — 6cd 
 
 3. 
 
 — 1 1 a — 5 mp^ 
 
 8a-{-nmp^ 
 
 — 9a— 7mp^ 
 
 4. 2a — 36 + 5c and 6 — 5 c-f 2d. 
 
 6. 9 mn^ -\- x^y, — mn^ + 3 xhj, and — 6 mn^ — Ix^y. 
 
 6. a?-2ab + b\ a'^ -{- 2 ab -^r b\ aQd 2a''-2b-. 
 
 7. 3a2 + 2a6H-462, r^a'-Sab+bS and -Qa^'+bab-bbK 
 
 8. 6aj«- 7a;-4, ar'-a;-2, and8a;-9a^-.T». 
 
 9. 4 mn -|- 3 a6 — 4 c, 3 a; — 4 a6 -|- 2 mn, and 3 m^ — 4 x. 
 10. ^x—2y — z,^y — bx — lz,8z—y — x, and Ax—'dy. 
 
 XI. 6a; — 32/ + 7m, 2n — x-{-y, 2y — 4:X — 5m — dn, 
 and m — 2x. 
 
 12. 2iB'-5ic2_3.^7^ 3a^_2-6ar + 8.r, x' + 3x'^-4, 
 
 and 1 -\-2oc^ — ox. 
 
 13. 2a — 36 + 4d, 2& — 3dH-4c, 2d — 3c + 4a -f-46, 
 
 and 2c — 3a. 
 
 14. 2a^-a'b-2b\ 8a^-8ab^-Sb\ Sa'b -ab'+b^, 
 
 aud6ab^-2a'b-oa\ 
 
 16. 4.x^-10a^-6ax'^ + 6a% 6a^ -hSx^ -\- 4:ax' -{-2a-x, 
 
 -17ar'+19aar'-15a2a;, and6a;«+7a2a;-|-5a3-18a«2 
 
20 ALGEBRA. 
 
 III. SUBTRACTION. 
 
 58. Subtraction, iu Algebra, is the process of taking one 
 quantity from another. 
 
 The Subtrahend is the quantity to be subtracted. 
 The Minuend is the quantity from which it is to be sub- 
 tracted. 
 
 The Remainder is the result of the operation. 
 
 59. It is evident from the above that the minuend is 
 equal to the sum of the subtrahend and the remainder. 
 
 60. Let it be required to subtract — h from a. 
 
 Using the interpretation of negative quantities as explained 
 in Art. 44, if a man cancels a debt of SlOO, we may regard 
 the transaction either as subtracting — $100 from his prop- 
 erty, or as adding $100 to it. That is, 
 
 Subtracting a negative quantity is equivalent to adding a 
 positive quantity of the same absolute value. 
 
 Thus, to subtract — b from a, we add b to a; or 
 
 a — (—b) = a-{-b. 
 
 Hence, to subtract one quaiitity from another^ change the 
 sign of the subtrahend,, and add the result to the minuend. 
 
 61. 1. Subtract ba from 2 a. 
 
 By Art. 60, the result is equal to the sum of —ba and 2a, 
 which is —3 a. 
 
 2. From — 2a subtract 5a. 
 
 The result is equal to the sum of — 2a and — 5 a, or — la. 
 
 3. From 5 a take — 2 a. 
 
 Result, 5a + 2a, or7a. 
 
 4. From —2a take —5a. 
 
 Result, — 2 a ^- 5 a, or 3 a. 
 
SUBTRACTION. 21 
 
 EXAMPLES. 
 62. Subtract the following : 
 
 1. -3 from 11. 
 
 3. 
 
 -8 from -3. 5. 23 from 10. 
 
 2. IG from -5. 
 
 4. 
 
 -11 from -17. 6. - 13 from 11. 
 
 7. 8. 
 
 
 9. 10. 11. 
 
 27a 17a; 
 
 
 — 13y —lOmn oa^b 
 
 l.Ta -llo; 
 
 
 4y -ISmn Ud'b 
 
 12. From 9 ab take — 2 ab. 16. From — ary- take 6a^y^. 
 
 13. From 0^2/ take —cd. 17. From — 70o6ctake — 52 abc. 
 
 14. From 17wi' take 41 m\ 18. From — 7m^ take — 8n^. 
 
 15. From - 5x take 3. 19. From - 33ary take 19ar»/. 
 
 20. From 5ab take the sum of dab and — 2a5. 
 
 21. From the sum of — lla^ and Sx^ take the sum of 
 
 -10ar^and4a^. 
 
 SUBTRACTION OF POLYNOMIALS. 
 
 63. When the subtrahend is a polynomial, each of its 
 terms is to be subtracted from the minuend. Hence, 
 
 To subtract one polynomial from another^ change the sign of 
 each term of the subtrahend^ and add the result to the minuend. 
 
 It will be found convenient to place the subtrahend under 
 the minuend, similar terms being in the same vertical column. 
 
 64. 1. Subtract 5 x^y — 3 a6 + m- from 3 ar?/ — 2 a6 + 4 ?i . 
 
 Changing the sign of each term of the subtrahend, and 
 adding the result to the minuend, we have 
 3x-y — 2a6 + 4?i 
 
 — 5x^y -{-Sab — m^ 
 
 — 2 ary + ab -\-4: n — ??r , Ans. 
 
22 ALGEBRA. 
 
 Note. The student should endeavor to perform mentally the opera- 
 tion of changing the sign of each term in the subtrahend, as shown in 
 the following example : 
 
 2. From bo^—lW—^a^h subtract ^a?h-4.ay^-2W+a\ 
 a^^Sa'b-4:ab--2W 
 
 EXAMPLES. 
 Subtract the following : 
 
 3. 4. 
 
 ab -{- cd— ax 7x-\-5y — Sa 
 
 4:ab — Scd — 4:ax a;— 7?/ + 5 a — 4 
 
 6. From a — 6 + c take a -\-b — c. 
 
 6. From a^ + 2 ab + b' take a^ - 2 ab + bK 
 
 7. From 7 abc— 11 x+6y— AS take lla6c+3a.'+7?/-f-100. 
 
 8. Subtract Sin -{-y^— 6a — 7 from dm — Sy- -j-7 a — 6. 
 
 9. Subtract 17 x^ -^ by"-— 4:ab + 7 from 31a^— 3/ + a6. 
 
 10. From 6a + 36— oc+l subtract 6a— 35 — 5c. 
 
 11 . From 3 m — 5 ?i + r — 2 s take 2r -\-3n — m — 5s. 
 
 12. Take 4a — & + 2c— 5cZ from d — 3b-^a — c. 
 
 13. From m^ + 3 w^ subtract — 4 7?i^ — 6 ?i^ + 71 a;. 
 
 14. From4c — 36-5d + 2a;take Sa-^8d—b — 6c. 
 
 15. From a — b — c take the sum of — 2a + & + c 
 
 and a — b-\-c. 
 
 16. From x^ -\-2x^ — 3x-\- 4 take 3 ar^ + 3 a?^ _^ 5 a? — 7. 
 
 17. From 4 a^ - 3 ab'^ -bW subtract 6 a^b - ab^ + 4 b^ 
 
 18. Froma^- 8-f2a^-3a3 take 6<^_ n _ 5<^2_ 2(^4^ 
 
SUBTRACTION. 23 
 
 19. Take 2a^ — y^ from the sum ofx^ — 2xy-\- 3/ 
 
 and xy — iiy^. 
 
 20. From the sum oi x-\-2y —'^z and 3y — 4a;-f-2 
 
 take 2 — 50; + 5 y. 
 
 21. From 7a^H-3 — 5a^4-a-5a2 
 
 subtract 2a— Ga^— 2a^ + 9 — Ha*. 
 
 22. From — 72/^ + 3a^2/ — 2a:'^ + 6ay 
 
 subtract 8 a^y — 2 a^ -}- ar* — 9 ?/*. 
 
 23. From the sum of 2 a? — x-\-b and a;^ 4- 8 a; — 11 take the 
 
 sum of a:^ — 9 ar^ — 11a; and — 4a;^-f-3a;^— 6. 
 
 24. From the sum of or + ab-{- b^ and (r — 4 a6 + 5 ^^ take 
 
 the sum o^Aa^ + lI/— 2ab and Sab — a- — 2b^. 
 
 26. From3ar^— Ty — 2 4-a^— 5?/= 
 
 subtract — 5 a;y + 6 a; — 2 ar — 8 + 2 2/^. 
 
 26. From3ar^-8a.'*4-3a;^-5a;2_2a; 
 
 subtract - 3a;* + 4ar'* + Ga;^ - 6a; -f 2. 
 
 27. From the sum of 2a;^ — x^y — 5xy^ and Sa^y — bxy^ — 4:y^ 
 
 take the sum of —2x^—7x^y—Qy^ and —6xy^-{-5^. 
 
 28. From the sum of a* — 1 and 2a^—l0a^—7a subtract the 
 
 sum of —Sa*-\-2a^ — oa and — oa^ —I2a^ -{-3. 
 
 Note. In Arithmetic, addition always implies augmentation, and 
 subtraction diminution. In Algebra tliis is not always the case; for 
 example, in adding — 2 to 5, the sum is 3, which is less tlian 5. Again, 
 in subtracting — 2 from 5, the remainder is 7, which is greater than 5. 
 
 Thus the terms Addition, Subtraction, Sum, and Remainder have a 
 much more general signification in Algebra than in Arithmetic. 
 
24 ALGEBRA. 
 
 IV. USE OF PARENTHESES. 
 
 65. The use of parentheses (Art. 20) is very frequent in 
 Algebra, and it is necessary to have rules for their removal 
 or introduction. 
 
 66. The expression 
 
 2a-Sb-\-(5b-c-\-2d) 
 
 indicates that the quantity 5& — c-j-2d is to be added to 
 2a — Sb. If the addition be performed, we obtain (Art. 55) 
 
 2a-Sb + 5b-c + 2d. 
 
 Again, the expression 
 
 2a-Sb-{5b~c + 2d) 
 
 indicates that the quantity bb — c-\-2d is to be subtracted 
 from 2 a — 3 6. If the subtraction be performed, we obtain 
 (Art. 63) 2a-36-56 + c-2d. 
 
 67. It will be observed that in the first case the signs of 
 the terms within the parenthesis are unchanged when the 
 parenthesis is removed ; while in the second case the sign 
 of each term within is changed, from + to — , or from — 
 to +. 
 
 We have then the following rule for removing a paren- 
 thesis : 
 
 A parenthesis preceded by a + sign may be removed without 
 altering the signs of the enclosed terms. 
 
 A parenthesis preceded by a — sign may be removed, if the 
 sign of each enclosed term be changed, from -\- to —, or from 
 — to -\-. 
 
 68. Since the brackets, the braces, and the vinculum 
 (Art. 20) have the same signification as the parenthesis, 
 the rule for their removal is the same. 
 
PARENTHESES. 25 
 
 It should be observed in the case of the vinculum that the 
 sign apparently prefixed to the first term underneath, is in 
 reality the sign of the vinculum. Thus, -\-a — b and —a — b 
 are equivalent to + (a — b) and — {a — b) , respectively. 
 
 EXAMPLES. 
 
 69. 1. Remove the parentheses from 
 
 2a- 36 - (5a - 46) -f (4a -ft). 
 
 By the rule of Art. G7, the expression becomes 
 
 2a — 36 — 5a-f46-}-4a — 6 = a, Ans. 
 
 Parentheses are often found enclosing others. In this 
 case they may be removed in succession by the rule of Art. 
 67, and it is better to remove first the innermost pair. 
 
 2. Simplify the expression 
 
 4:X — \Sx-}-{—2x — x — a)l. 
 
 We remove the vinculum first, and the others in succession. 
 Thus, 
 
 4a;— \3x-\-(—2x — x — a)\ 
 = 4:X—\3x-\-(—2x — x-{-a)l 
 — Ax— \Sx — 2x — x-\-al 
 = 4:X — Sx-}-2x-\-x — a = 4iX — a, Ans. 
 
 Reduce the following expressions to their simplest forms 
 by removing the parentheses, etc., and uniting similar terms : 
 
 3. a — (6 — c) + (— c?4-e). 
 
 4. DX — \2x — Syl — [—2x + 4y']. 
 
 5. a — 6 + c — a + 6 — c — c — 6 
 
 6. ni^ —2n-{-\a — n-\-3 nr J — 5 a -f- 3 n — m^. 
 
 7. a- - 62 - (a^ - 2 a6 + 6-) - [a- + 2 a6 + 6^] . 
 
 8. 3a-(2a- \a-{-2\). 
 
26 ALGEBRA. 
 
 9. a-{b + \ -c-j-dl -e). 
 10. a-[(-6 + c)-(d-e)]. 
 
 11. 3x-l2y + x-y-]-{-[3ij-2x-^y^. 
 
 12. Ux-{ox-d)-\4.-3x-{2x~3)l. 
 
 13. 2m-[n-j3m-(2n-m)J]. 
 
 14. 3x — {5x-{-[-4:X-y-x'])-{-x-Sy) 
 
 15. 3c4-(2a-[5c-j3a+c-4«J]) 
 
 16. 5a-(4a-J-3a-[2a-a-l]J). 
 
 17. 8x-l5x-{Sx-A)-\7x-^(-9x-i-2)l2. 
 
 18. 2m - [3?7i - \m - (2m - 3m-f-4.) S - (5m -2)]. 
 
 19. c - [2 c - (6 a - 6) - J c - (5 a + 2 6) - (a - 3 6) J ] . 
 
 20. Sa-{b-lb-(a + b)-\-b-{b-a-b)l']\. 
 
 70. To enclose any number of terms in a parenthesis, we 
 take the converse of the rule of Art. 67 : 
 
 Any number of terms may be enclosed in a parenthesis pre- 
 ceded by a -\- sign, ivithout altering their signs. 
 
 Any number of terms may be enclosed in a parenthesis pre- 
 ceded by a — sign, if the sign of each term be changed, from 
 -^ to —, or from — to -\-. 
 
 71. 1. Enclose the last three terms of a — b-{-c — d-{-e 
 in a parenthesis preceded by a — sign. 
 
 Result, a — b— (— c + d — e) . 
 
 In each of the following expressions, enclose the last 
 three terms in a parenthesis preceded by a — sign : 
 
 2. a-\-b + c-{-d. 5. a^y — afy^ — xy^ + y*. 
 
 3. 3a-264-5c-4d. 6. x* -Sa^-]-2x' -ox-8. 
 
 4. m^ + 5m--6m + 3. 7. a^ -b^ - c^ + 2ab -{-2ac. 
 
 8. In each of the above results, enclose the last two terms 
 in an inner parenthesis preceded by a — sign. 
 
multiplicatio:n^. 27 
 
 V. MULTIPLICATION. 
 
 72. Multiplication, in Algebra, is the process of taking 
 one quantity as many times as there are units in another. 
 
 Thus, the multiplication of a b}- 6, which is expressed ab 
 (Art. 10), signifies that the quantity a is to be taken b 
 times. 
 
 73. The Multiplicand is the quantity to be multiplied or 
 taken. 
 
 The Multiplier is the quantity which shows how many 
 times it is to be taken. 
 
 The Product is the result of the operation. 
 
 The multiplicand and multiplier are called factors. 
 
 74. In Arithmetic, the product of two numbers is the 
 same in whatever order they are taken ; thus, we have 3x4 
 or 4 X 3, each equal to 12. 
 
 Similarly, in Algebra, we have a x & or 6 x a, each equal 
 to ab. 
 
 That is, the product of the factors is the same in whatever 
 order they are taken. 
 
 75. Required the product of c and a — h. 
 
 In Arithmetic, if we wish to multiply 87 by 98, we may 
 express the multiplier in the form 100 — 2 ; we should then 
 multiply 87 by 100, and afterwards by 2, and subtract the 
 second result from the first. 
 
 Similarly, in Algebra, to multiply c by a — b, we should 
 multiply c by a, and afterwards by 6, and subtract the sec- 
 ond result from the first. Thus, the required product is 
 
 ac — be. 
 
 76. Required the product of a — b and c — d. 
 
 As in Art. 75, we should first multiply a — ^ by c, and 
 
28 ALGEBRA. 
 
 afterwards by d, and subtract the second result from the 
 first. 
 
 The product of a — 6 and c is ac — be (Art. 75) . 
 
 The product of a — b and d is ad — bd. 
 
 Subtracting the second result from the first, the required 
 product is ac-bc-ad + bd. 
 
 77. We observe, in the preceding article, that the product 
 is formed by multiplying each term of the nmltiplicand by 
 each term of the multiplier, with the following results in 
 regard to signs : 
 
 The product of the terms -j- a and + c gives the term + ac. 
 The product of the terms — b and + c gives the term — be. 
 The product of the terms + a and — d gives the term — ad. 
 The product of the terms — b and — d gives the term + bd. 
 
 From these considerations we may state what is called the 
 Rule of Signs in Multiplication, as follows : 
 
 -f- Ttudtiplied by -f? ^^''^d — multiplied by — , produce -j- ; 
 + multiplied by — , and — midtiplied by -\-^ produce — . 
 
 Or, as it is usually expressed with regard to the product of 
 two terms. 
 
 Like signs produce + , a7id unlike signs produce — . 
 
 78. Required the product of 7a and 2b. 
 
 Since the factors ma}- be written in any order (Art. 74), 
 
 we have 
 
 7ax26 = 7x2X(XX& = 14a6.. 
 
 That is, the coefficient of the product is the product of the 
 coefficients of the factors. 
 
 79. Required the product of a^ and a^. 
 
 By Art. 13, a^ = axaxa, and a^ = ax a. Hence, 
 
 a"" X a 
 
 2 
 
MULTIPLICATION. 29 
 
 That is, the exponent of a letter in the product is the sum of 
 its exponents m the factors. 
 
 Thus, a^ X «■'' X « = a''+^^ = a®. 
 
 MULTIPLICATION OF MONOMIALS. 
 
 80. We derive from Arts. 77, 78, and 79 the following 
 rule for the product of two monomials : 
 
 To the product of the coefficients annex the literal quantities^ 
 giving to each letter an exponent equal to the sum of its expo- 
 nents in the factors. Make the product -f- when the factors 
 have the same sign, and — when they have different signs, 
 
 EXAMPLES. 
 
 1. Multiply 2 a' by 7a\ 
 
 By the rule, 2a' x 7a* = 14a'+* = 14a^ Ans. 
 
 2. Multiply a^b^c hy -6a^bd. 
 
 a^b^c X ( — 5 a^bd) = — 5 a'^b^cd, Ans. 
 
 3. Multiply — 7a;'" by bx^. 
 
 — Ix"" X 5x^ = — SSaj^+^j Ans, 
 
 4. Multiply — 11a;'" by — 8af*. 
 
 -lla;'»x(-8a;"') = 88ar""', Ans. 
 
 Multiply the following : 
 
 5. 13 by -19. 11. - lln^y hy -Dn'^z. 
 
 6. - 18 by 12. 12. - Qa^bc by a^bm. 
 
 7. -22 by -51. 13. - 12 a^a; by - 2 o^y. 
 
 8. 15mV by 3mn. 14. — 2a'"6'* by 5aV. 
 
 9. 1 7 abc by - 8 abc. 15. 3 aV/ by 1 1 aa^y . 
 10. -17aVby3aV. 16. 3 a'»6« by - 5 a''6^ 
 
30 ALGEBRA. 
 
 It is evident from the Rule of Signs (Art. 77) that the 
 product of three negative terms is negative ; of four negative 
 terms, positive ; and so on. 
 
 Hence the product of three or more monomials will be pos- 
 itive or negative, according as the number of negative factors 
 is odd or even. 
 
 17. Required the product of — 2a-lf^ 6bcP^ and — 7c^d. 
 
 - 2a-b^ xQbc'x- lc-d = 8^a'b'&d, Ans. 
 
 In this case the product is positive, as there are two nega- 
 tive factors. 
 
 Multiply the following : 
 
 18. 5a, —6b, and 7c. 
 
 19. _2a2, -Ua^, and -9a. 
 
 20. -3a&2, -26r, and 7cd2. 
 
 21. 4.x"'y'', — a^"?/V, and \byh\ 
 
 22. -2a, — 3a2, -4a^ and -5a^ 
 
 23. - o?bc, 2 bhd, - 5 o?cd, and - 3 ab'd\ 
 
 24. — 7m'*fl:^, m'^x^'if, 2a^, and — Sm?/^'". 
 
 25. 6iC2/^, — a^2;, 3?/^, —2x7^, and —4:yz. 
 
 MULTIPLICATION OF POLYNOMIALS BY MONOMIALS. 
 
 81. In Art. 75 we showed that the product of a — 6 and c 
 was ac — be. We have then the following rule for the prod- 
 uct of a polynomial by a monomial : 
 
 Multiply each term of the multiplicand by the multiplier, 
 and connect the residts with their projjer signs. 
 
 EXAMPLES. 
 1. Multiply 2a;2_5^_ 7 i,y g^^ 
 
 By the rule, the product is 1G«^ — 4()a;* — SGa.*^, Ans. 
 
MULTIPLICATION. 31 
 
 2. Multiply -5ab* by Sa^b - 4:ab\ 
 
 3a-b-4:ab^ 
 — ba¥ 
 
 -15a*'6^H-20a26", Arts, 
 Multiply the following : 
 
 3. 3 a; — 5 by 4 a;. 8. m^ -\- mn + n^ hj m^n^ . 
 
 4. a^b -\- ab^ by — ab. 9. — 2m by Sm^— 5mn—n^. 
 
 5. Sa'bc - d hy 5a(P, 10. - a;*- 10x^+5 by - 2a». 
 
 6. .^2- 2a; -3 by -4a;. 11. a- -\- IS ab - Gb"" by 4: ab\ 
 
 7. -2ar^by3ar' + 6a;-7. 12. - Ga^c by 5 - 6ac- 8a^ 
 
 13. 5ar^-4a.-2-3a; + 2 by -Gar'. 
 
 14. a'b- by a" - 3 a'b -{-Sab'- b\ 
 
 MULTIPLICATION OF POLYNOMIALS BY POLYNOMIALS. 
 
 82. In Art. 76 it was shown that the product of a — b and 
 c — d was ac — bc — ad-{- bd. We have then the following 
 rule for the product of two polynomials : 
 
 Multiply each term of the multiplicand by each term of the 
 multiplier, and add the partial products. 
 
 EXAMPLES. 
 
 1. Multiply 3a-26 by 2a-5&. 
 
 In accordance with the rule, we multiply 3 a — 26 by 2 a, 
 and then by —56, and add the partial products. A convenient 
 arrangement of the work is shown below, similar terms being 
 in the same vertical column. 
 
 3a - 26 
 2a - 56 
 
 6 a^ — 4 a6 
 
 -15a6 + 1Q6'^ 
 6a2-19«6-f-106-, Ans. 
 
32 ALGEBRA. 
 
 2. Multiply x^-\-l-x^-xhy x + 1. 
 
 It is convenient to have both multiplicand and multiplier 
 arranged in the same order of powers (Art. 36), and to write 
 the partial products in the same order. 
 
 Arranging the expressions according to the ascending 
 powers of x, we have 
 
 1 —x + x^ — x"^ 
 
 1 -\-x 
 
 1 —x-\-x- — 0(^ 
 
 x — x"^ -{- X^ — X* 
 
 1 —x*. Ans. 
 
 3. Multiply eab-8b' + 4a' by _ 4 6- + 2 a^ - 3 ab. 
 Arranging according to the descending powers of a, we 
 
 have 
 
 4.a'-^6ab-8b^ 
 
 2a--^ab-W 
 8a^ + V2d%-l&d'b^ 
 
 -l2o?b-\8a'b'-^24.aW 
 
 -16a^6--24a6-^H-326^ 
 
 8a^ -bOd'b' +325^ Ans. 
 
 Note. The correctness of the answers may be tested by working the 
 examples with the multiplicand and multipUer interchanged. 
 
 Multiply the following : 
 
 4. 3aj + 2 and 5a;-7. 6. 3a-26 and - 2a + 46. 
 
 5. 6aj— 5 and 3 — 2a;. 7. 3 — 5a??/ and — 6 — 10a.'2/. 
 
 8. 0-2 + a6 + &^ and 6 — tt. 
 
 9. 2 a-6 - 3 ab'^ and 5 a=& -f 6 abK 
 
 10. \ -\- X -\- x^ -\- x^ and ax — a. 
 
 11. 3a;^— 2a.-?/ — ?/2 and 2a; — 42/. 
 
 12. m^ — mn — Sn^ and 2 m^ — 6 mn. 
 
 13. a;2^2a;+l anda;2-2a; + 3. 
 
MULTIPLICATION. 33 
 
 14. 5a2-|-462_3a6and 6a-56. 
 
 15. 4ar'^ + 6a;-7 and 2ar-3. 
 
 16. a-hb —c and a — b-\-c. 
 
 17. 2a^-3x + 5 anda^ + i»-l. 
 
 18. 3ar-7a; + 4 and 2ic2 + 9a;-o. 
 
 19. 2a^-Sx^-6x-l andSx-o. 
 
 20. Gm-2m^ — f)-m^ sindm--\-10 — 2m. 
 
 21. 2a^4-5ar — 8a;— 7 and 4 — 5a; — 3a^. 
 
 22. a'b - a-62 _ 4 a6-^ and 2 a-6 - ab\ 
 
 23 . aj^+'^z — Sxy""-^ and 4 aj'^+y — 4 xV- 
 
 24. ar 4- ?/ — a;?/ and a;?/ + 2/' + ^• 
 
 25. 2a6 4-6'+4a2aud4a2-2a6 + 62. 
 
 26. 6a;*-3a;3_a^^_^6a;-2 and2a:2_j_^^2. 
 
 27. m* — wi'^n 4- rnrn^ — m?i^ -h w* and m^ — 2 ?»w — 3 ?i^. 
 
 28. 27ar^4-0a;2?/4-3a;/ + y^and9a.'2-6aTy4-/. 
 
 29. a-^-3a264.3a62_6'»anda--2a6 + 62. 
 
 30. 3ir -{- y- -{- z^ — xy — yz — zx and a; + ?/ 4- 2. 
 
 31. 2ar'^-3ar + 5a;-l and 3ar^-ar-2a;-5. 
 
 32. ah + cd -\- ac -\- bd and ab + cd —ac — bd. 
 
 33. 2a' - 5a- - 6a -f 4 and 4a' 4- lOa^ - 12a - 8. 
 
 Find the product of the following : 
 
 34. a; — 3, a; 4- 4, and a; — 7. 
 
 35. a4-6, a2-a64-^>'; and a'-6'. 
 
 36. 2m — 1 , Sm + 4, and Gm, — 5. 
 
 37. a; 4- 1, 3a; — 2, and 3a;- — x — 2. 
 
 38. a;^ 4- a? + 1, ^ — a; + 1 , and a;^ - ar + 1 . 
 
84 ALGEBRA. 
 
 39. a + b, a — b, a^ + b'^, and a^ + b\ 
 
 40. m + 1, m— 1, m + 2, and m — 2, 
 
 41. 2a!— 1, 3c»+2, 4a; — 3, and 5a; + 4. 
 
 42. a + ^, ci-^, a4-2&, and a'^ - 2 a-ft - aft^ _^ 2 6^ 
 
 83. The product of two or more polynomials may be indi- 
 cated by enclosing each of them in a parenthesis, and writing 
 them one after the other. 
 
 Thus, the product of a; + 2, a; — 3, and 2a; — 7 is indicated 
 
 by 
 
 (a; + 2)(a;-3)(2a;-7). 
 
 Similarly, the expression (a-\-b-{-cy indicates that a +&+c 
 is to be multiplied by itself (Art. 13). 
 
 When the operations indicated are performed, the expres- 
 sion is said to be expanded or simplified. 
 
 EXAMPLES. 
 1. Simplify the expression (a — 2a;)^— 2(a; + 3a)(a — a;). 
 To simplify the expression, we should expand (a— 2a;)^ 
 and 2 (a; -}- 3 a) (a — a?) , and subtract the second result from 
 
 the first. 
 
 (a — 2a;)^ = a^ — 4 aa; -j- 4 ar^ 
 
 2(a; + 3a)(a-a;) =&a^ - 4:ax-2x^ 
 Subtracting the second result from the first, we have 
 a^ — 4 aa; + 4 a/-^ — 6 a^ + 4 aa; + 2 a/"^ = 6 a;^ — 5 a-, Arts. 
 Simplify the following : 
 
 2. (a + 6 4-c + cZ)-. 
 
 3. {a-b){c-d) -\-{a-G){b-d). 
 
 4. (2a;-3)-+ (l-a;)(3a;-9). 
 
 5. (a + 6 + c)-- (a-5 + c)2. 
 
 6. (2a-55)2-4(a-2Z>)(a-36). 
 
MULTIPLICATION. 35 
 
 7. {a -by {a -{-by, 
 
 8. {l-{-x){l-^x'){l ~x-{-ar-x'). 
 
 9. (l+a)^-(l-a)(l +a2). 
 
 10. [x- (27J -{-Sz)-][_x- (2y-Bz)]. 
 
 11. {x - y) {x" -{- f)lar - y(x -y)^, 
 
 12. (a-f 6)(/> + c) - (c + (Z)(d-f a)-(a + c)(6-d). 
 
 13. (a + 6 + c)2 + (a - 6 - c)2-f (6 _ c-a)2-H (c-rt-6)2. 
 
 14. {a-~b){b-c) + (6_c)(c-a) + (c-a)(a-t). 
 
 15. a;(a;-22/)+2/(2/-22;)+2:(2;-2a;)-(a;-^-2:)2. 
 
 16. x{x-\-l){x + 2){x-\-S) H- 1- (ar-h3xH-l)-. 
 
 17. (a + & -f c)2 — (a — 6 - c)-+ (6-o-a)-- (c— «— 6)-'. 
 
 18. [(m 4-270' - (2m - n)2][(2m + w)^ - (m - 27i)2]. 
 
 19. (a;4-2/ + 2;)«-(ar^ + ^4-^) -3(y + ^)(^ + ;c)(x-f 2/). 
 
 84. Since ( + a) ( -f- 6) = a6, and ( — a) ( — 6) = «6, it fol- 
 lows that in tiie indicated product of two factors all the signs 
 of both factors may be changed without altering the value of 
 the expression. Thus, 
 
 {a — b)(c — d) is equal to {b — a){d — c) . 
 
 Similarly, we may show that in the indicated product of 
 any number of factors, the signs of any even number of factors 
 may be changed without altenng the value of the exjyression. 
 
 Thus, (a — 6) (c — d) {e —f) , by changing the signs of the 
 second and third factors, may be written in the equivalent 
 form (a — b) (d — c) (/— e) . 
 
36 ALGEBRA. 
 
 VI. DIVISION. 
 
 85. Division, in Algebra, is the process of finding one of 
 two factors, when their product and the other factor are 
 given. 
 
 Hence, Division is the converse of Multiplication. 
 
 Thus, the division of 14 a6 by 7 a, which is expressed 
 
 la 
 
 (Art. 15), signifies that we are to find a quantity which, 
 when multiplied by 7 a, will produce 14 a6. 
 
 86. The Dividend is the product of the factors. 
 The Divisor is the given factor. 
 
 The Quotient is the required factor. 
 
 87. Since the dividend is the product of the divisor and 
 quotient, it follows, from Art. 77, that : 
 
 If the divisor is + , and the quotient is + , the dividend is + 
 If the divisor is — , and the quotient is + , the dividend is — 
 If the divisor is + , and the quotient is — , the dividend is — 
 If the divisor is — , and the quotient is — , the dividend is -|- 
 
 In other words, if the dividend and divisor are both +, or 
 both — , the quotient is + ; and if the dividend and divisor 
 ^re one +, and the other — , the quotient is — . Hence, in 
 Division as in Multiplication, 
 
 Like signs produce +, and unlike signs pi^oduce — . 
 
 88. Required the quotient of 14 a& divided by 7 a. 
 
 By Art 85, we are to find a quantity which, when multi- 
 plied by 7a, will produce 14 a6. That quantity is evidently 
 2 h ; hence 
 
 Uah 
 
 = 26. 
 
 V a 
 
 That is, the coefficient of the quotient is the coefficient of the 
 dividend, divided by the coefficient of the divisor. 
 
DIVISION. 37 
 
 89. Required the quotient of a^ divided by a?. 
 We are to find a quantity which, when multiplied by a^, 
 will produce a^. That quantity is evidently a^ ; hence 
 
 — = a^ 
 
 That is, the exponent of a letter in the quotient is equal to 
 its exponent in the dividend minus its exponent in the divisor. 
 
 For example, — = a*""**. 
 
 DIVISION OF MONOMIALS. 
 
 90. We derive from Arts. 87, 88, and 89 the following 
 rule for the division of monomials : 
 
 To the quotient of the coefficients annex the literal quantities, 
 giving to each letter an exponent equal to t^ exponent in the 
 dividend minus its exponent in the divisor. Make the quotient 
 + when the dividend and divisor have like signs, and — when 
 they have unlike signs. 
 
 EXAMPLES. 
 
 1. Divide 54 a' by — 9 a*. 
 
 By the rule, ^^^ = - 6 a'-* = - 6 a^ Ans. 
 •^ -da' 
 
 2. Divide -2 a^b'cd^ by abd*. 
 
 -2a^b-cd' 
 
 abd^ 
 
 == — 2a-6c, Ans. 
 
 Note. A literal quantity having the same exponent in the dividend 
 and divisor, as d^ in Ex. 2, is canceled by the operation of division, and 
 does not appear in the quotient. 
 
 3. Divide — 91x'"2/"2;'"by — ISaj^t/V. 
 
 ^-- = Ix'^-^z'^-^, Ans. 
 
38 ALGEBRA. 
 
 Divide the following : 
 
 4. 84 by -12. 14. - 18s(^fz by dx'z. 
 
 5. - 343 by 7. 15. - GSa^ftV by -5ab^c\ 
 
 6. -324 by -18. 16. 72m^n by -12m2. 
 
 7. 444 by -37. 17. 12a;Y ^J ^^Y- 
 
 8. 12a^ by 4a. 18. - 18a'"6 by Qah. 
 
 9. - a?c by ac. 19. - lUc^d'e^ by - ZQc'd^e. 
 
 10. 2 mV by - mn^. 20. - 3 a'^^^ by ci'»+i. 
 
 11. — 8a^2/^by — 4a^. 21. a'^+^ft"^'* by a^ft". 
 
 12. 30a-^&3 by bd'h. 22. - Ola^^fz^ ^^ -i^y^y\ 
 
 13. 14m^7i^ by —Imn^. 23. 18m^wy by —2m^np^. 
 
 DIVISION op POLYNOMIALS BY MONOMIALS. 
 
 91. The operation being simply the converse of Art. 81, 
 we have the following rule : 
 
 Divide each term of the dividend by the divisor, and connect 
 the results with their proper signs. 
 
 EXAMPLES. 
 1 . Divide 9 a^6 - 6 a^c + 1 2 a-hc by - 3 a\ 
 By the rule, 
 
 9a^6- 6 a*c+ 12 a^&c 
 
 3a2 
 
 — 3 a6 H- 2 a^c — 4 5c, Ans. 
 
 Divide the following : 
 
 2. 8 a%c +16 a'hc - 4 aV by 4 ah. 
 
 3. 9a;*+27ic3-21ic2by -3a^. 
 
 4. 30a«-75a^6by 15 a^ 
 
 5. 2 arV^z - 12 ajyV by - 2 ic/j;. 
 
DIVISION. 39 
 
 6. 5 a%G— 5 ah\ -\- 5 ahcr by — 5 dbc. 
 
 7. 4a/-8a;^-14a;^ + 2a;*-6ir3by 2a^. 
 
 8. -12a^'6' - ^Oa}-W + 108a"5" by - 6a"'5\ 
 
 9. 20.T^-12a^-28a;by 4a;. 
 
 10. — a?l^c — aire- + arhcr by — dbc. 
 
 11. 9 a'6c - 3 a-6+ 18 a-^6c by -3a6. 
 12.15 Tfy^'z'' - 35 a^^+y''^ by 5 x^'y'^z. 
 13. 20a*6c + 15a6(f — lOa'6 by — 5a6. 
 
 DIVISION OF POLYNOMIALS BY POLYNOMIALS. 
 
 92. Required the quotient of 12 + lOar' - 11 a; - 21 a;^ 
 divided by 2 ar^ — 4 — 3a;. 
 
 Arranging both dividend and divisor according to the 
 descending powers of x (Art. 37), we are to find a quantity 
 which, when multiplied b}' the divisor, 2ar — 3a; — 4, will 
 produce lOar'^- 21a;2_ iia._f. 12. 
 
 It is evident, from Art. 82, that the term containing the 
 highest power of x in the product, is the product of the 
 terms containing the highest powers of x in the factors. 
 Hence lOa:^ is the product of 2a;2 and the term containing 
 the highest power of x in the quotient. Therefore the term 
 containing the highest power of x in the- quotient is lOa;^ 
 divided by 2a;^, or 5 a;. 
 
 Multiplying the divisor by 5 a;, we have the product 
 lOar" — loa;^— 20a;; which, when subtracted from the divi- 
 dend, leaves the remainder — 6ar^ + 9a;-}-12. 
 
 This remainder is the product of the divisor by the rest 
 of the quotient ; heuce, to obtain the next term of the quo- 
 tient, we proceed as before, regarding — 6a.'^ + 9a; + 12 as a 
 new dividend. Dividing the term containing the highest 
 power of .T, — Qx;^^ by the term containing the highest power 
 
40 ALGEBRA. 
 
 of X in the divisor, 2x'^, we have —3 as the second term of 
 the quotient. 
 
 Multiplying the divisor by — 3, we have — G x^ -{- 9 x -\- 12 ; 
 which, when subtracted from the second dividend, leaves no 
 remainder. Hence 5x — 3 is the required quotient. 
 
 It is customary to arrange the work as follows : 
 
 10a^-21x^-Ux-^12 
 
 2a^-3a;- 
 
 -4, Divisor. 
 
 10ar^-15a.'2_20ic 
 
 5a; -3, 
 
 Quotient. 
 
 — 6ar+ 9a; + 12 
 
 - 6x'-{- 9i»+12 
 
 
 Note. We might have solved the example by arranging the divi- 
 dend and divisor according to the ascending powers of x, in which case 
 the quotient would have appeared in the form — 3 + 5 r. 
 
 93. From Art. 92, we derive the following rule for the 
 division of polynomials : 
 
 Arrange both dividend and divisor in the same order of 
 powers of some common letter. 
 
 Divide the first term of the dividend by the first term of the 
 divisor^ giving the first term of the quotient. 
 
 Multiply the whole divisor by this term, and subtract the 
 product from the dividend, arranging the remainder in the 
 same order of powers as the dividend and divisor. 
 
 Regard the remainder as a new dividend, and proceed as 
 before; continuing until there is nj remainder. 
 
 Note. The work may be verified by multiplying the quotient by 
 the divisor, which should of course give the dividend. 
 
 EXAMPLES. 
 1. Divide 21^2^-22 a;?/ -8 by 3 a;?/ -4. 
 
 2\xY-22xy 
 21x'y^-28xy 
 
 Sxy — 4 
 
 lxy-\-2, Ans. 
 
 Qxy — 8 
 exy — 8 
 
DIVISION. 41 
 
 2. Divide 84-18ic'-56ic2by -6a,-2 -f4+8ic. 
 Arranging according to the ascending powers of if, 
 
 8 + lGa; -12a^ 
 
 4^Sx-Gar 
 
 2 — 4a; — 3ar, Ans. 
 
 -IQx -44«2^igaj4 
 -Ux-d2x^-^24.x^ 
 
 -V2xr-'2ia^-^18x* 
 
 3. Divide 9 ab^ + a-* - 9 6^ - 5 a'b by Sb^ + a^-2 ab. 
 
 Arranging according to the descending powers of a, 
 
 a^-2ab + Sb^)a^ - 5a*6 + 9a6* - 96=» (a - 36, Ans. 
 c^^2a^b-{-Sab' 
 
 -3d'b-{-6ab^-9b^ 
 -Sd'b + eab^-db^ 
 
 Divide the following : 
 
 4. 6x^-x-S5 by 3a;-f7. 
 
 6. 2 - 3oa; - 2aV by 1 -2ax, 
 
 6. a2-4a6 + 462by a-26. 
 
 7. 59a;-56-15arby 3a;-7. 
 
 8. 36^ + 3ab^ -4a'b-4.a^ by 6 + a. 
 
 9. 2a^x— 2 aa;' by oa; — a-. 
 
 10. 18ar»-5a; + lby6a;2^2a;-l. 
 
 11. 8m3 + 35 -36m by 5 + 2m. 
 
 12. 27a;3 + 2/«by 3a; + 2/. 
 
 13. 16m*- 1 by 2?/i-l. 
 
 14. a^ - 62 _(_ c2 - 2ac by a + 6 - c. 
 
 16. 8a3 + 36a-6 + 54a62-h2763by 2a + 36. 
 
 16. x*-\-y' + a^ifhyx^-{-f-\-xy. 
 
42 ' ALGEBRA. 
 
 17. 2af^-ldx' + 9hy 2a^-\-6x^-x-3. 
 
 18. 8 m^ + 3 w"'' — 4 mhi — 6 mv^ by 2m — n. 
 
 19. 4a^-8ar'-6a;- + 24 by 2ar-4. 
 20.^23a^-48 + 6ic*-2a;-31ar^by 6+3a^-5aT. 
 
 21. 4a^ + 27-a3by 9-3a-^ + 4a2_j-2a4_6a. 
 
 22. x'-^iT-Qxy-y'hy 7?-{-3x + y. 
 
 23. a«-816*by a2-j_3^,. 
 
 24. x^ — y- -\- 2yz — z- hy X + y — z. 
 
 25. 3a;''- Mar' 4- 8 by a; -2. 
 
 26. / 4- arV by a; + y- 
 
 27. 15m'' + 50m2 + 15-32m-32m-'by Sm^-f 5-4m; 
 
 28. l-i-4a^ + 3aj*by (a.-+l)2. 
 
 29. 21a^-216'by la-lh. 
 
 30. 64a;^ + l by 8a;2_4^_^l^ 
 
 31. 50a;4-9a;'' + 24-67a^by a; + ar^-6. 
 
 32. a;* + ^ — 4a;2/^ — 4a^2/ + ^x^y'^ by .t^ + 2/^ — 23;?/. 
 
 33. a;*-4a^ + 2a;2 + 4x+l by (aj- 1)2-2. 
 
 34. 9a;^ + 42/^-37.Ty by 3a^-22/2 + 5ar2/. 
 
 35. a^ + a^fes 4.2554 by {a-h){a-bh) + 'dah. 
 
 36. 3ar' + 4a; + 6.r' - 11 a.-^- 4 by 3a;- -4. 
 
 37. 6ar''4-15a;^4-51a;-18 by 2a;3_4^_^7^_2. 
 
 38. 2a^-lla;-4a^-12-3a;3by 4 + 2a.'24-a;. 
 
 39. m* - 48 - 17m3 + 52m + 12m2 by m - 2 + m^. 
 
 40. 3;''+^ + x^'y — ocy'' — y""^^ by x'' — y". 
 
 41. a^y — xy^hya:^-\-y^-\-xy^-\-3ify. 
 
 42. a;^ — 6a;2 — a; — 6 by a.-2 + 2a; + 3. 
 
 43. 2a* + 53a263- 496^ - Id'b'' -da^b bv 2a' - r>ab-7bK 
 
DIVISION. 43 
 
 44. af-^ex* + 6x^-lhy x'-h2x'-x-l. 
 
 45. 2x'^ - Qy- - I2z^ -^ xy - 2xz + 17 yz by 2x-^4:Z-Sy. 
 
 46. a^H _ 52^ _|_ 2 ft-c*- - c^'- by a** + 6" - c^ 
 
 47. a;6_i_63,4_3^l^y _2a:2_a.^a.i_l^ 
 
 48. 12 a'-Ua*b + lOa^'ft^ _ a%^ _ 8a6^+ 46^ 
 
 by Ga^- 4 a-6 - 3 a^^ + 2 ^^ 
 
 The operation of division may be abridged in certain cases 
 by the use of parentheses. 
 
 49. Divide {a^-{-ab)x^-{-{2ac-\-bc-^ad)x-\-c{c + d) 
 
 by ax + c. 
 
 ax-\-c 
 
 {a^+ ab)si^ -\-(2ac-{-bc-\-ad)x-{-c{c-{-d) 
 
 (a^-{-ab)x^-{-( ac-\-bc )x | {a+b)x+{c-\-d), 
 
 ( ac -\-ad)x-\-c{c-j-d) Arts, 
 
 ( ac -\-ad)x-\-c(c-\-d) 
 
 Divi^le the following : 
 ^/'^ '^- . 
 
 50. ^ +Xa -\-b-\- c)x^ + (ab + 6c + ca)x + abc 
 
 by ic2+(6+c)a; + 6c. 
 
 51. (6 + c)a2 + (&^ + 36c + c2)a + «>c(6 + c) by a + ft+c. 
 
 62. (x + y)2_5(x-f2/) + 6by (a; + y)-2. 
 
 63. (a + 6)3+lby (aH-6)H-l. 
 
 64. a^ + (a -\-b — c)x^ + {ab — be — ca)x — a6c 
 
 by ar^ + (6 — c)a; — 6c. 
 
 55. (m-n)^-2(m-n)2+l 
 
 by (m — n)2— 2(m — n) + l. 
 
 66. a^-i-(a — b-^c)x^ + {axi — ab — bc)x — abc by a;4-c. 
 
 67. a;* + (3 - 6)a:^ + (c - 3 6 - 2) a^ + (2 6 + 3 c) a; - 2 c 
 
 by ar^ + 3a;-2. 
 
 58. a'-(6 + c)H-a(62-f.6c + c-)-6c(6 + c) by a + 6+c. 
 
44 ALGEBRA. 
 
 VII. FORMULAE. 
 
 94. A Formula is the algebraic expression of a general 
 rule. 
 
 95. The following results are' of great importance in 
 abridging algebraic operations : 
 
 a +6 
 
 a —h 
 
 a +b 
 
 a +& 
 
 a —b 
 
 a —b 
 
 aH cib 
 
 o?—ab 
 
 a--\-ab 
 
 ah +y 
 
 ^ah +62 
 
 -ab-b' 
 
 a^+2a6-f ^' 
 
 a--2a6 + 62 
 
 a' ~b' 
 
 In the first case, we have (a+ by = a^ -{- 2 ab -\- b^. (1) 
 
 That is, the square of the sum of two quantities is equal to 
 the square of the first, plus twice the product of the two, plus 
 the square of the second. 
 
 In the second case, we have (a — by = a^ — 2 ab -{- b^. (2) 
 
 That is, the square of the difference of two quantities is 
 equal to the square of the first, minus twice the product of the 
 two, plus the square of the second. 
 
 In the third case, we have (a -\-b){a — b) =o? — b^. (3) 
 
 That is, the product of the sum and difference of two quan- 
 tities is equal to the difference of their squares. 
 
 EXAMPLES. 
 
 96. 1. Square 3« +25c. 
 
 The square of the first term is Oa^, twice the product of 
 the terms is 12a6c, and the square of the second term is 
 4 6V. Hence, by formula (1), 
 
 (3a -f 26c)2 = 9a2 -}- I2a6c + 46^02. 
 
FORMULA. 45 
 
 Note. The following rule for the square of a monomial is evident 
 from the above : 
 
 Square the coefficient, and multiply the exponent of each letter by 2. 
 
 Thus, the square of 5 a% is 25 a^b^. 
 
 2. Square 4a; — 5. 
 
 By formula (2), (4a;- 5)^ = 16ar^- 40.T + 25, Ans. 
 
 3. Multiply 6 a2 -f 6 by G a2 _ 6. 
 
 By formula (3), (joa' + b)(fia? -h) = ?>(ja' -h\ Ans. 
 
 Write by inspection the values of the following : 
 
 4. (a; -4)2. 16. (3a;»4-13)2. 
 
 6. (3 + a)2. 17. {(Sd'-h^cy. 
 
 6. (K + 3)(a;-3). 18. {ba+llr){6a-lW), 
 
 7. ^(3 a + 5)2. 19. (13a64-5ac)2. 
 
 8. (2a;+l)(2a;-l). 20. (ar^H-5a;)(a;«- 5 a;). 
 
 9. (7-2a;)2. 21. {\-V2xyzY- 
 
 10. (2m + 37i)2. 22. {4.7? -\-Zf){4.x' -^^f). 
 
 11. (4a6-a;)2. 23. (10ar^4-9a;')'. 
 
 12. (5 + 7a;) (5 -7a;). 24. {4.a^-b¥y, 
 
 13. (a;*-/)2. 25. (a"* + «'')(«'" -a"). 
 
 14. (3a;+ll)(3a;-ll). 26. (7a,'3 + lla;)^. 
 
 15. (a;V + 4)'''. 27. {bar-a'')\ 
 
 28. Multiply a + ^ + cbya + ft— c. 
 
 (a + 6 + c)(a + 6-c) = [(a + 6) + c][(a + 6)-c] 
 
 = {a^-hy - c\ by formula (3) 
 = a^ + 2ah-{-h- — c^, Ans. 
 
40 ALGEBRA. 
 
 29. Multiply a-\-b — chya — b-\-c. 
 
 (a 4- 6 - c) (a - 6 + c) = [a + (Z> - c)] [a - (5 - c)] 
 = a'-(b-cy 
 
 = a2-(&2_26c4-c2) 
 
 = a^-b'-h2bc-c', Ans. 
 
 Expand the following : 
 
 30. {x + y-\-z){x-y + z). 32. (1 + a- 6) (l-a + ft). 
 
 31. (x-j-y-\-z){x-y-z). 33. (x^ -\-x-\-l){x^-x-l). 
 
 34. (a + 6-c)(a-&-c). 
 
 35. (a2 + 2a4-l)(a'-2a + l). 
 
 36. (a^ + 2a;-3)(a^-2iB + 3). 
 
 37. (m^ + mn + n^) (m^ — mn + n^) . 
 
 97. 
 
 ''e find by multiplication : 
 
 
 X +6 
 
 X -5 
 
 X +3 
 
 oj -3 
 
 x'-^-bx 
 
 a^-5x 
 
 + 3a;+15 
 
 -3aj+15 
 
 a^ + 8a;+15 
 
 a^ — 8a5+15 
 
 X +5 
 
 a; -5 
 
 X -S 
 
 X +3 
 
 a-2 + 5i« 
 
 a^ — 5a; 
 
 -307-15 
 
 + 3a7-15 
 
 0^ + 207-15 
 
 a^-2a;-15 
 
 We observe in these products the following laws : 
 
 I. The coefficient of x is the algebraic sura of the numbers 
 in the factors. 
 
 II. The last term is the product of the numbers. 
 
 By aid of the above laws the product of two binomials of 
 the form ic + a, x-{-b may be written by inspection. 
 
FORMULA. 47 
 
 1. Required the value of {x — 8) (a; + 5) . 
 
 The coefficient of a; is — 3 ; and the last term is —40. 
 Hence, (x-8)(x-\-5) = x- -i\x-4.0, Ans. 
 
 EXAMPLES. 
 Write by inspection the values of the following : 
 
 2. {x-\-7)(x+o). 10. {x + 9){x-o). 
 
 3. (a;-3)(aj-4). 11. {x-8)(x-9). 
 
 4. {x-\-S)(x-2). 12. {x4-4.m)(x-^6m). 
 6. (a; -3)(a;4-l). 13. (x-oa)(x-\-a). 
 
 6. {x-o){x-\-(J). 14. (a-f 6)(a-46). 
 
 7. (a; + l)(a;4-12). 15. {a-\- ob){a + 8b). 
 
 8. (a;-7)(a;-f-2). 16. (ar'-3) Oc^- 7). 
 
 9. {x-S){x-G). 17. (a:'' + 2a)(ar'^-6a). 
 
 98. The following results may be verified by division : 
 
 (1) t:z^ = a-b. (3) ^±^=a^-ab + b'. 
 ^ ' a + 6 ^ ' a + 6 
 
 (2) ^L:z:^ = a + 6. (4) ^^-:^ = «2^a6 4-&'. 
 
 a— 5 a— 
 
 Formulae (3) and (4) may be stated in words as follows : 
 
 If the sum of the cubes of two quantities be divided by the 
 sum of tJie quantities^ the quotient is equal to the square of 
 the first quantity, minus the product of the two, plus the square 
 of the second. 
 
 If the difference of the cubes of two quantities be divided by 
 the difference of the quantities, the quotient is equal to the 
 square of the first quantity, plus the product of the two, plus 
 the square of the second. 
 
48 ALGEBRA. 
 
 EXAMPLES. 
 
 1. Divide 36 t/V - 9 by 6 yz" + 3. 
 
 By formula (1 ) , ^^-^T"'^ = Qyz^-^, Ans. 
 62/2;' + 3 
 
 2. Divide 1 + 8«^ by 1 + 2a. 
 
 By formula (3) , -^ = 1 — 2 a + 4 a^, Ans. 
 
 8. Divide 27a»- 6^ by 3a -5. 
 
 By formula (4) , ^^^'~^' = ^a? + 2,ah -\-h\ Ans. 
 
 EXAMPLES. 
 Write by inspection the values of the following : 
 
 4 ^-81 9 27+a?^ 14 ^ -^ 
 
 ' 3 + x' ' x'-f 
 
 c zo^iocr jQ a^ — 16x-^ ^g 27 + flry 
 
 ic^ + 4 a; * 3 + aJ2/^ 
 
 c ..-rx .. 0.-3-64 i« 49a2-121&4 
 
 7. 
 
 a?-9 
 
 25^16a2 
 
 5 + 4a 
 
 a^+1 
 
 x+l 
 
 l-m"^ 
 
 1 — m 
 
 a^-S 
 
 -o l_-8m^^ j,^ 64m^4-n^ 
 
 1 — 2m 47?i + 7i^ 
 
 8 «'-^ 13 «!±34§. 18 ^±125/. 
 
 a — 2 * a + 7 ' x-{-by 
 
 Divide the following : 
 
 19. '21i^y^-U^hy 3xy-4.z. 
 
 20. 25 a^ - 81 Vc^ by 5 a- - 9 6c«. 
 
 21. 343 + 125ar>3by 7 + 5a;2/- 
 
FORMULAE. 49 
 
 22. Q4:m^-216n^hy 4m-6n\ 
 
 23. 729 a^f +512z^hy doc^y-^ 8^. 
 
 99. By actual division we obtain : 
 
 a+b 
 a*-b' 
 
 = CfcS _ ^25 ^ ^^2 _ 53^ 
 
 ^^-±-^' = a^ _ a-^ft + a^ft^' - a&8 + 6*. 
 a + 6 
 
 ^^^:J^ = a' -ha'b -\-a'b' + ab' + b* ; etc. 
 a — 
 
 In these results we observe the following laws : 
 
 I. The number of terms is the same as the exponent of a 
 in the dividend. 
 
 II. The exponent of a in the first term is less by. 1 than 
 the exponent of a in the dividend, and decreases b}' 1 in 
 each succeeding term. 
 
 III. The exponent of b in the second term is 1, and 
 increases by 1 in each succeeding term. 
 
 IV. The terms are all positive when the divisor is a — 6, 
 and are alternately positive and negative when the divisor 
 is a + 6. 
 
 100. In connection with Art. 99, the following principles 
 are of great importance : 
 
 If n is any whole number, 
 
 (1) a** -f 6" is divisible by a-\-b if n is odd, and by neither 
 a + 6 nor a — b if n is even. 
 
 (2) a'' — h^ is divisible by a — b if n is odd, and by both 
 a-\-b and a — b if n is even. 
 
50 ALGEBRA. 
 
 EXAMPLES. 
 101. 1. Divide dJ - V by a - h. 
 Applying the laws of Art. 99, we have 
 
 a — h 
 
 2. Divide iK^- 81 by a; + 3. 
 Since 81 = 3'', we have 
 
 ^^-^^ = a.-3-3a;2-|-32a;-33 = a^-3a^ + 9a;-27, Ans. 
 a; + 3 
 
 Write by inspection the values of the following : 
 3. 
 
 4. 
 6. 
 6. 
 
 7. 
 
 a« 
 
 -Jf 
 
 a 
 
 -b 
 
 x' 
 
 -f 
 
 X 
 
 4-2/ 
 
 m! 
 
 + n^ 
 
 m 
 
 + n 
 
 wJ 
 
 -n^ 
 
 m 
 
 — 7i 
 
 1- 
 
 -a;^ 
 
 8. 
 
 x'-lQ 
 x-2 
 
 13. 
 
 a;^-32 
 fl;-2 
 
 9. 
 
 1-a' 
 1 -a 
 
 14. 
 
 a«-64 
 
 a + 2 
 
 10. 
 
 a^ + 1 
 a + 1 
 
 15. 
 
 a^ + b' 
 
 11. 
 
 1 -7i« 
 
 1-71 
 
 16. 
 
 a;^-128 
 
 a;-2 
 
 12. 
 
 a;* -81 
 
 17. 
 
 a.-^+243 
 
 1— a; a; — 3 a;4-3 
 
 Divide the following : 
 
 18. m^-16n^hym-2n\ 20. 32a^4- &* by 2a + &. 
 
 19. a^ — y^s^ by x — yz. 21 . m^ — 243 n^ by m — 3 n. 
 
 22. 2bQx^-f by 4x + 2/^ 
 
FACTORIXG 
 
 VIII. FACTORING. 
 
 102. Factoring is the process of resolving a quantity into 
 its factors. (Art. 11.) 
 
 103. The factoring of monomials may be perfoimed by 
 inspection ; thus, 
 
 \2a'b-c = 2'2'3aaabbc. 
 
 A polynomial is not always factorable ; but there are 
 certain forms which can always be factored, the more 
 important of which will be considered in the succeeding 
 articles. 
 
 Case I. 
 
 104. ^V7len the terms of the polynomial have a common 
 monomial factor. 
 
 1. Factor a'* 4- 3 a. 
 
 Each term contains the monomial factor a. 
 Dividing the expression by a, we have a^-|-3. B^ence, 
 a^ + 3a=^a{a^-{-Z), Ans. 
 
 2. Factor Uxy^ - 35 oi^f. 
 
 Uxy*-3Da^f=7xf{2y^-6x^), Aiis, 
 
 EXAMPLES. 
 Factor the following : 
 
 3. x^-\-5x. 8. 5a^ + 10^2 + 15a;. 
 
 4. 3m3-12wr. 9. a' -2a*-\-Sa^ -a\ 
 
 5. lGa*-12a. 10. 36a^y -GOx'y' -84:xY. 
 
 6. 27 c*d--\-dc^d. 11. 21m^n-\-Som7i^-Umu. 
 
 7. 60mV-12m8. 12. S^scf^f -140 a^y*-\-70xy. 
 
52 • ALGEBRA. 
 
 13. Factor the sum of 54a%^ - 72 aV, and - dOa^d. ' 
 
 14. Factor the sum of 96c^c/% 120cW, and - IMc^ti^ 
 
 Case II. 
 
 105. Wlien the polynomial consists of four terms, of which 
 the first two and the last two have a common binomial factor. 
 
 1 . Factor am — 6m -f- an — bn. 
 
 Factoring the first two and last two terms as in Case I, 
 we have 
 
 m (a — b)-{-7i(a — b). 
 
 Each term now contains the binomial factor a — b. Divid- 
 ing the expression by a — 6, we obtain m-\-7i. Hence, 
 
 am — bm-\-an — bn = (a — 6) (m -f- n) , Ans. 
 
 2. Factor a7n — bm — an ■}- bn. 
 
 am — bm — an-\- 6n = am — bm — {an — bn) 
 = m{a — b) — n(a — b) 
 = (a — b) {m — n) , Ans. 
 
 Note. If the third term is negative, as in Ex. 2, it is convenient, 
 before factoring, to enclose the last two terms in a parenthesis preceded 
 by a — sign. 
 
 EXAMPLES. 
 Factor the following : 
 
 3. ab-^bx-\-ay -\- xy. 8. a^ — a^b — ab"^ -f W. 
 
 4. ac — cm -{-ad — dm. ^. a?-{-ax — bx — ab. 
 
 5. a^ -{- 2 X — xy — 2 y. 10. ma^ — my^ + 7ix- — ny^. 
 
 6. x^ — ax — bx-{-ab. 11 . a^ -\-x^ + x~\- 1. 
 
 7. a^-a'b + ab^-b^ 12. 6af + 4cx'-9x-6. 
 
 13. 8cx—12cy+2dx — 3dy. 
 
 14. 6n-21mhi-8m-{-28mK 
 
FACTOmXG. 53 
 
 106. If a quantity can be resolved into two equal factors, 
 it is said to be a perfect square^ and one of the equal factors 
 is called its square root. 
 
 Thus, since 0a''&2 equals 3a-6 X 3a-6, it is a perfect square, 
 and 3 arh is its square root. 
 
 Note. 9 a'^b- also equals — 3 a-i X — 3 a-b, so that its square root is 
 cither 3 a-b or — 3 a-b. In the examples in this chapter we shall con- 
 sider the positive square root only. 
 
 107. The following rule for extracting the square root of 
 a monomial is evident from Art. lOG : 
 
 Extract the square root of the coefficient^ and divide the 
 exponent of each letter by 2. 
 
 For example, the square root of 'Ihv^ifz- is bot^y^z. 
 
 108. It follows from Art. 95 that a trinomial is a perfect 
 square when its first and last terms are perfect squares and 
 positive, and the second term is twice the product of their 
 square roots. 
 
 Thus, 4a;^ — 12^-^ + 9^ is a perfect square. 
 
 109. To find the square root of a perfect trinomial square, 
 we take the converse of the rules of Art. 95 : 
 
 Extract the square roots of the first and last terms, and 
 connect the results by the sign of the second term. 
 
 Thus, let it be required to find the square root of 
 
 ^ix'-Uxy-^df. 
 
 The square root of the first term is 2x^ and of the last 
 term Sy; and the sign of the second term is — . Hence 
 the required square root is 
 
 2x — Sy. 
 
54 ALGEBRA. 
 
 Case III. 
 HO. Wlien a trinomial is a perfect square (Art. 108). 
 
 1. Factor a^ 4-2 a6'' + 6^ 
 
 By Art. 109, the square root of the expression is a-\-Jr. 
 Hence, 
 
 a^ + 2 aW + 5^ = (a + W) (a + W) , or (a -f h-y, Ans. 
 
 2 . Factor 4 a;^ — 1 2 cc^/ + 9 2/^. 
 
 4a^ - 12^2/ + 9/ = (2a; - 3?/) (2a; - 32/) 
 
 = (2a; — 32/)^, ^ns. 
 
 Note. The given expression may be written d y"^ — \2xy -\- 4:X'\ 
 whence, 
 
 9^2 _i2:ry + 40-2= (3?/-2a:)(3y-2x)r:. (3?/-2ar)2; 
 
 which is another form of the answer. 
 
 EXAMPLES. 
 Factor the following : 
 
 3. x' + ^xy + f-. 16. 36m2-36mn + 9n2. 
 
 4. 4 + 4m + m-. 17. 4a2_j_44a6 4- 12162. 
 
 5. a^- 14a; 4- 49. 18. a;^ + 8a;^ + 16a;^ 
 
 6. a^-\0a + 2b. 19. a^ft^ + 18a6-c + Slc^. 
 
 7. 2/2 + 22/ + 1. 20. 25a;--70a;2/>-f 49 2/V. 
 
 8. m2-2m4-l. 21. 9 a;«- GGx« 4- 121 a;^ 
 
 9. a;^4-12a;2 + 36. 22. 9a* +60 a^^c^d +100 &V(^2^ 
 
 10. n«- 20^3 + 100. 23. 64a;8_ig0a;^ + 100a;^ 
 
 11. a;y+16a;2/ + 64. 24. 4a*6'-^ + 52a«6^ + lG9ci26^ 
 
 12. l-10a& + 25a-6-. 25. 16a;^-120mwar+225m-yr. 
 
 13. lQ,m''-d>am + a'. 26. (a - 6)2 + 2(a- 6)+ 1. 
 
 14. a* + 2a3 + a2. 27. (a; + 2/)'- 16 (a; + 2/) + 64. 
 
 15. a;«-4a;'' + 4a;2. 28. (a;^ - a;)2 + 6(ar - a;) + 9. 
 
FACTORING. 55 
 
 Case IV. 
 
 111. When an expression is the difference of two jjerfect 
 squares. 
 
 Comparing with the third case of Art. 95, we see that 
 such an expression is the product of the sum and difference 
 of two quantities. 
 
 Therefore, to obtain the factors, we take the converse of 
 the rule of Art. 95 : 
 
 Extract the square root of the first term and of the last 
 term; add the results for one factor, and subtract the second 
 result from the first for the other. 
 
 1. Factor36a:2_492^^ 
 
 The square root of the first term is 6 a;, and of the last 
 term ly. Hence, by the rule, 
 
 36ic2-49/ = (6a;-f 72/)(6ic-72/), Ans, 
 
 2. Factor (2a; -3 2/)'- (a;- ?/)^ 
 {2x-^yy-{x-yy 
 
 = [(2a;-3y)4-(^-2/)][(2a:-32/)-(a;-2/)] 
 = {2x-^y + x-y){2x-^y-x + y) 
 = (dx — 4ty){x — 2y), Ans. 
 
 EXAMPLES. 
 Factor the following : 
 
 3. x'-y-. 7. da^-l6y\ 11. 4.9 ni" - 100 7i\ 
 
 4. a^-1. 8. 25a'-b*. 12. S6x*-Sly\ 
 
 5. 4-a2. 9. l-4:9a^y\ 13. Gia^ -I21b-c'. 
 
 6. 9m2-4. 10. a^b^-c*d^ 14. 144 a;^^^* - 225 ««. 
 
5(5 ALGEBRA. 
 
 15. (a-\-by-{c + dy. 19. {x-cy-(y-dy. 
 
 16. (a-cy-b'. 20. {a-3y-{b + 2y. 
 
 17. m^-{x-yy. 21. (2a; + m)2-(aj-m)2. 
 
 18. m^-(m-l)2. 22. (3 a + 5)2 -(2 a -3)2. 
 
 It is sometimes possible to express a polynomial in the 
 form of the difference of two perfect squares, when it may 
 be factored by the rule of Case IV. 
 
 23. Factor 2 mn -{- m^ — 1 -{- 7i^. 
 
 The expression may be written m^ -j- 2mn -j-^r — 1, which, 
 by Case III., is equivalent to {m-\-ny— 1. Hence, by the 
 rule, 
 
 (m -\-ny — 1 = (m + ?i + 1) (m + ?i — 1) , Ans. 
 
 24. Factor 2xy -{-1 —£- — y'^. 
 
 2xy -\-\—Q^ — y'=\—'2i?-\-2xy — y- = \ — {x^— 2xy-{-y-) . 
 
 By Case III., this maj^ be written \ —{x — yy. Hence 
 the factors are 
 
 l\+{x-y)^[l-ix-y)-]=={\+x-y){\-x + y), Ans. 
 
 25. Factor 2xy -\-W — x^ — 2ab —y^ -\-a^. 
 
 2xy ^b"" -a? -2ab -y- + o? 
 
 ^a^-2ab-\-y'-Q^-\-2xy-y^ 
 ^a?-2ab + b''-{:t?-2xy + y-) 
 = (a _ 5)2 _ (a; - 2/)2, by Case III. 
 = [(a-6)+(a._2/)][(a-6)-(x-2/)] 
 = {a — b-{-x — y){a — b — x-^y)^ Ans. 
 
 Factor the following : 
 
 26. a^ + 2a;y + ?/2-4. 28. o? -b'' + 2bc-c'. 
 
 27. a''-2ab^b^-(^. 29. a? -b^ -2bc-c'. 
 
FACTORING. 57 
 
 30. c2-l+(F + 2cd. 32. 4:b-l-U- + 4.m\ 
 
 31. 9_a^_2/2_j_2a^. 33. 4:0^ -^b^ -9d' -4.ab. 
 
 34. a^ — 2a7n-\-m^ — b' — '2 bn — ir. 
 
 35. ar^_2/2^_c2_d2_2ca; + 2d?/. 
 
 36. a- -I)- + 11V -n^ -\-2am -\-2bn. 
 
 37. a2-62_|_c2_d2-f-2ac-26d. 
 
 Case V. 
 
 112. TWien an expression is a tnnomial of the form oi?-{-ax-\-h. 
 
 In Art. 97 we derived a rule for the product of two biuo- 
 mials of the form a: + a, a; + 6, by considering the following 
 cases in multiplication : 
 
 1. (ic + 5)(a;4-3) = ar-f 8a; + 15. 
 
 2. {x-b){x-^) = x--Sx+lb. 
 
 3. (a; -h 5) (a; - 3) = or H- 2a; -15. 
 
 4. (;f — 5)(a; + 3) = ar — 2a;4-lo. ' 
 
 . In certain cases it is possible to reverse the operation, and 
 resolve a trinomial of the form a,*^ -^-ax-^-b into the product 
 of two binomial factors. 
 
 The first term of each factor will obviously be x ; and to 
 obtain the second terms, we take the converse of the rule of 
 Art. 97 : 
 
 Find two numbers whose algebraic sum is the coefficient of a;, 
 and whose 2)roduct is the last term. 
 
 Thus, let it be required to factor a^ — 5 a; — 24. 
 
 The coefficient of a; is — 5, and the last term is — 24 ; we 
 are then to find two numbers whose algebraic sum is — 5, 
 and product — 24. By inspection we determine that the 
 numbers are — 8 and 3. Hence, 
 
 a;2 _ 5 a; - 24 = (a; - 8) (a; -h 3) . 
 
58 ALGEBRA. 
 
 113. The work of finding the numbers may be abridged 
 by the following considerations ; 
 
 1. Wlien the last term of the product is +, as in Exs. 1 
 and 2, the coefficient of x is the sum of the numbers ; both 
 numbers being -|- when the second term is -f-> and — when 
 the gecond term is — . 
 
 2. When the last term of the product is — , as in Exs. 3 
 and 4, the coefficient of x is the difference of the numbers 
 (disregarding signs) ; the greater number having the same 
 sign as the second term, and the smaller number the opposite 
 sign. 
 
 We may embody these observations in two rules, which 
 will be found more convenient than the rule of Art. 112 in 
 the solution of examples : 
 
 1. If the last term is -}-, jind two numbers ivhose sum is the 
 coefficient of x, and whose product is the last term ; arid give to 
 loth numbers the sign of the second term. 
 
 II. If the last term is — , find two numbers whose difference 
 is the coefficient of x, and whose jyroduct is the last terrn ; give 
 to the greater number the sign of the second term^ and to the 
 smaller number the opposite sign. 
 
 Note. By the expressions " coefficient of x" and " last term," in tlie 
 above rules, we understand their absolute values, without regard to sign. 
 
 EXAMPLES. 
 
 114. 1. Factor a^+ 14a; + 45. 
 
 According to Rule I., we find two numbers whose sum is 
 14, and product 45. The numbers are 9 and 5 ; and as the 
 second term is +, both numbers are -f-. Hence, 
 
 ar-f-14a; + 45 = (a;-j-9)(a; + 5), Ans. 
 
 2. Factor x^—(jx-\-b. 
 
 By Rule I., we find two numbers whose sum is 6, and 
 
FACTORING. 59 
 
 product 5. The numbers are 5 and 1 ; and as the second 
 term is — , both numbers are — . Hence, 
 
 a^— Qx-\-D ={x — 5)(x — l), Ans. 
 
 3. FactoriB2_|.5^_14, 
 
 By Rule II., we find two numbers whose difference is 5, 
 and product 14. The numbers are 7 and 2 ; and as the 
 second term is +, the greater number is +, and the smaller 
 number — . Hence, 
 
 a^^5x-U = (x-\-7){x-2), Ans, 
 
 4. Factor ic2_53._ 24. 
 
 By Rule H., we find two numbers whose difference is 5, 
 and product 24. The numbers are 8 and 3 ; and as the 
 second term is — , the greater number is — , and the smaller 
 number +. Hence, 
 
 a^-5x-24: = (x-S){x-^3), Ans. 
 
 Factor the following : 
 
 5. a^_|_5a;^6. 17. x'-Gx-U, 
 
 6. ar^ — 3a;-|-2. 18. m^-flGm + GS. 
 
 7. y^^2y-8. 19. a2-15a-h44. 
 
 8. m2-7m-30. 20. y--i-7y-G0. 
 
 9. a'-lla+lS. 21. a^-llir + 10. 
 
 10. a^^x-6. 22. m2 + 2??i-80. 
 
 11. c2 + 9c + 8. 23. n2 + 23?i+102. 
 
 12. 2/2 _ 22/ -35. 24. x^-dx-[)0. 
 
 13. a-4-13a-48. 25. a'-na-26. 
 
 14. ar-10a; + 21. 26. x'-\-x-^2. 
 16. a^-{-13x-{-SG, 27. (r-18c + 32. 
 16. n^-n- 90. 28. m^-Bm- 33. 
 
60 ALGEBRA. 
 
 29. a^ + 20aj-f 75. ' 37. a^* - 19a;2_ 120. 
 
 30. a.-2 + 4aj-96. 38. c« + 12(r^ + ll. 
 
 31. 2/2- 172/- 110. 39. ixrf-\-2xf-U0: 
 
 32. a^- 19 a; +78. 40. a-b^ - 7 ab'^ - lU. 
 
 33. 05- + 7a; — 98. 41. n^x^ + 26nx-\-100. 
 
 34. a2 + 22a + 105. 42. 2/8-20/ + 91. 
 
 35. a^- 23a; + 130. 43. a^^*- 2a26^-48. 
 
 36. a* + 10a^-U4:. 44. m^ + 26m2-87. 
 
 45. FsLGtorx^-i-^aba^-SiaFbK 
 
 We find two numbers whose difference is 5, and product 
 84. The numbers are 12 and 7 ; and, by the rule, the 
 greater is +, and the smaller — . Hence, 
 
 a-* + 5 abx- - 84 arb'- = (or -j- 12 ab) {x- - 7 ab) , Ans, 
 
 46. Factor 1 - 6a- 27 a^. 
 
 The numbers whose difference is 6, and product 27, are 9 
 and 3. Hence, 
 
 1 -Ga-27o"= (1 -9a)(l+3a), Ans. 
 
 Factor the following : 
 
 47. a^-3ax + 2x'. 56. (a + 5)^ + 5(a + 6) + 4. 
 
 48. a^ + 5a;2/-66?/-. 57. l-9rt + 8«2. 
 
 49. l + 13a + 42a2. 58. b* + dab--62a\ 
 
 50. m2-15mri + 567i2. 59. (m- ?i)2+ (m-n) - 2. 
 
 51. a^-ab-5Gb\ 60. x^-i)x^-bOx^. 
 
 52. d'b^ + 4:abc-4.5(^. 61. a' + Sab + ^2bK 
 
 53. 1- 3a; -10^2^ 62. 1 - 13a;?/ + 40ary. 
 
 54. a' + 15a^ + Ua'. 63. {a-by -S(a-b) - 4. 
 
 55. ;22_iOa;2/-2;-39a;V. 64. a^y + 8a;2^-2-482;2. 
 
factori:n"g. 61 
 
 115. If a quantity can be resolved into three equal fac- 
 tors, it is said to be a perfect cube, and one of the equal 
 factors is called its cube root. 
 
 Thus, since 27 a^b^ equals Sa^b x 3o-6 x 3a-?>, it is a per- 
 fect cube, and Ba^b is its cube root. 
 
 116. It is evident from the above that the cube root of a 
 monomial may be found by extracting the cube root of the 
 coefficient and dividing the exponent of each letter by 3. 
 
 Thus, the cube root of l2oa^t/^z^ is bx^y^z. 
 
 Case VI. 
 
 117. ^VJlen an expression is the sum or difference of two 
 perfect cubes. 
 
 By Art. 98, the sum or difference of two perfect cubes is 
 divisible by the sum or difference of their cube roots ; and 
 in either case, the quotient may be written by inspection by 
 aid of the rules of Art. 98. 
 
 EXAMPLES. 
 
 1. Factor a^-fl. 
 
 The cube root of a^ is a, and of 1 is 1 ; hence, one factor 
 is a + 1. 
 
 Dividing the expression by a-\-\, we have the quotient 
 a^-a + l (Art. 98) . Hence, 
 
 a3+l = (a-f-l)(a2-a-f 1), Ans. 
 
 2. Factor 27a^- 642/3. 
 
 The cube root of 21a:^ is 3a;, and of 64^/^ is 4?/; hence, 
 one factor is 3aj — 4y. By Art. Q8, the other factor is 
 9ar^-f 122^ + 16?/^. Hence, 
 
 27ar^-r)4?/» = (3a;-42/)(9a^-f 12aj?/4-ir)jr). Ans. 
 
>2 
 
 
 ALGEBRA. 
 
 
 
 Factor the following : 
 
 
 
 
 3. a^ + x\ 
 
 8. 
 
 a' + b'. 
 
 13. 
 
 m^-64:n\ 
 
 4. m^ — n^. 
 
 9. 
 
 o^-hl. 
 
 14. 
 
 Ux'-Uo. 
 
 5. af-1. 
 
 10. 
 
 27a^-l. 
 
 15. 
 
 125a3 + 27m«. 
 
 6. a%^ + c«. 
 
 11. 
 
 8c«-(f. 
 
 16. 
 
 U(f€lF + 27. 
 
 7. l-8aT^ 
 
 12. 
 
 27 + 8a^ 
 
 17. 
 
 126- 8 a^b\ 
 
 • 
 
 
 Case VII. 
 
 
 
 118. When an expression is the sum or difference of two 
 equal odd powers, of two quantities. 
 
 B}' Art. 100, the sum or difference of two equal odd powers 
 is divisible by the sum or difference of the quantities ; and 
 in either case, the quotient may be written by inspection by 
 aid of the laws of Art. 99. 
 
 EXAMPLES. 
 1. Factor a* + 6^ 
 
 By Art. 100, one factor is a + 6. Dividing the expression 
 hj a + b, the quotient is a* — a^b-^a-b- — ab^ + b* (Art. 99) . 
 Hence, 
 
 a^ - 6-5 = {a-\-b) (a* - a^b + a^b- -alf + h), Ans. 
 
 Factor the following: 
 
 2. a^-b"". 5. m'+n^ 8. c^-mV. 
 
 3. ic^+l. .6. x^-/. 9. l+32w^ 
 
 4. \-a\ 7. a^-1. 10. 243ar'-?/^ 
 
 11. a;'^ + 128. 12. 32-243tt*. 
 
 119. B}^ applying one or more of the rules already given, 
 an expression ma^' often be separated into more than two 
 factors. 
 
FACTORING. G3 
 
 1 . Factor 2 ax^y- — 8 axi/. 
 
 By Case I. , 2 ax^- — 8 axy* = 2 axy-{x^ — ^ /) • 
 Factoring the quantity in the parenthesis by Case IV., 
 2 aoi^y^ — Saxy* = 2 axy-(x -\-2y){x — 2y), Ans. 
 
 2. Factor m^ — 7t^ 
 
 By Case IV. , m^ — n^ = {m^ 4- n-^) (m^ — 7t^) . 
 By Case VI. , tif + n^ = (m 4- n) (»i^ — W7i + ir) , 
 and ??i'' — li^ =i(^m — n) (m^ -f- mn -\- n^) . 
 
 Hence, 
 m^—n^= (??i4-?i) (7?i — 7i) (7;r — mn + zr) (??r4-m;i + )i-), Ans. 
 
 3. Factor .r^ — y*. 
 By Case IV., 
 
 a^-f = (x'-^y*){x*-y*) 
 
 = (x* -\-y*)(x^-^ f) (x -f y) {x — y) , Ans. 
 
 MISCELLANEOUS EXAMPLES. 
 
 120. In factoring the following expressions, the common 
 monomial factors should be first removed, as shown in 
 Example 1 of the preceding article. 
 
 1. 
 
 6aV-6rt*a;. 
 
 8. 
 
 5 a'' — 5. 
 
 2. 
 
 l-4a; + 4ar. 
 
 9. 
 
 a%^ - (fd\ 
 
 3. 
 
 x'-\. 
 
 10. 
 
 x'-\Q>. 
 
 4. 
 
 a- + 9a + 18. 
 
 11. 
 
 a^ — a^-{- a^ — a-. 
 
 6. 
 
 ar + ax -\-hx + ab. 
 
 12. 
 
 3ar + 27if + 42. 
 
 6. 
 
 rn? — Im — 8. 
 
 13. 
 
 x^-(2y-Szy. 
 
 7. 
 
 2x'-\-x. 
 
 14. 
 
 a'-h^Oab + lOOb- 
 
64 ALGEBRA. 
 
 15. baJ^bc— 10 ab^c- lb abc~. 21. 1 -^12x + 27^j^. 
 
 16. 3a^-21a^ + 30a'. 22. ISor^ - 2a;?/«. 
 
 17. a^ + Sf^. 23. x^-x". 
 
 18. 2a^-2a. 24. 4x^7/ -{-2Sxf -{-49. 
 
 19. i_a2_52_f_2c^^,. 25. a^ + Ga-^- 40. 
 
 20. a^-8a;+7. 26. a^- 18a5- 405^. 
 
 27. 2xhj-{-2xy^~2xyz^Mf.4a^y\ 
 
 28. 12m3n-18mV + 24mn3. 
 
 29. 32a*6 + 4a6^ 40. (ar^ + /-;22)2_ 43^22^2^ 
 
 30. a;^-81. 41. o?bc - a(rd - abhl -\- bed' . 
 
 31. 2^-2^^ 42. a2-14a6 + 3362. 
 
 32. Q^ + 2x'^-x~2. 43. 3a;V4-3i»/. 
 
 33. a?+lix?-ZOx\ 44. 4m^- 20m2n + 25^2. 
 
 34. (3a; + ?/)2-(a;-2?/)'. 45. 3 a'b -\- Z o?^ - Q, ab\ 
 36. mV-8ma^- 65. 46. a V - a-y - 6^0^ + &y . 
 
 36. 135a^-5a^. 47. (a- 26)--2(o.-26)-8. 
 
 37. 2a^?/-2a^/ + 60a;/. 48. lOOor?/^- 81^2. 
 
 38. 80x^f-6x^y. 49. a«-64. 
 
 39. 3a36 + 18a2& + 27a5. 50. x'-(x-6y. 
 
 51. (a2 4-3a)2-14(a2 + 3a)+40. 
 
 52. (4m + ny - (2m - Sny. 
 
 53. (a2-?>2_c2)2_ 452^2^ 57. a2+62-c2-d2-2a6-2cd. 
 
 54. 1000 + 27m«. 58. (a^^-f 4)^- 16a^. 
 
 55. x''-x'-x-\-l. 59. x^-f~3xy(x-y). 
 66. 3(a2_&2)_(a_6)2. 60. (a^^ a -4)^-4. 
 
HIGHEST COMMON FACTOR. 65 
 
 IX. HIGHEST COMMON FACTOR. 
 
 121. A Common Factor of two or more quantities is a 
 quantity which will divide each of them without a remainder. 
 
 Thus, 2a;/ is a common factor of I2x^f and 20 o^y*. 
 
 122. A prime quantity is one which cannot be divided, 
 without a remainder, by any integral quantity except itself 
 or unitv. -. 
 
 For example, a, 6, and a-\-c are prime quantities. 
 
 123. Two quantities are said to be prime to each other 
 when they have no common factor except unity. 
 
 Thus, 2 a and 36^ are prime to each other. 
 
 124. The Highest Common Factor of two or more quanti- 
 ties is the product of all the prime factors common to those 
 quantities. 
 
 It is evident from this definition that the highest common 
 factor of two or more quantities is the expression of highest 
 degree (Art. 33) which will divide each of them without a 
 remainder. 
 
 Thus, the highest common factor of «^y and a^y* is x^y^. 
 
 125. In determining the highest common factor of alge- 
 braic quantities, it is convenient to distinguish three cases. 
 
 Case I. 
 
 126. TF7ien the quantities are monomials. 
 
 1. Find the H.C.F. of 420^62, lOa'bc, and d8a*I/cP. 
 
 42c^b^ = 2'S-7'(fb^ 
 
 70a^bc=2-5-7'a-bc 
 
 98a'b^d' = 2'7'7'a'b^cF 
 
 Hence, the H.C.F. = 2 • 7 • a=6(Art. 124) = Ua% Ans. 
 
66 ALGEBRA. 
 
 RULE, 
 
 To the highest common factor of the coefficients^ annex the 
 common letters^ giving to each the lowest exponent with which 
 it occurs in any of the given quantities. 
 
 EXAMPLES. 
 Find the highest common factors of the following : 
 
 2. a?x^, la'x. 5. 18m7i% 45mhi, 72mV. 
 
 3. 16cd\ 9cU 6. lUxfz^ loAafyz^, 
 
 4. 54 a^^, 90«c2. 7. ISa^a;, 45 ay, 60 aV. 
 
 8. 108a^2/V, U4:xfz\ 120a^2^V. 
 
 9. 96a^5^ 120a«6^ U8a'b\ 
 
 10. 51 a-m%, SoaVx, 119 aVyK 
 
 Case II. 
 
 127. Wlien the quantities are polynomials which can be 
 readily factored by ins^iection. 
 
 EXAMPLES. 
 
 1. Find the H.C.F. of 
 
 5a^2/ ~~ 15a^y ^^^ 10x\i + 40a^2/ — 210a;2/. 
 By the methods of Chapter VIII., 
 
 b x^y — lb s(?y = b x^y {x — ^) 
 lOx^y + AOx^y — 210xy = 10a;?/(a^ + 4a; — 21) 
 = 10a;?/(aj + 7)(a;-3). 
 
 In this case the common factors are 5, x^ y, and x-~3. 
 Hence, the H.C.F. = 5iC2/(a; — 3), ^ns. 
 
 2. Find the H.C.F. of 
 
 4ic2_4aj4-l, 4a^— 1, and 2ax- — a — 2bx-\-b. 
 
HIGHEST COMMON FACTOR. 67 
 
 4ar'-l=(2a; + l)(2a;-l) 
 2ax-a-2bx + b= {a-b){2x-l) 
 Hence, the H C.F. = 2.t — 1, Aus. 
 
 Find the highest common factors of the following : 
 
 3. 3aoi:^ — 2a^xsLuda^x^-3abx, 
 
 4. a:^ — ^ and a^ + y*. 
 
 6. 9a*- 462 and (3a2- 26)2. 
 
 6. 2a^-2x^sindGa^-Gx. 
 
 7. 3cx + 21c — 3dx — 21dandx^ — 3x—70. 
 
 8. m^n + 2m^n^ + win" and m*/t 4- »^'**' 
 
 9. 3x^-^dx^ — 120xaud3a3^ — 9ax — 30a. 
 
 10. 3icy-4yH-3a:2;-42and 9if2_16. 
 
 11. iy2-a;-42, ar^- 4a; -60, and ic^^ 12a; + 36. 
 
 12. a^-l, a^+l", anda2 4_2a + l. 
 
 13. 437^-123; + 9, 4.-^-9, and 4 ? Aa; — 6 m^w. 
 
 14. a:^ — a;, a.'^ + 9a:2_ xoa;, and a;^ — a;. 
 
 15. a^ - 86^ a' -ab- 2b\ and a^ - 4ab + W. 
 
 16. 2x^-j-2a^-4x,3x*-^Ga^-9a^,a.nd 4ar^-20a;*+16a:\ 
 
 17. 8m3-125, 4m2-25, and 4 771^-20 m -|- 25. 
 
 18. a;* - 16, ar^- a; -6, and (3^^-4)2. 
 
 19. 3 ax"^ — 3ax^, ax^ ~ 9 axr^ -\- S a.r, and 2 oar* — 2 aa;. 
 
 20. a^ -b^^ab-b^-^-ac- be, and a^ - a^b + ab- - bK 
 
 21. 12aa;-3a + 8ca;-2c, 16ar^-l, and 16a;2_32._^i 
 
68 ALGEBRA. 
 
 Case III. 
 
 128. WJien the quantities are polynomials which cannot he 
 readily factored by inspection. 
 
 The rule in Arithmetic for the H.C.F. of two numbers, is 
 
 Divide the greater number by the less; if there is a re- 
 mainder^ divide the divisor by it; and so on; continuing the 
 operation until there is no remaiyider. Then the last divisor 
 is the highest common factor required. 
 
 For example, required the H.C.F. of 169 and 546. 
 
 169)546(3 
 507 
 39)169(4 
 156 
 
 13)39(3 
 39 
 
 Therefore 13 is the H.C.F. required. 
 
 129. We will now prove that a similar rule holds for the 
 H.C.F. of two algebraic quantities. 
 
 Let A and B be two expressions, the degree of A being 
 not lower than that of B. Suppose that B is contained in 
 A p times with a remainder C ; that C is contained in B q 
 times with a remainder D ; and that D is contained in C r 
 times with no remainder. To prove that D is the H.C.F. of 
 A and B. 
 
 The operation of division is shown as follows : 
 
 B)A(p 
 pB 
 
 ~C) B{q 
 
 D) C(r 
 rD 
 
 We will first prove that 2) is a common factor of A and B. 
 
HIGHEST COMMON FACTOR. 69 
 
 From the nature of subtraction, the minuend is equal to 
 the sum of the subtrahend and remainder (Art. 59) . 
 
 Hence, A=pB-\-C (1) 
 
 B = qC+D (2) 
 
 C = rD 
 Substituting the value of C in (2) , we have 
 
 B = qrD + D = D{qr + l) (3) 
 
 Substituting the values of B and O in (1), we have 
 
 A=pD{qr-i-l)-\-rD = D (pqr -\-p + r) (4) 
 
 From (3) and (4) we see that D is a common factor of A 
 and B. 
 
 We will next prove that every common factor of A and B 
 is a factor of D. 
 
 Let K be any common factor of A and B, such that 
 A = mK^ and B = nK. From the operation of division, we 
 see that 
 
 C = A-pB (5) 
 
 D = B-qC (G) 
 
 Substituting the values of A and B in (5), we have 
 
 C = mK — pnK. 
 
 Substituting the values of B and (7 in (6), we have 
 
 D = nK— q (mK—pnK)= K{n — qm -\-pqn) . 
 
 Hence K is a factor of D. 
 
 Therefore, since every common factor of A and B is a 
 factor of D, and since D is itself a common factor of A 
 and -B, it follows that D is the highest common factor of ^1 
 andB. 
 
 130. Hence, to find the H.C.F. of two algebraic exprcs- 
 sions, A and 5, of which the degree of ^4 is not lower than 
 that of jB, 
 
70 ALGEBRA. 
 
 Divide A by B; if there is a remainder, divide the divisor 
 by it; and continue thus to make the remainder the divisor, 
 and the preceding divisor the dividend, until there is no re- 
 mainder. Then the last divisor is the highest common factor 
 required. , 
 
 Note 1. Each division should be continued until the remainder 
 is of a lower degree than the divisor. 
 
 Note 2. It is important to keep the work in the same order of 
 powers of some common letter, as in ordinary division. 
 
 1. Find the H.C.F. of 
 
 ISa^-hlx' + nx + b and6a^-13a;-5. 
 
 18 ar^— 39 ar- 15 a; 
 
 -12 
 
 x^-{-2Sx 
 
 '+ 5 
 
 
 
 
 -12 
 
 ar+2Ga; 
 
 4-10 
 
 
 
 
 
 2x 
 
 - 5)6ar2- 
 
 -13a;- 
 -loa; 
 2a;- 
 2a;- 
 
 -5^ 
 -5 
 
 ,3a; -fl 
 
 Hence, 2a; — 5 is the H.C.F. required. 
 
 Note 3. Either of the given expressions may be divided by any 
 quantity which is not a factor of the other, as such a quantity can 
 evidently form no part of the highest common factor. Similarly, any 
 rejnainder may be divided by a quantity which is not a common factor 
 of the given expressions. 
 
 2. Find the H.C.F. of 
 
 6a;^ — 25a;2-f-14a; and Gaa;^^ 11 aa;— 10a. 
 
 Dividing the first expression by x, and the second by a, 
 
 we have 
 
 6a.-2-25a;-h 14)6 ar^H- 11a;- 10(1 
 6a;^ — 25 a; -f- 14 
 36.T-24 
 
HIGHEST COMMON FACTOR. 71 
 
 Dividing the remainder b}' 12, 
 
 3x-2)6a^-2bx-^U(2x-7 
 
 QX^- 4:X 
 
 -2lx-{-U 
 -21a; +14 
 
 Hence, 3a; — 2 is the H.C.F. required. 
 
 Note 4. If the first terra of a remainder is negative, the sign of 
 each term may be changed. 
 
 3. Find the H.C.F. of 2a^ -Sx-2 and 2a^ - 5x -S, 
 
 2x'-3x-2)2a^-6x-3{l 
 2ar-3x—2 
 -2a;-l 
 
 Changing the sign of each term of this remainder, 
 
 2x-\-l)2x^-Sx-2{x-2 
 2ar+ X 
 -4a;-2 
 -4x-2 
 
 Hence, 2a; + 1 is the H.C.F. required. 
 
 Note 5. If the first term of the dividend or of any remainder is 
 not divisible by the first term of the divisor, it may be made so by 
 multiplying the dividend or remainder by any quantity which is not a 
 factor of the divisor. 
 
 4. Find the H.C.F. of 
 
 2a^-7ar-{-6x-G and 3ar^- 7ar'- 7a; + 3. 
 
 Since Sar* is not divisible by 2ar^, we multiply the second 
 quantity by 2. 
 
 2a^-7cr-\-5x-6)Gx^-Ux^-Ux-{- 6(3 
 6af^- 210.-^+ loa;- 18 
 7a;2-29a;-|-24 
 
72 ALGEBRA. 
 
 Since 2aj^ is not divisible by 7a;^ we multiply each term of 
 the new dividend by 7. 
 
 7x'-20x-j-2^)Ux^-4:da^-{-S5x-42{2x 
 
 9x2-13a;-42 
 
 Multiplying this by 7 to make its first term divisible by 
 
 7a^, 
 
 7a^-2dx+2A)6Sa^- 91a;-294(9 
 
 63x'-2e\x-^2W 
 
 170a;-510 
 
 Dividing by 170, 
 
 x-3)7af-29x-\-24:{7x-8 
 7a;^-21a; 
 
 - 8aj+24 
 
 - 8a; + 24 
 
 Hence, ic— 3 is the H.C.F. required. 
 
 Note 6. li the given quantities liave a common factor which can 
 be seen by inspection, remove it, and find the II.C.F. of the resulting 
 expressions. This result, multiplied by the common factor, will give 
 the H.C.F. of the given quantities. 
 
 5. Find the H.C.F. of 
 
 6a^-ax^-6a^x8iud21s(^-2Qax'-\-oa^x. 
 
 Removing the common factor x^ we find the H.C.F. of 
 6x'-ax-5 a^ and 2\x^ — 2Qax-\- bo?. Multiplying the lat- 
 ter by 2, 
 
 ^;f?^ax-6a?)A2o^-b2ax-^\0a\7 
 4:2 x^ — 7ax — S6a^ 
 — 4:5ax-\- 45 a^ 
 Dividing by — 45 a, 
 
 x — a)6x?— ax — 5a^(6x + 6a 
 
 6ay^ — Gax 
 
 dax — 5a^ 
 6 ax — 5 a^ 
 
HIGHEST COMMON FACTOR. 73 
 
 Multiplying x — a by x, the common factor, we have 
 x{x — a) or ar — ax as the H.C.F. of the given expressions. 
 
 EXAMPLES. 
 131. Find the highest common factors of the following : 
 
 1. a^ + a;-6 and 2a;2-lliK + 14. 
 
 2. 6x^-7x-24cSindl2a^-\-Sx-15. 
 
 3. 2a2_5a4.3 and4a^-2a2-9a + 7. 
 
 4. Ua^-{-nax-28a^SLnd40a^-61ax-{-Ua^ 
 
 5. 8a«-22a2 + 5aand 6a26-23a6 + 206. 
 
 6. ar* — 5mar -f- 4m^a; and x* — 7na^ -^-Smraf — 3m^x. 
 
 7. 5mV + 58 wn^ + SSn^ and lOm^ -j- 31 m- - 20??i - 21. 
 
 8. 2a* + 3a-'a;-9aVand6a«-17a2a; + 14aar^-3a:3. 
 
 9. (B^-S anda.-3-6.x-^ + lla;-6. 
 
 10. 2i«»-3ar-a; + l and 6ar^-ar + 3a; — 2. 
 
 11. 8?;r-227^7i+5«2 and Gm*-2dm^n-}-43mhi^-20mn\ 
 
 12. oar* -f- 2 aa;2 -I- aa;-f 2 a and 3 a."* — 12ar^ — 3ar^ — 6a;. 
 
 13. ax*— oa:''— 2aa^H-2aa; and aa^—Sax*-\-2ax^-{-aa^—ax. 
 
 14. 2a;* — 2a:^ + 4ar^ + 2a; + 6 and3a^ + 6a^-3a;-6. 
 
 15. a* + a^ — 6a'-{-a-{-3 and a"* + 2a3 — Ga^— a-f-2. 
 
 16. a^ — a^ — dx^ -^ 2x^ + Gx and x^ + x^ — x^ — 2x^ — 2x. 
 
 17. 15a-ar''-20a-a;2_(;5^2^_30^2 
 
 and 12ba^-\-20bx^-16bx-l6b. 
 
 18. a* + a»a;H-aV + aa;3_ 4^.4 
 
 and a* + 2a3a;4-3aV-|-4aa;3-10a^. 
 
74 ALGEBRA. 
 
 19. x*-{-oc^-i-i^-\ andx^-\-3x'-}-2x. 
 
 20. i»*-arV-3a^/ + 5iC2/'-62/' 
 
 and 3 iC* — 5 oc^y — a?y^ — 7 .ry^ + 10?/^. 
 
 21. 2a;*— 5ar^+5a;^— 5a;+3 and 2ic*— 7a;^+4ar + ox- — 3. 
 
 22. 3a*-2a26 + 2a262_5a63_2// 
 
 and 6 a* - a^6 + 2 a^ft^ _ 2 aft^ _ 6^^ 
 
 132. To find the H.C.F. of three or more quantities, find 
 the H.C.F. of two of them ; then of this result and the third 
 quantity, and so on. The last divisor will be the H.C.F. of 
 the given quantities. 
 
 EXAMPLES. 
 Find the highest common factors of the following : 
 
 1. 2a^-5a;-42, 4ar^ + 8a;-21, and Gar^ -f 23a; + 7. 
 
 2. 12a^-28aj-5, 14a^- 39a;4-10, and 10fl^-lla;-35. 
 
 3. 6 7?!^ + 7 m/i + 2 71^, 3m^— 7m^7i— 12m7i^ — 47i\ 
 
 and 15 m^ -f 4m?i — 4ri^. 
 
 4. 6a2+13a-5, 6a^4-19a^H- 8a- 5, 
 
 and 3a^ + 2a- + 2a-l. 
 
 5. ar^ + Sa^-Gic-S, a^4-5a^ + 2ic-8, 
 
 anda^-3a^-16a; + 48. 
 
 e. a^-7a;-F6, x« + 3a^-16a;H-12, 
 and 0^-5^2 + 7a; -3. 
 
 7. 2a3-3a2_5a + 6, 2a^ + 3a.=^-8a-12, 
 and2a3-a2-12a-9. 
 
LOWEST COMMON MULTIPLE. 75 
 
 X. LOWEST COMMON MULTIPLE. 
 
 133. A Common Multiple of two or more quantities is a 
 quantity which can be divided by each of them without a 
 remainder. 
 
 Hence, a common multiple of two or more quantities must 
 contain all the prime factors of each of the quantities. 
 
 134. The Lowest Common Multiple of two or more 
 quantities is the product of their different prime factors, 
 each being taken the greatest number of times which it 
 occurs in any one of the quantities. 
 
 It is evident from this definition that the lowest common 
 multiple of two or more quantities is the expression of 
 lowest degree which can be divided by each of them without 
 a remainder. 
 
 Thus, the lowest common multiple of ary, '{fz^ and ar^^^ is 
 
 When quantities are prime to each other, their product is 
 their lowest common multiple. 
 
 135. In determining the lowest common multiple of alge- 
 braic quantities, we may distinguish three cases. 
 
 Case I. 
 
 136. When the quantities are monomials. 
 
 1. Find the L.C.M. of 36a% 60ay, and 84ciB*. 
 36a^x =2.2.3.3.a-^x 
 60aY = 2'2'S'O'aY 
 84ca^ =2'2'3'7-c3^ 
 
 Hence, the L.C.M. = 2.2.3.3.5.7. a^ca^f (Art. 134) 
 = 12Q0 a^cxy, Ans, 
 
76 • algp:bra. 
 
 RULE. 
 
 To the lowest common multiple of the coefficients^ annex all 
 the letters which occur in the given quantities^ giving to each 
 the highest exponent which it has in any of the quantities. 
 
 EXAMPLES. 
 Find the lowest common multiples of the following : 
 2. 6a^6, a"h\ 6. a'h\ 9a^b\ 12 a'b\ 
 
 ^. 10x-y,V2fz. 7. 16x^y,42fz. 
 
 4. 30m2, 27n\ 8. 8c^d^ lOac, 18aU 
 
 5. Qab, lObc, Uca. 9. 24:m^it', ^On-y, ?>2xy\ 
 
 10. Uxy^^, &^a^yz\ 2%a?fz, 
 
 11. 40a26c?3, OOoc^dS bWcd?. 
 
 Case II. 
 
 137. When the quantities are polynomials which can be 
 readily factored by inspection. 
 
 1. Find the L.C.M. ot x^+x-&,a? - 4:X-\- ^, and a^- 9a;. 
 
 ic2_f.aj_6= (a;+3) {x-2) 
 ic2_4a; + 4= {x-2y 
 
 ix?-dx = x{x-{-^) (a; -3) 
 
 Hence, the L.C.M. =^x{x- 2y{x + 3) (aj - 3) , (Art. 134) 
 = a;(a;-2)2(a^-9), Ans. 
 
 EXAMPLES. 
 Find the lowest common multiples of the following : 
 2. x^ — y"^ and xy — ?/^. 
 ^. a?-! anda;2-7aj-8. 
 
LOWEST COMMON :MULTIPLE. 77 
 
 4. 8 a^b-h 8 ab' and 6a -Gb. 
 
 5. m^ — n^ and m^ — ii^. 
 
 6. a- 6 and cr-4:ab-hSb\ 
 
 7. 3C^ — 2xy -\-i/ and a^y — rcy^. 
 
 8. 2a2 + 2a6, 3a6-362, and4a2c-462c. 
 
 9. x'-j-2ax-35a^ a.nda^-2ax-15a\ 
 
 10. ^171 -|- ?i^, m?i — ?j^, and m^ — ?i^. 
 
 11. ax-2a + 6x-26 and a^ — 2a6-362. 
 
 12. aaf + a^a;, ar^ — a^, and a^ — a*. 
 
 13. 8(rt2-?)2), G(a + 6)2, and 12(a-by. 
 
 14. a;»-10a^ + 21a?andaa:2 + 5aa;-24a. 
 16. ar'-l, ar'-2a;+l, anda^ + 2a;+l. 
 
 16. 2-2a^, 4-4a;, 8 + 8a;, and 12 + 12a^. 
 
 17. 3.-2 + 5a; 4- 4, ar' 4- 2a; -8, and ar^ 4- 7.^' 4- 12. 
 
 18. a{x — b) (x— c),b(x — c)(x — a), and c{x — a){x — 6). 
 
 19. (2m-l)^ 4m2-l, and8m^-l. 
 
 20. a^ 4- a, a* - a\ and a« 4- a^- 
 
 21. a2-4a4-3, a2 4-a-12, and a2-rt-20. 
 
 22. 1 -a;S l + 2ar^4-a;*, and 1 -2a;2 4-a;\ 
 
 23. (a 4- 5)2 - c^ and {a - cY - b\ 
 
 24. ax — ay — bx + bij, {x — yY, and 3 a-b — 3 al^, 
 
 25. 9a;3 4-12ar'4-4a;, 18aa;*-12aa;3_^3^ar^^ 
 
 and 27ar''4-8. 
 
 26. y? — y"^ — z^ -\-2yz and v? — y^ -\-'^ -\-2xz. 
 
78 ALGEBRA. 
 
 Case III. 
 
 138. When the quantities are polynomials ichich cannot 
 be readily factored by inspection. 
 
 Let A and B be two expressions ; let F be their highest 
 common factor, and 3f their lowest common multiple. Sup- 
 pose that A = aF and B=bF; then, 
 
 Ay.B = abF^ (1) 
 
 Since a and b can have no common factor, the L.C.M. of 
 ai^ and 6i^is abF\ that is, Jf= abF\ whence, 
 
 FxM=abF- (2) 
 
 From (1) and (2) we have AxB = Fx M (Art. 42, 7). 
 That is, the product of any two qua^itities is equal to the 
 
 product of their highest common factor and lowest commo7i 
 
 multiple. 
 
 Hence, to find the L.C.M. of two quantities. 
 
 Divide their product by their highest common factor ; or, 
 
 Divide one of the quantities by their highest common factor, 
 
 and multiply the quotient by the other quantity. 
 
 139. 1. Find tlie L.C.M. of 
 
 6ar-17a;-f 12 and Ux'-4.x-21. 
 6a^ 
 
 17a; + 12)12ar^- 
 12a;2- 
 
 - 4ic-21(2 
 
 -34a; + 24 
 
 
 
 30a; -45 
 2a;- S)6x'-17x-hl2(Sx- 
 Gx^- 9x 
 
 -4 
 
 
 - 8.T+12 
 
 - 8a;+12 
 
 
 That is, the H.C.F. of the quantities is 2a; — 3. Dividing 
 6x^ — 17a; + 12 by 2a; — 3, the quotient is 3a; — 4. 
 Hence, the L.C.M. = (3a;-4) (12ar^- 4a;- 21) 
 
 = 36a;3-60a;2_47^^84, Ana, 
 
LOWEST COMMON MULTIPLE. 79 
 
 EXAMPLES. 
 Find the lowest common multiples of the following : 
 
 2. 2x'^-\-x — 6 iind4:Xr-Sx-{-S. 
 
 3. 6a;--f 13a;-28 and 12ar-31a; + 20. 
 
 4. Sx' + S0x-\-7 and 12x^-20 X- 8. 
 
 6. 6a^-8x^-30x Sind6ax^-^19ax-\-15a. 
 
 6. a^ - Sab + 7b- and a^ - 9a^b + 23ab^ - 15b\ 
 
 7. 2m2|i — 3m» — 2?i and 2m'' — Gm^ + Gm^ — 8m + 8. 
 
 8. 6aa^ - a^x -12a^ and lOax^- 17 a^x + Sa\ 
 
 9. a» + a2-8a-6 and 2a»-5a2 — 2aH-2. 
 
 10. 2x''-hx'-x-^3 cind2x^ + ox^-x-6. 
 
 11 . a^ - 2 a^ft + 2 ai^ _ h^ and a^ + a-6 - ah' - b\ 
 
 12. x*-{-2oi^ -^2x^ -\-x and aa:^ — 2a^ — a. 
 
 13. 2x'*-lla^4-3a:2_^10a;and 3a;^- Ua^- Gar^ + Sa;. 
 
 14. a;*-af'-8a; + 8 anda;*-8a.-H-9a;-2. 
 
 140. To find the L.C.M. of three or more quantities, find 
 the L.C.M. of two of them; then of this result and the 
 third quantity ; and so on. 
 
 EXAMPLES. 
 Find the lowest common multiples of the following : 
 
 1. ar^— 1, 2a^ — 9a; + 7, and 2a:2_|_ 3^. _ 5 
 
 2. 3a2_2a-l, 6a2-a-l, and9a--3a-2. 
 
 3. 2ar-5a;-h2, ix^ -\- ix-S, and lOa.-^- 7a;+ 1. 
 
 4. 4a^ — 6a:— 18, 4a^ + 4a^ — 3a;, and 6 a;^ -|- 5 a,-^ — 6 a*. 
 6. a^-Ca^+lla-e, a3-a2-Ua4-24, 
 
 anda3 + a2_17a+15. 
 
80 ALGEBRA. 
 
 XI. FRACTIONS. 
 
 141. The expression - signifies a-i-b ; in other words, - 
 b b 
 
 denotes that a units are divided into b equal parts, and that 
 
 one part is taken. 
 
 Or, what is the same thing, - denotes that one unit is 
 
 b 
 
 divided into b equal parts, and that a parts are taken. 
 
 . 142. The expression - is called a Fraction ; a is called the 
 b 
 
 numerator, and b the denominator. 
 
 By Art. 141, the denominator shows into how many parts 
 the unit is divided, and the numerator shows how man}" parts 
 are taken. 
 
 The numerator and denominator are called the terms of the 
 fraction. 
 
 143. An Entire Quantity or Integer is one which has no 
 fractional part ; as 2xy, or a-\-b. 
 
 Every integer may be considered as a fraction vthose de- 
 nominator is unity ; thus, a = — 
 
 144. A Mixed Quantity is one having both entire and 
 fractional parts ; as a + -, or x + 
 
 2 tj-hz 
 
 GENERAL PRINCIPLES. 
 
 145. If the mimerator of a fraction be multiplied, or the 
 denominator divided, by any quantity, the fraction is multi- 
 plied by that quantity. 
 
FRACTIONS. 81 
 
 I. Let - be any fraction. Multiplying its numerator by 
 
 b 
 
 c, we have — • To prove that — is c times -• 
 b b b 
 
 In each of these fractions the unit is divided into b equal 
 parts ; in the first case ac parts are taken, and in the second 
 case a parts. Since c times as many parts are taken in 
 
 — as in -, it follows that 
 b b 
 
 f = cxf (1) 
 
 b b 
 
 II. Let — be any fraction. Dividing its denominator by 
 
 be 
 
 c, we have -• To prove that ^ is c times ^' 
 b b be 
 
 In each of these fractions a parts are taken ; but since in 
 the first case the unit is divided into b equal parts, and in the 
 
 second case into be equal parts, the parts in - will be c times 
 
 o 
 
 as great as in — • Hence, 
 be 
 
 146. If the numerator of a fraction be divided, or the de- 
 nominator multiplied, by any quantity, the fraction is divided 
 by that quantity. 
 
 ac 
 I. Let — be anv fraction. Dividins; its numerator by c, 
 
 we have -• To prove that ^ is — divided by c. 
 b b b -^ 
 
 By Art. 145, (1), ^X^ = f- 
 
 Whence it follows that - = — - -f- c. 
 
 b 
 
82 ALGEBRA. 
 
 II. Let - be any fraction. Multiplying its denominator 
 h 
 
 by c, we have — • To prove that — is - divided by c. 
 -^ he he h 
 
 By Art. 145, (2), c X ^ = ^• 
 
 he b 
 
 Whence it follows that — = - -f- c. 
 
 he b 
 
 147. If the numerator and denominator of a fraetion he 
 hoth multiplied, or hoth divided, hy the same quantity, the 
 value of the fraction is not altered. 
 
 For, by Arts. 145 and 146, multiplying the numerator mul- 
 tiplies the fraction, and multiplying the denominator divides 
 it. Hence, the fraction is both multiplied and divided by 
 the same quantity, and its value is not altered. 
 
 Similarly we may sliow that if both terms are divided by 
 the same quantity, the value of the fraction is not altered. 
 
 TO REDUCE A FRACTION TO ITS LOWEST TERMS. 
 
 148. A fraction is in its lowest terms when its numerator 
 and denominator are prime to each other. 
 
 Case I. 
 
 149. When the numerator and denominator can he readily 
 factored hy inspection. 
 
 Since, dividing both numerator and denominator b}^ the 
 same quantity, or canceling equal factors in each, does not 
 alter the value of the fraction (Art. 147), we have the fol- 
 lowing rule : 
 
 Resolve hoth numerator and denominator into their prime 
 factors, and cancel all which are common to hoth. 
 
FRACTIONS. 83 
 
 1 8 a^h^c 
 
 1. Reduce — to its lowest terms. 
 
 45 cth^x 
 
 Abd^b^x 3-3;5-a-6V 
 
 2qc 
 Dividing both terms by 3 • 3 • a-&^, we have — — , Arts. 
 
 D X 
 
 2j3 27 
 
 2. Reduce — to its lowest terms. 
 
 a^-2x-S 
 
 a^-27 ^ {x-S)(a^ + Sx + d) ^ x'-^Sx-^-d ^^^^ 
 a^_2aj-3~ (a;-3)(a; + l) x-\-l ' 
 
 Note. If all the factors of the numerator be removed by cancella- 
 tion, unity (which is a factor of all algebraic expressions) remains to 
 form a numerator. 
 
 If all the factors of the denominator be removed, the result is an 
 entire quantity ; this being a case of exact division. 
 
 EXAMPLES. 
 
 g x^'fz g 32 mn g 15 mayy^ 
 
 . 2a-b'c » (jbarjfz* -^ 115 c^ar^ 
 
 * baW ' 2i^ ie'fz'' ' 23 c^ar^ * 
 
 5 11^. 8 M^. 11 l^±^. 
 
 ' 32ar^* * 72 a'bc ' 8Sm^xf' 
 
 j2 2a^cd-{-2abcd ^g 6a^6 + 3a-6- 
 
 6arxy-^Qabxy ' 3 d'b^ ■+■ 6 ab^' 
 
 j^g Sx'-Gx^y . jy 4c^-20c4-25 
 
 6a^7f—12xy^ ' 4c^ — 25 c 
 
 14 2x^y-Gx'y jg 7?i^-10m + 16 
 
 x'.-Sx-^lb ' m2-fm-72 * 
 
 -e g^ — 2 a— 15 ^g 9071^ — 4 a 
 
 a^Jul0a + 2l 9&n2- 12671 + 46 
 
84 ALGEBRA. 
 
 a^ -\-ab- 
 
 -6b' 
 
 8a^ + 2/^ 
 
 4ar^_2a^i/4-a^2/' 
 
 ac — ad 
 
 -bc-^bd 
 
 o?- 
 
 -b' 
 
 a^- 
 
 4:a 
 
 a^-9a^ 
 
 -\-Ux 
 
 21f- 
 
 -125 
 
 21. ^'-^-rr 26. 
 
 22 (^^ — ^^^ — bc-^bd n» 
 
 2ar* + ar-j-4a; + 2 
 
 ax"^ 4- 64 ctic 
 a^-(&-f-c)' 
 
 23 aa^-4a 28 (a^-4) (a^-3cg+2) 
 
 ' " " -^ ^ ' ' ' (a^-4a;+4)(a^+a;-2)* 
 
 24. ^^'^-^-^-^ . 29 
 
 (a-by-{G-dy 
 
 9^/2 _ 302/ 4- 25 (« - c)' - (& - dy 
 
 Case II. 
 
 150. TF^ew the numerator and denominator cannot be 
 readily factored by inspection. 
 
 Since the H.C.F. of two quantities is the product of their 
 common prime factors, we have the following rule : 
 
 Divide both 7iumerator arid denominator by their highest 
 common factor. 
 
 EXAMPLES. 
 
 1. Reduce ^ ~ — ^"l^ to its lowest terms. 
 6a^-a- 12 
 
 By the rule of Art. 130, the H.C.F. of 2a'-Da-{-S and 
 6a^ — a — 12 is 2a — 3. Dividing the numerator by 2 a — 3, 
 the quotient is a — 1 ; and dividing the denominator, the 
 quotient is 3 a + 4. Hence, 
 
 2a" — 5a + 3_ a — 1 
 
 6a2-a-12 3a + 4 
 
 Ans, 
 
FRACTIONS. 85 
 
 Reduce the following to their lowest terms : 
 2 a^ — 6^ + 5 y oif-\-a^ — 3x — 2 
 
 3a,-2 -1-40^-7 
 
 10a^-«-2 
 2a^-7a + 6 
 
 g 10a^-«-21 g 
 
 ^ 2m^-5m + 3 g 
 
 12^2-28771 + 15* 
 
 g2-2a;-3 
 a^^2sc^-2x-S 
 
 10. 
 
 a;=^- 
 
 4af' 4-20^4-3 
 
 6aT^- 
 
 -7x^-\-^x — 2 
 
 2a^4-5af*-2a; + 3 
 
 62^ 
 
 -19/ +72/ +12 
 
 ef- 
 
 -251/2 +172/ + 20 
 
 a'- 
 
 .3a2 + a + 2 
 
 2a^- 
 
 -aa2-a-2 
 
 a^- 
 
 .^x'y-^-Axf-f 
 
 g 12m2 + 16mn — 3^2 -- 
 
 10 m^ + m?i — 21 ?i2 ' ^ — 2x^y -\- 4:Xif — '6^f 
 
 151. Since a fraction represents the quotient of its 
 numerator divided by its denominator, it is positive when 
 its terms have the same sign, and negative when they have 
 different signs. 
 
 Thus, if- = a;, 
 h 
 
 then ^— = X, and ^^— = = — x. 
 
 — b b —b 
 
 152. It follows from Art. 151 that the fraction - can be 
 
 b 
 
 written in any one of the forms 
 
 — a — a a 
 
 or 
 
 -b b -b 
 
 That is, if the signs of both numerator and denominator 
 are changed^ the value of the fraction is not altered. But if 
 the sign of either one is chayiged^ the sign before the fraction 
 is changed. 
 
86 ALGEBRA. 
 
 153. If either numerator or denominator is a polynomial, 
 care must be taken, on changing its sign, to change the sign 
 of each of its terms. 
 
 Thus, the fraction ~ , by changing the signs of both 
 
 numerator and denominator, can be written in the form 
 
 — (a — h) b — a,.. pr-x 
 
 — ^^ -^ or (Art. 67). 
 
 -{c-d) d-G ^ ^ 
 
 154. It follows from Art. 151 that the fraction — can be 
 
 cd 
 written in any one of the forms 
 
 (-«)6 ( -«)(-&) (-«)(-&) etc- 
 
 c(-d)' cd ' (-c)(-d)' 
 
 or (-«)» «fe i-o)(-b) „,„ 
 
 cd {—c)d c(— a) 
 
 From which it appears that 
 
 If the terms of a fraction are composed of factors^ the 
 signs of any even number of factors may be changed ivithout 
 altering the value of the fra-ction. But if the signs of ajiy 
 odd number of factors are changed^ the sign before the frac- 
 tion is changed. 
 
 Thus, the fraction ^^^^ can be written in any 
 
 (x — y) (x — z) 
 one of the forms 
 
 a — b b^a b — a . 
 
 (y -x){z- x) {y -x){x- z) (y -x){z- x) 
 
 TO REDUCE A FRACTION TO AN ENTIRE OR MIXED 
 QUANTITY. 
 
 155. Since a fraction is an expression of division, we 
 have the following rule : 
 
 Divide the numerator by the denominator. 
 
FRACTIONS. 87 
 
 1. Reduce — ^^— to a mixed quantity. 
 
 O X 
 
 Dividing each term of the numerator by the denominator, 
 
 3a; 3x Sx Sx 3x' 
 
 2. Reduce ' to a mixed quantity. 
 
 4ar — 3 
 
 ^a^-S)8a^-12x^-9x-{-10{2x-S 
 
 Sx^ -6x 
 
 — V2xr — Sx 
 -Ux" +9 
 
 -3a; + l 
 
 A remainder whose first term will not contain the first term 
 of the divisor, may be written over the divisor in the form 
 of a fraction, and added to the quotient. Thus, the result is 
 
 — 3a; 4-1 
 
 2x-3 + 
 
 4a;- -3 
 
 Or, since the sign of each term of the numerator may be 
 changed, if at the same time the sign before the fraction is 
 changed (Art. 152), we have 
 
 8a;»-12ar'-9a;+10 .-, „ 3a;-l . 
 
 -L — =2a; — 3 ; , Ans. 
 
 4a;2_3 4ar-3 
 
 EXAMPLES. 
 Reduce the following to mixed quantities : 
 
 3 oa;g-10a; + 4 g 2a;^-41 
 
 6x ' x — 3 
 
 ^ Ga^-Sxr-\-dx-7 ^ a^-gi^q^^ 
 
 3x ' ar-\-a—l 
 
 ^ 3?-\-2f g 12ar^-8a; + 7 
 
 x-\-y ' 4a;— 1 
 
88 ALGEBRA. 
 
 g g^ -f h\ j2 flr^ + 2a-^ + 3a; + 4 
 
 a-\-b x^ -\-x-\-l 
 
 2m — Sn ^ + ^ 
 
 11 2a^-a^-9«^+14 ^^ 6ar-13a^+ 6a;- C 
 
 2a^-a-3 ' ' 3a;2-2a;+l 
 
 TO REDUCE A MIXED QUANTITY TO A FRACTIONAL 
 
 FORM. 
 
 156. The operation being the converse of that of Art. 155, 
 we have the following rule : 
 
 Multiply the integral part by the denominator; odd the 
 numerator to the product when the sign before the fraction is 
 -\- , and subtract it when the sign is — ; and write the result 
 over the denominator. 
 
 1. Reduce — — 1- ^ — 2 to a fractional form. 
 
 2x-3 
 
 By the rule, 
 
 ^ — 5 , ^ ^_ x — 5+(x—2)(2x — S) 
 
 2x-3 ~ 2x — 3 
 
 ^ x-5-\-2x^-7x-{-6 
 2a;-3 
 
 2x-3 
 
 2. Reduce a-^b to a fractional form. 
 
 a — b 
 
 ^ ^ a'-W-h ^ {a + b){a-b)-{a'-b-- 5) 
 a —.b a — b 
 
 a—b a—b 
 
 Note. If the numerator is a polynomial, it will be found convenient 
 to enclose it in a parenthesis, when the sign before the fraction is — . 
 
FRACTIONS. 89 
 
 EXAMPLES. 
 Reduce the following to fractional forms : 
 
 3. X+1+-1+1. 11. i±l'-l. ' 
 
 X x — y 
 
 4. X+1-4-. 12. ™_„ + ?!^l±iil 
 
 6. 70.-3-^^"-^^. 14. ^_3a.-M3^^. 
 
 8 a;-2 
 
 7. 1 ^ — 15. "i— -^---2-(w-?i). 
 
 8. a + 6-^-±^. 16. l+2x + 4.x'^^.±^. 
 
 a-\-h 2 a; — 1 
 
 9. _?— +3.r-2. 17. x-2y t^zM_. 
 
 10. a=-6= + ^^Mii±6). 18. a^_2*+3-5!±il^=^. 
 
 TO REDUCE FRACTIONS TO THEIR LOWEST COMMON 
 DENOMINATOR. 
 
 157. 1. Reduce — ^, — ^, and — ^ to equivalent frac- 
 3 a^h 2 a¥ 4 a^h 
 
 tions having the lowest common denominator. 
 
 The lowest common denominator is the lowest common 
 multiple of 3a-&, 2a6^ and 4a'^6, which is \2a^b^. 
 
 By Art. 147, both terms of a fraction may be multiplied 
 by the same quantity without altering its value. Hence, 
 
90 ALGEBRA. 
 
 Multiplying both terms of -^ by 4a&, we have ?^S^. 
 
 Multiplying both terms of ^^ by Go", we have 1^^^. 
 
 2ab^ 12 d-^b^ 
 
 Multiplyinsr both terms of — '■— by 3&, we have — ^• 
 
 Therefore the required fractions are 
 
 20 abed IS armx , dbni/ . 
 
 r-^, r, ana ^, Avs, 
 
 12 a^b" 12 d'b^ I2a'b^ 
 
 It will be observed that the terms of each fraction are 
 multiplied by a quantity which is obtained by dividing the 
 lowest common denominator by its own denominator. Hence 
 the following rule : 
 
 Find the lowest common multiple of the given denominators. 
 Divide this by each denominator separately^ multiply the cor- 
 responding numerators by the quotients^ and write the results 
 over the common denominator. 
 
 Note. Before applying the rule, each fraction should be in its 
 lowest terms. 
 
 EXAMPLES. 
 
 Reduce the following to equivalent fractions having the 
 lowest common denominator : 
 
 o 3a6 2ac ^ 56c 
 
 2. , , and 
 
 14 21 6 
 
 3 — — and — • 
 
 a^'X^ aa^^ d-x 
 
 - 4c-l , Sb-2 
 
 4. — ■ and -—• 
 
 8ab^ 12 a^c 
 
 15 6az 3bx , ley — m 
 
 Qoe'y 8yh lOxz^ 
 
FRACTIONS. 91 
 
 and 
 
 d^ -\-a — ij a^ — 4: 
 
 7. — i— and ^ 
 
 a^-l ar'-l 
 
 w? 1 mn^ 
 
 8IIV 1 I, 
 
 . m, -, and -— 
 
 mn — n' m? — ,t 
 
 9. , , and — -• 
 
 a-b a^b d' + b^ 
 
 10. ^^, _^^„ and ^-^ . 
 
 «« ab , 711 — n 
 
 11. : — and 
 
 am — bm -\-an — bn 2 a^ — 2 a6 
 
 12. -^^±i_, -^^±1-, and - + 2 
 
 ADDITION AND SUBTRACTION OF FRACTIONS. 
 
 150. It follows from the definition of Art. 141 that 
 
 a b a-^b ,a b a — b 
 
 - + - = — — , and = . 
 
 c c c c c c 
 
 Hence the following 
 
 RULE. 
 
 To add fractions, reduce tJiem, if necessary, to equivalent 
 fractions having the lowest common denominator. Add the 
 numerators of the resulting fractions, and write the sum over 
 the common denominator. 
 
 To subtract one fraction from another, reduce them to 
 equivalent fractions having the lowest common denominator. 
 Subtract the numerator of the subtrahend from thai of the 
 minuend, and write the result over the common denominator. 
 
 Note. The final result should be reduced to its simplest form. 
 
92 ALGEBRA. 
 
 1. Required the siira of ^^^ and ~ "^ — 
 
 4ac 6 b'G 
 
 The lowest common denominator is 12 ab^c. Multiplying 
 the terms of the first fraction by 36^, and of the second by 
 2 a, we have 
 
 4a-l 3-6b''^ 12ab^-Sb^ 6a-10ab^ 
 4:ac 6b'c 12 ab-c 12 ab''c 
 
 12ab^-3b''-i-6a-10ab^ 
 
 12 abh 
 2ab''-^b' + Qa 
 
 12 ab'c 
 
 , Ans. 
 
 2. Subtract i^^ from ^^^-^II^. 
 2x 3a 
 
 The L.C.D. is 6 ax. Hence, 
 
 6a — 2 _ ix — l __ 12aa; — 4a? _ 12aa;— 3a 
 3 a 2x 6 ax 6ax 
 
 _ 12ax — 4:X— (12ax — 3a) 
 
 6 ax 
 _ 12acc — 4a;— 12aa; + 3a 
 6 ax 
 Sa — 4x 
 
 Qax 
 
 -, Ans. 
 
 Note. If a fraction, whose numerator is a polynomial, is preceded 
 by a — sign, care must be taken to change the sign of each term in the 
 numerator before combining it with the others. It is convenient in 
 such a case to enclose the numerator in a parenthesis, as shown in 
 Ex.2. 
 
 EXAMPLES. 
 Simplify the following : 
 
 „ 2a;-5 3a; + H 4 _3 1_, 
 
 12 18 ' ' 5ab^ 2a^h 
 
FRACTIONS. 93 
 
 c '2a-\-3 3a4- 5 » 6 — 4a a4-56 
 
 6 8 * * 24a 306 
 
 A ^ — 2 2 — 3 mn^ « a— 6 . 2a-f& 3a— 6 
 
 2mn 3mV * * 4 6 8 " 
 
 9. 
 
 g^+l 6a^ + l 6-2 
 3a2 12a3 66 
 
 16. Simplify 
 
 10 2a;— 1 2.T4-3 6a;+l 
 12 "^ 15 20 
 
 -. mH-2 m + 2 m + 3 
 7 14 21 
 
 12 2 2a;-l Sar'+l 
 '3 6a; da^ ' 
 
 13 ^-2 , 3 a; 4-1 _ 6a;-5 _ 3^ 
 2 3 4 5* 
 
 14 3a + l 26-1 4c-l 6cZ + l , 
 12a 86 16c 24d 
 
 1 . 1 
 
 x-\-a^ x — x^ 
 
 The L.C.M. of a;4-a;^ anda; — a;^ is a;(l 4-a;) (1 — a;), or 
 a;(l— ar^). Multiplying the terms of the first fraction by 
 1 — a;, and of the second by 1 + a;, we have 
 
 1 1 _ 1— a; 14-a; 
 
 x-\-x^ x — a^ x{l—xr) x(l — a^) 
 
 __ 1 — a;4-l+a; 
 a;(l-ar') 
 
 = —- -, Ana, 
 
 x{l —x-) 
 
04 ALGEBRA. 
 
 le. Simplify ^^±*-^^-J^. 
 a — h a + h o? — Ir 
 
 The L.C.D. is o? - h\ Hence, 
 
 g + ^ g — 6 4 a5 
 a — h a 4- 6 o? — V^ 
 
 a2_62 ^2_^2 ^2_^2 
 
 ^ qg + 2a?; + 5^- gg + 2a& - 62_ 4^5 ^ ^ ^^^ 
 
 d^ — b^ ' ' 
 
 Simplify the following 
 
 17 
 
 1 
 
 1 1 
 
 
 \+x 
 
 ' 1-0. 
 
 18. 
 
 1 
 aj + 2 
 
 n4-. 
 
 19. 
 
 1 
 
 1 
 
 x + 1 
 
 aj + 8 
 
 30. 
 
 a 
 
 b 
 
 a-b 
 
 a-\-b 
 
 21, 
 
 a + b 
 
 a-b 
 
 
 a-b 
 
 a-\-b 
 
 22. 
 
 x-\-y 
 
 2xy + x^ 
 
 
 y 
 
 y(^ + y) 
 
 23. 
 
 l^x 
 
 l-x 
 
 24. ^ + ^ I 2m 
 
 (m — n)^ m^ — n^ 
 
 25. 1 ^ 
 
 a^ — 4a + 4 a^-{-a—6 
 
 26. 
 
 a. 3 a? 2 a??/ 
 
 x — y x-\-y a^ — y^ 
 
 27 <^ [ ^ I 2or6 
 
 a + 6 a — 6 a^ — b^ 
 
 28. -^?: 
 
 mn — 71^ ??i — n 11 
 
 29. ^±^4-^:11+2. 
 x—y x-\-y 
 
 30 ^ 3 2a?-3 
 
 l-« 1 +a; ' x 2x-l 4a^-l" 
 
FRACTIONS. 95 
 
 32 1 , __1 2a «2 1 X _ x^—\x 
 
 ' a^h a-h a^ + Z^* ' \-x {\-xy (1-a;)^' 
 
 33. ^ 1 ^^^ 
 
 a6 — cd ab-\- cd a-b^ — c^d^ 
 
 34 ^-3 a? + l . a; + 13 
 
 x-2 x-\-5 a^ + 3ic-10* 
 
 gg g; — g a; — 6 (a — 6)- 
 
 a; — 6 ic — a (x — a){x — b) 
 
 1 ^ X 
 
 a;(a;H-l) x{x—l) x^-l 
 
 37 « + ^ , 6 + c , c + g 
 
 (6-c) (c-g) (c-g) (g-6) (a-6) (ft-c)' 
 
 38. -^ ._^4._J_. 
 
 a;-l ar*-l ar*-l 
 
 «Q 2a; — 6 a;+2 a; + l 
 
 ar^-f3ic+2 x^-2x-S x^-x-6 
 
 (a;+2j) (2/+^) (a:-f?/)(a;+2:) (x+y){y-^z)' 
 
 In certain cases, the principles of Arts. 152 and 154 en- 
 able us to change the form of a fraction to one which is more 
 convenient for the purposes of addition and subtraction. 
 
 41. Simplify -A_ + ^^ + «. 
 ^ ^ a-b b'-a^ 
 
 Changing the signs of the terms in the denominator of the 
 second fraction, and at the same time changing the sign 
 before the fraction (Art. 152), we have 
 
 _3 25 + g 
 
 a-b a'-b'' 
 
96 ALGEBRA. 
 
 The L.C.D. is now a^ — W. Hence 
 
 _3 2& + a ^ 3(a4-6) 2b + a 
 
 a-b a?-h^ d'-J)' a?-W 
 
 ^{a + h) — {1h + a) 
 
 d' -b' a'- b' 
 
 \. Simplify 
 
 1 1 1 
 
 {x-y){x-z) {y-x)(y-z) {z-x){z-y) 
 
 By Art. 154, we may change the sign of the factor y — x 
 in the second denominator, at the same time changing the 
 sign before the fraction ; and we may change the signs of 
 both factors of the third denominator. The expression then 
 becomes 
 
 1 ^ 1 
 
 (x-y){x-z) {x-y){y-z) (x-z){y-z) . 
 
 The L.C.D. is now {x — y){x — z){y — z). Hence the 
 result 
 
 _ {y — z)-\-{x — z) — {x — y) _ y — z-j-x — z — x-\-y 
 
 ^ 2y-2z ^ ^(y-z) 
 
 ~ (x-y){x-z)(y-z) {x-y){x-z){y -z) 
 
 2 
 
 {x-y){x-z) 
 
 Ans. 
 
 Simplify the following : 
 
 43. -A- + -^-. 45. —1-, + ^. 
 
 d^ — ab b^ — ab 3 a; — xr x- — 9 
 
 44 5a + l 3a-l ^g __J 1_ 
 
 ' 3a — 3 2 — 2a * w" 
 

 FRACTIONS. 
 
 
 
 47. 
 
 ^ 1 ^ 
 
 
 
 (a_2)(a; + 2) (2-a)(a; + a) 
 
 
 48. 
 
 a . a 2a^ 
 
 
 
 a + 6 ' 6 — a ' or — Ir 
 
 
 49. 
 
 X X a? 
 
 
 
 l_|_a; l-a; x^-\ 
 
 
 50. 
 
 2_a; a;-3 a^-5a; + 6 
 
 
 
 51. 
 
 ^ 1 ^ 
 
 1 
 
 
 (a_Z>)(6_c) {h-a){a-c) (c- 
 
 -a)(c- 
 
 -6) 
 
 52. 
 
 2 3 
 
 1 
 
 
 97 
 
 (a;-2)(a;--3) (3 - a;) (4 - a;) {x-4:){2-x) 
 
 MULTIPLICATION OF FRACTIONS. 
 
 159. Required the product of - and -• 
 
 a 
 
 2 5 5 
 
 In Arithmetic, - times - signifies two-thirds of -• 
 
 a c c 
 
 Similarly, in Algebra, - x - signifies a 5ths of — That 
 b d d 
 
 is, tee divide -by b, and multiply the result by a. 
 d 
 
 By Art. 146, II., ^-5-6= A. 
 
 d bd 
 
 By Art. 145, I., h^<^ = ~ 
 
 bd bd 
 
 TT a ,, c ac 
 
98 ALGEBRA. 
 
 We have therefore the following rule for the multiplication 
 of fractions : 
 
 Multiply the numerators together for the yiumerator of the 
 product, and the denominators for its denominator. 
 
 Mixed quantities should be reduced to a fractional form 
 before applying the rule. 
 
 Common factors in the numerators and denominators 
 should be canceled before performing the multiplication. 
 
 EXAMPLES. 
 1. Multiply '-^ by m. 
 
 9 oar 4 aV 
 
 96a^ 4 ay' d-^-a^xy^ 
 5 b^x . 
 
 = , A71S. 
 
 a^—2x a^ — 9 . xr + x 
 
 2. Multiply together , ~^^' , V=^' and ^^ + ^ 
 ^ -^ "" x'-2x-3 x'-x x'^x-e 
 
 x'^~2x xP — 9 m? -\-x 
 X ~~z X 
 
 9? — 2x — 3 a?-x x?-\-x-& 
 
 x{x-2) (x + 3){x-?,) ^^^ x{x + l) 
 
 (a;-3)(if + l) x{x-l) {x + 3){x-2) 
 
 X 
 
 , Ans. 
 
 Multiply the following : 
 
 4. 3a6^ and -^. 6. «^. l^^l and ^, 
 
 5a/ 3«6a^ 9?/^' 162^ lOo^ 
 
FRACTIONS. 99 
 
 7. ^J^, ^, and «i^. 12. 2!^2«6 + 6^ ^^^, _^. 
 4cd 2M y^^; rt + 6 ax—bx 
 
 9. ^^^- and -i^-.- 14. ^^^!±^ and ^IIl-^- 
 
 -rt a^-16 , a^-25 .« . , 4 5 _^ Sx 
 10. , . , and — — — • 15. 1 H — and ^- -• 
 
 a^ + aic ar— 4a; a;ar ar + a; — 2 
 
 a^ + 2 ad a^ — ah I —y x-\-xr l—x 
 
 x^ — 4:xy — 21y^' x^ — 4:if 
 
 18. A-^y and ^-'^-''^ 
 (x — yf — z' xij + 2y 
 
 or — xy -\-y- or-i- xy -}- 1/- ;c — y 
 
 20. ^^-(^-^>^ and ^l^lHhc)!. 
 (a-{-cy-b' {a-cY-b- 
 
 DIVISION OF FRACTIONS. 
 160. Required the quotient of - divided by ^• 
 
 By Art. 85, we are to find a quantity which, when multi- 
 plied by -, will produce -• 
 
 That quantity is evidently -— ; hence, 
 
 be 
 
 g . c _ad 
 b ' d be 
 
100 ALGEBRA. 
 
 We observe that the quotient is obtained by multiplying 
 
 the dividend, -, by -, which is the divisor inverted. We 
 h c 
 
 have then the following rule for the division of fractions : 
 Invert the divisor, and proceed as in multiplication. 
 
 Mixed quantities should be reduced to a fractional form 
 before applying the rule. 
 
 If the divisor is an integer, it may be written in a frac- 
 tional form, as explained in Art. 143. 
 
 EXAMPLES. 
 
 - ^. ., Garb . 9a'b' 
 
 1. Divide — -— by — -^t-t- 
 
 By the rule, 
 
 6a'b . da^b^ ^ 6a'b lOa-y ^ 4y ^^^^ 
 5ary * lOary 5xY 9a'^^ 36^0;' 
 
 2. Divide by — ■ 
 
 lo 5 
 
 a:2_9 a^-f-2a;-3 ^ (a; + 3)(a;-3) 5 
 
 15 * 5 15 (a;-j-3)(^-l) 
 
 3(ic-l) 
 
 , Ans 
 
 Divide the following 
 
 3. — TT-. by lAab*. 4. — by ^ , _ • 
 
 5mV ^ 2ony^ bn^ 
 
 5 __L___b 1 
 
 6. l_iby^ + ^. 
 4: a^ ^ 12 3 
 
FRACTIONS. .101 
 
 „ ar^ — 25 a; , a;^ — oa; 
 • x' + x-Q, ^ gi^-x-Vl 
 
 o ab — b^ 1 b' 
 
 m^ — 2 m/i + ?r t??. — n 
 
 or — y^ x — y 
 
 a2-2a6-362 ^ • a-36 
 
 13. A-A + Abv-^ —• 
 
 ^f xy 2x' " 3/ 2a.'2 
 
 COMPLEX FRACTIONS. 
 
 161. A Complex Fraction is one having a fraction in its 
 numerator or denominator, or both. 
 
 It may be regarded as a case in division ; its numerator 
 answering to the dividend, and its denominator to the divisor. 
 
 EXAMPLES. 
 
 1 Reduce ^^ its simplest form. 
 
 d 
 
 h — - bd — (^ ' bd — c bd — c 
 
102 ALGEBRA. 
 
 It is often advantageous to simplify a complex fraction by 
 multiplying both numerator and denominator by the lowest 
 common multiple of their denominators. 
 
 2. Reduce -— - to its simplest fonn. 
 
 b ■ a 
 
 a — b a-{-b 
 
 The L.C.M. of a + b and a — 6 is (a + b)(a — b). Multi- 
 plying each of the component fractions by (a + 5)(a — ^), 
 we have 
 
 a{a-\-b) — a(a — b) a^ -\- ab — a' -^ ab _ 2ab . 
 b{a-\-b)-{-a{a-b) ^ ab -\- b' -\- a' - ab ~ a' -f- b^' 
 
 3. Reduce — to its simplest form. 
 
 X 
 
 1 1 x+l ^ + \^,,, 
 
 ^ ■ 1 , ■ a; x-j-l-\-x 2x-\-l 
 
 1+1 -+i 
 
 Reduce the following to their simplest forms : 
 
 ^-- ^ ^^ + i t±^.Ax 
 
 4. ^. 6. 1 8 
 
 x i^ 
 
 11 ..1 
 
 5. ^_^. 7. 
 
 a-2 + 
 
 a 
 
 a_ 6 ^ _ 1 
 
 b a a 
 
 
 1 2 
 
 y X 
 
 9 
 
 -^+¥ 
 
 
 » + 3-^ 
 
FRACTIONS. 103 
 
 m2 ^ J 20^2 
 
 ^Q n^ m. ^fj 1 — ar 
 
 m m — n i ^ , 4a^ 
 
 1 —Qr-\- 
 
 w mn l — x^ 
 
 a,_l__i?_ o_±b_^a-b 
 
 a;_5+^^ a + & a-b 
 
 x-^3 c —d c -{-d 
 
 12. 2 U— . 19. ^ 
 
 3 + — i- «+ ^ 
 
 ^-2 1 + ^^^ 
 
 2 3-x 
 
 a_l^ x-\-2y X 
 
 13. J_^. 20. " + ^ ^" 
 
 a J 6 x-\-2y x 
 
 ha y x-^y 
 
 '^-i + ^ .-3a+ *«' 
 
 14. ?! £.. 21. TIT—' 
 
 y X a^x 
 
 X 3?/ 2a2 
 
 1 1 a2_j_52 ^2_J2 
 
 ' 1.1' '*'^* a + 6 a-6 ' 
 
 1— ic \ +x a—b a-i-b 
 
 ^ _ 2 ?> — 2 c m — n 771' -{- n^ 
 
 16 _ii±lzif. 23 '"'^" m--?r 
 
 1 _i_ 2c m^ m-?i + 7i^ 
 
 a — h — c m — n {m — ny 
 
104 ALGEBRA. 
 
 MISCELLANEOUS EXAMPLES. 
 162. Reduce the following to their simplest forms : 
 
 1 p ■ 26 a + bx + cx^ g 10 a^-\- SO ab -\- 20 b^ 
 
 X a? ba? -\-\Od-b 
 
 ^ m3 + 4m^-5m , ^^ f^ + l+iV^.l + l 
 
 3m«-75m V ^7\ ^, 
 
 g x'jl + xy-x^l + xy g l-aa; + a(a;+a) 
 
 (l+a;)« * * {l-axY-{-{x-iray 
 
 o? — b^ G? — ab 
 
 '■ (^^+sKf4: 
 
 g 1 +2a^ 2 + a^ . Iq b{b — ax) -\-a(a-^bx) 
 
 ' 2 + 2ar^ 2 + 2a;* * (6 - aa;)^H- (a + 6a.")^ 
 
 ^Il ax _ b ax (3 b — ax) 
 
 ax-\-b ax — b a^af — W 
 
 ^2 Zx-Sx" ^^ 10a; + 10a^ 
 
 13. 
 
 2 + 4a; + 2i^ d-lSx-^-dx" 
 
 6n'-48 rv' 
 
 971^ + 18/1* + 36 ^/ 
 
 14. a^-2v^-^y'-^^'^ + ^-^'. 
 ^ «-3^ 
 
 le a + 6 g — 6 4b^ 
 a — b a-\-b a^ — b^ 
 
 16. L^^x + ^ + ^(x-^ 
 \ X xrj\ x_ 
 
 .„ 2x — \ 2x-\-\ 
 
 2oc^-2x-\-l 2a:2^2a; + l 
 
FRACTIONS. 105 
 
 18. ^ + Y-! L_Y 
 
 14-^ 2\^a; — a x-\-aj 
 
 ^-2 
 
 19. ^-y ^' + .y . 20. a:^-9^ + 26^a;-24 
 
 _j^^ — H — 
 
 r 
 
 ar*-12.x-^-(-47x-G0 
 
 21. a2-3a6-26--- ^'^^^ + ^^^ . 
 a — 36 
 
 22. ^^ + y 1 ^ ^ ■ 
 
 a^ + 2/^ + 2^ 
 
 24 _i ^- ^(^-^) . 
 
 • a;-2 (.r-2)=-' (a;-2)« 
 
 25. 
 
 a^-a^d-a^t^ + aft* 
 
 og (4a; + .v)^-(a^-2y)^ 
 {Zx-\yy-{:2x^-Zyy 
 
 27 ^_4,J_ + _i (« + ^> + c)' 
 
 a + 6 64-c c + a (a + 6)(6 + c)(c + a)* 
 
 28 2(1 -3.r) l-2a; 2 
 
 (l+a:)(14-9a;) (i -(-a;) (1 +4x) 1 +4a; 
 
 .T-1 ic+l a;»H-l x^-X 
 l^a-6 a^-^V * \a-\-h a« + 6V 
 
106 ALGEBRAr 
 
 XII. SIMPLE EQUATIONS. 
 
 163. An Equation is a statement of the equality of two 
 expressions. 
 
 Tlie First Member of an equation is the expression on the 
 left of the sign of equahty, and the Second Member is the 
 expression on the right of that sign. 
 
 Thus, in the equation 2ic — 3 = 3x — 5, the first member is 
 2a; — 3, and the second is 3a; — 5. 
 
 The sides of an equation are its two members. 
 
 164. A Numerical Equation is one in which all the known 
 quantities are represented by numbers ; as 
 
 2a;-3 = 3a;-5. 
 
 165. A Literal Equation is one in which some or all the 
 known quantities are represented by letters ; as 
 
 2x-\-^a = bx — 4:. 
 
 166. An Identical Equation is one whose two members are 
 equal, whatever values are given to the letters involved ; as 
 
 x^ r- a~ = {x-^ a) (x — a). 
 
 167. The Degree of an equation, in which there is but one 
 unknown quantity, is denoted by the highest power of the 
 unknown quantity in the equation. Thus, 
 
 ^— — ^— I j^j.g equations of the Jirst degree. 
 and a^x = bc —d) 
 
 3 a;^ — 2a; = 65 is an equation of the second degree, etc. 
 
 168. A Simple Equation is an equation of the first degree. 
 
SIMPi^E EQUATIONS. 107 
 
 169. The Root of an equation containing but one unknown 
 quantit}', is the value of the unknown quantity ; or, it is the 
 value which, when put in place of the unknown quantity, 
 makes the equation identical. 
 
 Thus, the equation oic— 7 = 3a;+l, wlien4 is put in place 
 of a, becomes 20 — 7=12-f-l, which is identical. Hence 
 the root of the equation, or the value of x, is 4. 
 
 Note. A simple equation has but one root ; but it will be seen here- 
 after that an equation may have two or more roots. 
 
 170. The solution of an equation is the process of finding 
 its roots. 
 
 A root is verified, or the equation satisfied, when, on sub- 
 stituting the value of the root in place of its symbol, the 
 equation becomes identical. 
 
 171. The operations required in the solution of an equa- 
 tion are based upon the following general principle, which is 
 derived from the axioms of Art. 42 : 
 
 If the same operations be performed upon equal quantities, 
 the results will be equal. 
 
 Hence, 
 
 Both members of an equation may be increased, diminished, 
 multiplied, or divided by the same quantity, without destroying 
 the equality. 
 
 TRANSPOSITION. 
 
 172. Any term may be transposed from one side of an 
 equation to the other by changing its sign. 
 
 For, consider the equation x-\-a = b. 
 
 Subtracting a from both members (Art. 171), we have 
 
 x-\-a — a = b — a', 
 or, by Art. 26, x = b — a. 
 
108 ALGEBRA. 
 
 where + a has been transposed to the second member by 
 changing its sign. 
 
 Again, consider the equation x — a^h. 
 
 Adding a to both members (Art. 171), we have 
 
 x — a-\-a = h + a\ 
 
 or, x = h-\- a. 
 
 where —a has been transposed to the second member by 
 changing its sign. 
 
 ITote. If the same term appear in both members of an equation 
 affected with the same sign, it may be suppressed. 
 
 173. The signs of all the terms of an equation may be 
 changed without destroying the equality. 
 
 For, consider the equation a — x=l) — c. 
 
 Transposing each term (Art. 172), we have 
 
 c — h = x — a\ 
 
 or, x — a=c — h^ 
 
 which is the same as the original equation with every sign 
 changed. 
 
 SOLUTION OF SIMPLE EQUATIONS. 
 
 174. 1 . Solve the equation bx — 1 — 2>x-\-\. 
 
 Transposing the unknown quantities to the first member, 
 and the known quantities to the second, we have 
 
 bx-^x = l -^1. 
 Uniting the similar terms, 2ic = 8. 
 Dividing both members by 2 (Art. 171), 
 
 ic = 4, Ans. 
 
SmPLE EQUATIONS. 109 
 
 Note. The result may be verified by substituting the value of .r in 
 the given equation, as shown in Art. 169. 
 
 We have then the following rule for the solution of a 
 simple equation containing but one unknown quantity : 
 
 Transpose the unknown terms to the first member^ and the 
 known terms to the second. 
 
 Unite the similar temis^ and divide both members by the 
 coefficient of the unknown quantity. 
 
 EXAMPLES. 
 
 2. Solve the equation 14 — 5 ;r =19 + 3 a;. 
 
 Transposing, — ox — 3a;= 19 — 14. 
 
 Uniting terms, —8 a; = 5. 
 
 Dividing b}- —8, a; = — -, Ans. 
 
 8 
 
 Note. To verify this result, put x = — - in the given equation. Then, 
 
 o 
 
 U-5(-|)=19 + 3(-|) 
 
 Or, 14+^=19-1^ 
 
 8 8 
 
 Or, 137 ^ 137 ^^^^ . identical. 
 
 8 8 
 
 Solve the following equations : 
 
 3. 8« = 5.x'-f-42. 9. 5a;+14=17-3;r'. 
 
 4. 7a;4-5 = -30. 10. 3a;-31 = llic- 16. 
 6. 7.'cH-5 = x-f 23. 11. 18 - 7a; == ISa;- 7. 
 
 6. 9a: + 7 = 3x-ll. 12. 27 + 10a; = 13a; + 23. 
 
 7. 3a;-8 = 5a; + 8. 13. 19a; - 11 = 15 + 6a;. 
 
 8. 5-6a;=l-4a;. 14. 32a;- 15 = 7 + 65a;. 
 
110 ALGEBRA. 
 
 15. 13a;-81 = 5a;-31a;-159. 
 
 16. 12a;-20a;4-13 = 9a;-259. 
 
 17. Solve the equation 
 
 {2x-Sy-x{x-^l) = S{x-2)(x+ 7) -5. 
 Performing the operations indicated, we have 
 
 4:X^—12x-\-9-x^-x = Sx^+15x-4t2-5, 
 Transposing, 
 
 Ax^-12x-af-x-Sx'-l5x==-4:2-o-9, 
 Uniting terms, — 28ic = — 56. 
 
 Dividing by — 28, x = 2, Ans, 
 
 Solve the following equations : 
 
 18. 3 + 2(2a; + 3) = 2a;-3(2ic + l). 
 
 19. 2x-(4tx-l)=:bx-{x-l), 
 
 20. 7{x-2)-5{x-}-3) = S{2x-6)-G{4:X-l). 
 
 21. 3(3aj + 5)-2(5a;-3) = 13-(5a;-16). 
 
 22. (2a;-l)(3a^ + 2) = (3ir-5)(2a; + 20). 
 
 23. (5-6a;)(2cc-l) = (3a; + 3)(13-4a;). 
 
 24. (x-Sy-{5-xy = -4.x. 
 
 25. (2a; - 1)2- 3 (oj - 2) + 5 (3 a; - 2) - (5 -2xy= 0. 
 
 26. 2(x-2y-S(x-iy+x'=l. 
 
 27. (a; - 1) (a; - 2) (a; + 4) = (a; + 2) (x + 3) (a; - 4) . 
 
 28. o(7+3a;)-(2a;-3)(l-2a;)-(2a;-3)2-(5+a^) = 0. 
 
 29. (5a;-l)2-(3a; + 2)2-(4a;-3)2 + 4 = 0. 
 
 30. (2a; + l)3 + (2a;-l)3=16a;(a^-4)-228. 
 
SIMPLE EQUATIONS. HI 
 
 SOLUTION OF EQUATIONS CONTAINING FRACTIONS. 
 
 175. 1 . Solve the equation = 
 
 ^ 3 4 G 8 
 
 The L.C.M. of 3, 4, 6, and 8 is 24. Multiplying each 
 term of the equation by 24, we have 
 
 16a;-30 =20a;-27 
 16a;-20a;=30 -27 
 -4a; = 3 
 
 x = , Ans. 
 
 We have then the following rule for clearing an equation 
 of fractions : 
 
 Multiply each term by the loicest common multiple of the 
 denominators. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 2. a;4-^ + ^ = -ll. 7. ^-^-1^=^. 
 
 2 3 5 20 10 4 
 
 4 6 18 a; 2a; 2a; 
 
 4 2x — — = — — -' 9 -4- — — - = --.^ 
 
 4 14 7* *26 3~6 4* 
 
 6. 1^-7 = ^^-^. 10. a;-^+20 = ^ + ^ + 26. 
 
 4 3 4 7 2 4 
 
 * 6 2a; 4 12a;' * a; 2a; 12 3a;* 
 
112 ALGEBRA. 
 
 12. Solve 'the equation ^^id _ i^Zl J =. 4 + ^ ^ + ^ 
 
 4 5 ^ 10 
 
 Multiplying through by 20, the L.C.M. of 4, 5, and 10, 
 
 15aj - 5 - (16 a; - 20) = 80 4- Ua? -H 10 
 15aj -5 -16a; + 20 = 80 + 14a: + 10 
 15 a; - 16a; - 14a.' = 80 + 10 + 5 - 20 
 
 a7= — 5, Ans. 
 
 Note. If a fraction whose numerator is a polynomial is preceded by 
 a — sign, care must be taken to change the sign of each term of the 
 numerator when the denominator is removed. It is convenient, in such 
 a case, to enclose the numerator in a parenthesis, as shown in the above 
 example. 
 
 13. 3x + ^^±^ = 7^. 14. x-^-^±l = 5x-^^ 
 
 7 2 5 3 
 
 • 15. 7a,_ll£zi5 = 3a;+7. 
 
 16. 4a;-?-^^+-(a;-9) = 5a;. 
 o 2 
 
 17. a;-(3a;-4)-^^=l^=2. 
 
 18 2a; ^^ . y 1 x + S ^g a; + l a; + 4 ^ a;-4 
 
 '21 15 * * 2 5 7 * 
 
 20. 2-I^^ = 3a;-li^±^. 
 6 4 
 
 21 ^^~^ _ ^^ + ^ _ 7a; + 2 _ a;— 10 
 
 22 ^ (a: f- 1^ 2a; — 5 ^ 11a; + 5 a;— 13 _ 
 
SIMPLE EQUATIONS. 113 
 
 3 9 2^ ^ 6 
 
 7 2^ ^ 14 3^ ^ 
 
 o- 2a; + l 4a;-f 5 8+.t , 2a; + 5 
 
 26 ^^~^ _ 7 — 3a; _ 10a; — 3 _ 3 — 5a; 
 2 3a; ~ 4 2a; ' 
 
 gy 3a; + 7 4(ar^-2) a;«+16 ^7 
 2 3a; 6ar^ 2* 
 
 28. Solve the equation — -i — = 0. 
 
 a;-l a;-|-l x^-l 
 
 Multiplying each term by a;^— 1, the L.C.M. of the denom- 
 inators, 
 
 2(a; + l) -3(a;-l) -1=0 
 
 2a;4-2 — 3a; + 3 — 1 = 
 
 2a;-3a; = -2-3-|-l 
 — a; = — 4 
 X = 4, Ans. 
 
 Oft o 1 *u 4.' 6a; +1 2a; — 4 2a;— 1 
 
 29. Solve the equation ' = . 
 
 ^ 15 7a;-16 5 
 
 Multiplying each term by 15, 
 
 g _30^-6p^g^_3 
 
 7a; -16 
 
 Transposing and uniting terms, 4 = — — ^^-^. 
 
 Multiplying by 7a; — 16, 28a;— 64 = 30a; — 60 
 
 -2a; = 4 
 
 x= —2, Ans. 
 
114 ALGEBRA. 
 
 Note. If the denominators are partly monomial and partly polyno- 
 mial, it is often advantageous to clear of fractions at first partially ; 
 multiplying by a quantity which will remove the monomial denomi- 
 nators. 
 
 Solve the following equations : 
 
 30. 
 
 1 ^ -0 
 
 34. 
 
 x a^ — DX 2 
 
 3x-l 2>x-\-l 
 
 3 Sx-7 3 
 
 31. 
 
 2x-l 2x + l 
 3i» + 4 3a,^-}-2 
 
 35. 
 
 {x-j-oy_ox+i 
 
 x-3 5 
 
 32. 
 
 Qa?-lx-\-b _2 
 2a^-f5x-13 
 
 36. 
 
 1 2 _ 3 
 x-\-l x-\-2 aj-f-3 
 
 33. 
 
 bx — 2 _bx-\-l 
 
 37. 
 
 3x + 2 2x—l X 
 6 3x-7 2 
 
 
 f^R 2 1 
 
 
 1 
 
 x — 2 x — 3 XT — 5x-\-ii 
 
 39 6.-^ + 7 2(a;-l) ^ 2a;-f 1 
 15 7x-G 5 
 
 40. -^ ^ = 0. 
 
 1 —X 1 -\-x 1 —a^ 
 
 la^4-3a? 1 
 2a;4-l 3x 
 
 41. ?^±^ + ± = x+l. 
 
 42. 2f-^+lU3fii±^V5. 
 \x-^2J \x-^lj 
 
 43. 
 
 44. 
 45. 
 
 3a;-fl x-i-1 2x-l 
 
 X _ a?-f-l _ 7 — 2 a^ 
 9" 3 l-9a;* 
 
 (a; + 2)2 a; -2* 
 
46. 
 
 47. 
 
 48. 
 
 49. 
 
 SIMPLE EQUATIONS. 115 
 
 3 
 
 a^ -\-a 
 
 ;-l 
 
 3a;+l 
 
 X 
 
 -1 1 
 
 x-\-l 
 
 2(jc2 + 4a;4-l) 
 
 X 
 
 -2 ' 
 
 x-h2 
 
 (a; + 2)2 
 
 4. 
 
 r + 3 
 10 
 
 12a;- 
 
 -5 2x-\_Q 
 
 
 ox- 
 
 •29 5 
 
 X 
 
 -1 
 
 x-2 
 
 x — S X — 4: 
 
 x — 2 x — 3 x — 4: x — 6 
 
 SOLUTION OF LITERAL EQUATIONS. 
 
 176. 1 . Solve the equation 2ax — Sb = x-\-c — S ax. 
 
 Transposing and uniting terms, 5ax — x=3b-\-c. 
 
 Factoring the first member, a;(5a— l)=36 + c. 
 
 Dividing by 5 a — 1 , x= "^^ Ans. 
 
 5a— 1 
 
 2. Solve the equation (6 — cxy — {a — cxy = b{b — a) . 
 
 Performing the operations indicated, 
 
 b^-2bcx + c^x^-(a^-2acx + <fx') = b^-ab 
 
 b''-2bcx-\-c'x^-a^ -\- 2 acx - (^:(^ =b--ab 
 
 2acx — 2bcx = a^ — ab 
 
 Factoring both members, 2cx{a —b) = a{a — b) 
 
 Dividing by 2c(a - b\ x=: ^(^^~^) 
 
 ^ ^ ^ ^' 2c{a-b) 
 
 = — , Ans. 
 2c 
 
116 ALGEBRA. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 3. 2ax-\-d = 3c — bx. 
 
 4. Qbmx — 6 a7i= 15 am — 2 bnx. 
 
 5. x + 1 = 2ax-a\x-l). 
 a a^ , b 4:b^ , a 
 
 D. 1 = 1 • 
 
 x 2 X 4: 
 
 7. {a^-2xy = {4.x-3a')(x + a'). 
 
 8. {2m + 3x){2m-Sx)=zn"-{Sx-ny, 
 
 b a 
 
 10. (x — a — by-(x — a){x — b)-\-ab = 0. 
 
 11. 
 
 X x + 2b a^-\-b'- 
 
 x — a X -{-a x~ — a^ 
 
 j2 (b-Sx)(c-\-2x) ^^ 
 2{x-c)(b-3c-3x) 
 
 13. (x+ay-{x-ay-a{3x-a)(2x+a) =x{a + l)-\-S. 
 
 ^^ (n^-^)(n-\-x) ^_^^^^^^,^ 
 x-^2n 
 
 15. {a-x)(b-x) -a(b-^l)=^--hcif. 
 
 2a id' 3a' ^ ^ 
 
 -- X . 1 — 2ax 2x — l _f. 
 
SIMPLE EQUATIONS. 117 
 
 18. i_^±^' = ^_(m^l), 
 mn 3 7i Zn 
 
 19. a; + 2a ^ — ^^ _2 
 x — a x-\-a 
 
 20. 4^-<^ ^ + ^ =1, 
 2x — a x — a 
 
 o< £ _ g — Sea; _ ^ ac — 4bx 
 
 2 2bc "60 36c 
 
 22 ax + b _ 36 ^ aV + 6^ 
 ax — b ax-{-b arxr — b^ 
 
 nnox — b bx — a a — b 
 
 ax-{-b bx-^-a {ax-\-b){bx-\-a) 
 
 ic — w ar — mx — ^ _ ^ , 
 
 m mx — inr ma? — «'' 
 
 SOLUTION OF EQUATIONS INVOLVING DECIMALS. 
 
 177. 1. Solve the equation .2a;-. 01-. 03x = .113a;+. 161, 
 Changing the decimals into common fractions, 
 2a; 1 3a; 113a; , 161 
 
 10 100 100 1000 1000 
 
 Clearing of fractions, 
 
 200 a; -10- 30a; = 113a; H- 161 
 hlx — 171 
 a; = 3, Ans. 
 
118 ALGEBRA. 
 
 Or, we may solve the equation as follows : 
 Transposing, .2 a; — .03 aj — . 113 a; = .01 + .161. 
 Uniting terms, .Oo7ic=.171. 
 
 Dividing b}' .057, a? = 3, Ans. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 2. .23a; -2. 05 =.02 a; -1.882. 
 
 3. .001a;-. 32=. 09a;-. 2a;-. 653. 
 
 4. .3a;-.02-.003a; = .7-.06a;-.006. 
 
 5. .3(1.2a5-5)=14+.05a;. 
 
 6. .7(a;+.13) = .03(4a;-.l)+.5. 
 
 7. 3.3a;- -'^^^~"^'' =.la; + 9.9. 
 
 8. 4.25-^=11-1^^. 
 X 4 X 
 
 g .6a; + .044 _ .5a;-. 178 ^ ^g 
 .4 .6 * * 
 
 10 2 — 3a; 5a; 2a; — 3 ^ a; — 2 25 
 * 1.5 1.25 9 1.8 9° 
 
PROBLEMS. 119 
 
 XIII. PROBLEMS. 
 
 LEADING TO SIMPLE EQUATIONS CONTAINING ONE 
 UNKNOWN QUANTITY. 
 
 178. For the solution of a problem by Algebra no gen- 
 eral rule can be given, as much must depend on the skill 
 and ingenuity of the student. A few suggestions, however, 
 may be found of service : 
 
 1. Express the unknown quantity, or one of the unknown 
 quantities, by one of the final letters of the alphabet. 
 
 2. From the given conditions, find expressions for the other 
 unknown quantities, if any, in the problem. 
 
 3. Form an equation in accordance with the conditions of 
 
 the problem. 
 
 4. Solve the equation thus formed. 
 
 PROBLEMS. 
 
 179. 1. What number is that to which if four-sevenths 
 of itself be added, the sum will equal twice the number 
 diminished by 27? 
 
 Let X = the number. 
 
 4a: 
 Then, — = four-sevenths of it, 
 
 and 2x = twice it. 
 
 4ar 
 By the conditions, x ■\ = 2x —21 
 
 7 
 
 7^. + 4.r= 14^-189 
 - 3 x = - 189. 
 Whence, x = 63, the number required. 
 
 2. A is three times as old as B, and eight years ago he 
 was seven times as old as B. Required their ages at 
 present. ^ 
 
120 ALGEBRA. 
 
 Let X = B's age. 
 
 Then, 3 a- = A's age. 
 
 Also, a: — 8 = B's age 8 years ago, 
 
 and 3 X — 8 = A's age 8 years ago. 
 
 By the conditions, 3a: — 8=7(z — 8) 
 3x-8=7a:-56 
 _4a:=-48. 
 Whence, x = 12, B's age, 
 
 and 3x = 36, A's age. 
 
 Note. In the ahove solution we say " Let x = B's age," meaning 
 "Let a: = the number of years in B's age." Abbreviations of this 
 nature are often used in Algebra; but it should be remembered that 
 they are in fact abbreviations, and that x can only represent an ab- 
 stract number. 
 
 3. A had twice as much money as B ; but, after giving B 
 $35, he had only one-third as much as B. How much had 
 each at §rst? 
 
 Let X = what B had at first. 
 
 Then, 2 x = what A had at first. 
 
 After giving B $35, A had left 2 x — 35 dollars, while B had ar + 35 
 dollars. Then, by the conditions, 
 
 a: + 35 = 3(2 a: -35) 
 a: + 35 =r 6 a: - 105 
 -5a: =-140. 
 Whence, x = 28, B's money at first, 
 
 and 2 ar = 56, A's money at first. 
 
 4. What number is that whose double exceeds its half by 
 45? 
 
 5. Divide 34 into two parts such that four-sevenths of 
 one part may be equal to two-fifths of the other. 
 
 6. What number exceeds the sum of its third, tenth, and 
 twelfth parts by 58 ? 
 
 7. Divide 59 into two. parts such that the sum of one- 
 seventh the greater and one-third the less shall be equal to 
 13. 
 
PROBLEMS. 121 
 
 8. A is four times as old as B, and in 30 years he will 
 be only twice as old as B. What are their ages ? 
 
 9. A is 62 years of age, and B is 36. How many years 
 is it since A was three times as old as B ? 
 
 10. A had one-half as much money as B ; but after B had 
 given him $42, he had four times as much as B. How much 
 had each at first? 
 
 11. Divide 207 into two parts such that one-fourth the 
 greater shall exceed two-sevenths the less by 3. 
 
 12. What two numbers are those whose difference is 3, 
 and the difference of whose squares is 51 ? 
 
 13. A drover paid $1428 for a lot of oxen and cows. 
 For the oxen he paid $55 each, and for the cows $32 each ; 
 and he has twice as many cows as oxen. How many has 
 he of each? 
 
 14. Divide 80 into two parts such that if the greater is 
 taken from 62, and the less from 48, the remainders are 
 equal. 
 
 15. A gentleman left an estate of $1872 to be divided 
 between his wife, three sons, and two daughters. The wife 
 was to receive three times as much as either of the daugh- 
 ters, and each son one-half as much as each of the daugh- 
 ters. How much did each receive? 
 
 16. Divide $70 between A, B, and C, so that A's share 
 may be thi-ee-eighths of B's, and C's share two-ninths of 
 
 A's. 
 
 17. In a garrison of 2744 men, there are 12^ times as 
 many infantry as cavalry, and twice as many cavalry as 
 artillery. How many are there of each kind? 
 
 18. A is 34 years older than B ; and he is as much above 
 50 as B is below 40. Required their ages. 
 
122 ALGEBRA. 
 
 19. A man travelled 3036 miles. He went four-sevenths 
 as many miles on foot as by water, and two-fifths as many 
 miles on horseback as by water. How many miles did he 
 travel in each manner? 
 
 20. Divide a into two parts such that m times the first 
 part shall be equal to n times the second. 
 
 Let X = the first part. 
 
 Then, a — x= the second part. 
 
 By fhe conditions, mx = n{a — x). 
 Or, inx + nx = an. 
 
 Whence, x = ■, the first part. 
 
 m -\- n 
 
 Therefore, a — x = a ^ = , the second part. 
 
 wi + n m -{- n 
 
 21. Divide a into two parts such that m times the first 
 shall be equal to the second divided by w. 
 
 22. Find four consecutive numbers whose sum is 94. 
 
 23. Divide 43 into two parts such that one of them shall 
 be three times as much above 20 as the other lacks of 17. 
 
 24. Divide $47 between A, B, C, and D, so that A and B 
 together may have $27, A and C $25, and A and D $23. 
 
 25. If a certain number is increased b}' 15, one-half the 
 result is as much below 80 as the number itself is above 
 100. Required the number. 
 
 26. Divide 205 into four parts such that the second is 
 one-half of the first, the third one-third of the second, and 
 the fourth one-fourth of the third. 
 
 27. Eleven years aojo, A was 4 times as old as B, and in 
 13 years he will be only twice as old. Required their ages 
 at present. 
 
 28. Find two consecutive numbers such that the differ- 
 ence of their squares added to three times the greater num- 
 ber exceeds the less number bv 92. 
 
PROBLEMS. 123 
 
 29. What number is that, five-sixths of which as much 
 exceeds 25 as one-ninth of it is below 9 ? 
 
 30. A is m times as old as B, and in a years he will be n 
 times as old. Required their ages at present. 
 
 31. Divide a into three parts such that the first may be n 
 times the second, and the second n times the third. 
 
 32. A can do a piece of work in 8 days which B can 
 perform in 10 days. In how many days can it be done by 
 both working together ? 
 
 • Let X = the number of days required. 
 
 Then, - = what both can do in one day. 
 
 X 
 
 Also, - = what A can do in one day, 
 
 o 
 
 and — = what B can do in one day. 
 
 By the conditions, - 4 — = — 
 •^ 8 10 a: 
 
 5 X + 4 r = 40 
 
 9a: = 40. 
 
 Whence, x = 4|, the number of days required. 
 
 33. A can do a piece of work in 15 days, and B can do 
 the same in 18 days. In how many days can it be done by 
 both working together? 
 
 34. A can do a piece of work in 3| hours which B can 
 do in 2| hours, and C in 2^ hours. In how many hours can 
 it be done by all working together ? 
 
 35. The stones which pave a square court would just 
 cover a rectangular area whose lenp^th is 6 yards longer, 
 and breadth 4 yards shorter, than the side of the square. 
 Required the area of the court. 
 
 36. A, B. and C found a sum of money. It was agreed 
 that A should receive $15 less than one-half, B $13 more 
 than one-fourth, and C the remainder, which was $27. 
 How much did A and B receive? 
 
124 ALGEBRA. 
 
 37. A can do a piece of work in a hours which B can do 
 in h hours. In how many hours can it be done by both 
 working together? 
 
 38. A vessel can be filled by three taps ; by the first 
 alone it can be filled in a minutes, by the second iu h 
 minutes, and by the third in c minutes. In what time will it 
 be filled if all the taps are opened ? 
 
 39. A sum of money, amounting to $4.32, consists 
 entirely of dimes and cents, there being in all 108 coins. 
 How many are there of each kind ? 
 
 Let X = the number of dimes. 
 
 Then, 108 — a: = the number of cents. 
 
 Also, 10 a: = the value of the dimes in cents. 
 
 By the conditions, 
 
 10 a: +108 -a: = 432 
 9x = 324. 
 Whence, x = 36, the number of dimes, 
 
 and 108 — x=12, the number of cents. 
 
 40. A man has S4.04 in dollars, dimes, and cents. He 
 has one-fifth as many cents as dimes, and twice as many 
 cents as dollars. How many has he of each kind? 
 
 41. A man has 3 shillings 7 pence in two-penny pieces 
 and farthings ; and he has 19 more farthings than two- 
 penny pieces. How many has he of each kind? 
 
 42. I bought a picture for a certain sum, and paid the 
 same price for a frame. If the frame had cost $ 1 less, and 
 the picture 75 cents more, the. price of the frame would 
 have been only half that of the picture. Required the cost 
 of the picture. 
 
 43. A laborer agreed to serve for 36 days on condition 
 that for every day he worked he should receive $1.25, and 
 for ever}' day he was absent he should forfeit 50 cents. At 
 the end of the* time he received $17. How many days did 
 he work, and how many was he absent? 
 
probl!:ms. 125 
 
 44. A has $105, and B $83. After giving B a certain 
 sum, A has only one-tliird as much money as B. How 
 much was given to B ? 
 
 45. A has a dollars, and B b dollars. After giving B a 
 certain sum, A has c times as much money as B. How 
 umch was given to B ? 
 
 46. A vessel can be emptied by three taps ; by the first 
 alone it can be emptied in 80 minutes, by the second in 200 
 minutes, and by the third in 5 hours. In what time will it 
 be emptied if all the taps are opened ? 
 
 47. The second digit of a number exceeds the first by 2 ; 
 and if the number, increased by 6, be divided by the sum 
 of its digits, the quotient is 5. Requh-ed the number. 
 
 Let X = the first digit. 
 
 Then, a: + 2 = the second, 
 
 and 2 ar + 2 = the sum of tlie digits. 
 
 The number itself is equal to 10 times the first digit, plus the second, 
 which is 10 X + a: + 2, or 11 a: + 2. Hence, by the conditions, 
 
 11 a: + 2 + 6 - 
 
 rr O 
 
 2ar+2 
 llar + 8=10a:+10. 
 Whence, ar = 2. 
 
 Therefore, 11 r + 2 = 24, the number required. 
 
 48. The first digit of a number exceeds the second by 4 ; 
 and if the number be divided by the sum of its digits, the 
 quotient is 7. Required the number. 
 
 49. The first digit of a number is three times the second ; 
 and if the number, increased by 3, be divided b}' the differ- 
 ence of its digits, the quotient is 16. I^equired the number. 
 
 50. A merchant has grain worth 9 shillings per bushel, 
 and other grain worth 13 shillings per bushel. In what pro- 
 portion must he mix 40 bushels, so that the mixture may be 
 worth 10 shillings per bushel? 
 
126 ALGEBRA. 
 
 51. Gold is 19J times as heav}- as water, and silver 10|- 
 times. A mixed mass weighs 41 GO ounces, and displaces 
 250 ounces of water. How many ounces of each metal 
 does it contain ? 
 
 52. The second digit of a number exceeds the first by 3 ; 
 and if the number, diminished by 9, be divided by the sum 
 of its digits, the quotient is 3. Required the number. 
 
 53. Two persons, A and B, 63 miles apart, start at the 
 same time and travel towards each other. A travels 4 miles 
 an hour, and B 3 miles an hour. How far will each have 
 travelled when they meet? 
 
 Let X ■= the distance A travels. 
 
 Then, 63 — a: = the distance B travels. 
 
 Also, - = the time A takes to travel x miles, 
 
 4 
 
 and 63-x^ 
 
 3 
 
 By the conditions. 
 
 
 X _ 
 4~ 
 
 (S'S-x 
 
 3 
 
 Whence, 
 and 
 
 63 
 
 Zx = 
 7x = 
 
 X - 
 — X = 
 
 = 252 -4a: 
 
 .252. 
 
 r 36, the distance A travels, 
 
 : 27, the distance B travels. 
 
 54. A person has 4^ hours at his disposal. How far can 
 he ride in a' coach which travels 5 miles an hour, so as to 
 return home in time, walking back at the rate of 3J miles an 
 hour ? 
 
 55. A courier who travels a miles daily is followed after 
 n days by another, who travels h miles daily. In how many 
 days will the second overtake the first? 
 
 5Q. Two men, A and B, 26 miles apart, set out, B 30 
 minutes after A, and travel towards each other. A travels 
 3 miles an hour, and B 4 miles an hour. How far will each 
 have travelled when thev meet? 
 
PROBLEMS. 127 
 
 57. A capitalist invests | of a certain sum in 5 per cent 
 bonds, and the remainder in 6 per cent bonds ; and finds 
 that his annual income is $180. Required the amount in 
 each kind of bond. 
 
 58. What principal at r per cent interest will amount to 
 a dollars in t years ? 
 
 59. In how many years will 2^ dollars amount to a dollars, 
 at r per cent interest ? 
 
 60. Separate 41 into two parts such that one divided by 
 the other may give 1 as a quotient and 5 as a remainder. 
 
 Let 
 
 X = the divisor. 
 
 Then, 
 
 41 — X = the dividend. 
 
 By the conditions, 
 
 «— =1 + 5 
 
 X X 
 
 41-x = r+5 
 
 
 -2x = -36. 
 
 Whence, 
 
 X = 18, the divisor, 
 
 and 
 
 41 - a: = 23, the dividend. 
 
 61. Separate 37 into two parts such that one divided by 
 the other may give 3 as a quotient and 1 as a remainder. 
 
 62. Separate 113 into two parts such that one divided by 
 the other may give 2 as a quotient and 20 as a remainder. 
 
 63. A general, arranging his men in a solid square, finds 
 he has 21 men over. But attempting to add 1 man to each 
 side of the square, he finds he wants 200 men to fill up the 
 square. Required the number of men on a side at first, 
 and the whole number of troops. 
 
 64. Separate a into two parts such that one divided by 
 the other ma}- give 6 as a quotient and c as a remainder. 
 
 65. The denominator of a fraction exceeds the numerator 
 by 6 ; and if 8 is added to the denominator, the value of 
 the fraction is ^. Required the fraction. 
 
128 
 
 ALGEBRA. 
 
 66. The sum of the digits of a number is 6, and the num- 
 ber exceeds its first digit b}' 4G. What is the number? 
 
 67. At what rate of interest will p dollars amount to a 
 dollars in t 3-ears ? 
 
 68. A man bought a picture for a gertain price, and paid 
 three-fourths the samj2 amount for a frame. If the frame 
 had cost $2 less, and the picture GO cents more, the price of 
 the frame would have been one-third that of the picture. 
 How much did each cost? 
 
 69. The denominator of a fraction exceeds the numerator 
 by 1. If the denominator be increased by 2, the resulting 
 fraction is less by unity than twice the original fraction. 
 Required the fraction. 
 
 70. At what time between 3 and 4 o'clock are the hands 
 of a watch opposite to each other? 
 
 Let OM and OH represent the positions of the minute and hour- 
 M hands at 3 o'clock, and OM' and OW their 
 
 positions when opposite to each other. 
 
 Let X — the arc MIIH'M' over which the 
 minute-hand has passed since 3 o'clock. 
 
 Then, — = the arc HH' over which the hour- 
 12 
 hand has passed since 3 o'clock. 
 
 Also, the arc MH= 15 minute-spaces, 
 and the arc H'M' — 30 minute-spaces. 
 
 arc MHH'M' = arc MH+ arc H'3I' -\- arc HII'. 
 x=Vo+SO+^ 
 
 1.A 
 
 12 ar = 540 -f a: 
 nx = 540. 
 
 X = 49xV. 
 Hence the required time is 49j\ minutes after 3 o'clock. 
 
 71. At what time between 7 and 8 are the hands of a 
 watch opposite to each other?' 
 
 Now, 
 That is. 
 
 Whence, 
 
PROBLEMS. 129 
 
 72. At what time between 2 and 3 are the hands of a 
 watch opposite to each other? 
 
 73. At what time between 5 and 6 are the hands of a 
 watch together ? 
 
 74. At what time between 1 and 2 are the hands of a 
 watch together? 
 
 75. A woman sells half an egg more than half her eggs. 
 Again she sells half an egg more than half her remaining 
 eggs. A third time she does the same ; and now she has 
 sold all her eggs. How many had she at first? 
 
 76. A man has two kinds of money, dimes and half- 
 dimes. If he is offered $1.35 for 20 coins, how many of 
 each kind must he give ? 
 
 77. A man has a hours at his disposal. How far can he 
 ride in a coach which travels b miles an hour, so as to return 
 home in time, walking back at the rate of c miles an hour? 
 
 78. At what time between 6 and 6.30 o'clock are the 
 hands of a watch at right angles to each other ? 
 
 79. At what times between 10 and 11 o'clock are the 
 hands of a watch at right angles ? 
 
 80. A banker has two kinds of money. It takes a pieces 
 of the first kind to make a dollar, and b pieces of the sec- 
 ond kind. If he is offered a dollar for c pieces, how many 
 of each kind must he give ? 
 
 81. A alone can perform a piece of work in 12 hours; A 
 and C together can do it in 5 hours ; and C's work is two- 
 thirds of B*s. The work must be completed at noon. A 
 commences work at 5 a.m. ; at what hour can he be relieved 
 by B and C, and the work be just finished in time? 
 
 82. At what time between 4 and 5 is the minute-hand of 
 a watch exactlv 5 minutes in advance of the hour-hand? 
 
130 ALGEBRA. 
 
 83. A man buys a certain number of eggs at the rate of 
 
 3 for 10 cents. He sells one-third of them at the rate of 2 
 for 7 cents, and the remainder at the rate of 4 for 15 cents ; 
 and makes 16 cents by the transaction. How many eggs 
 did he buy ? 
 
 84. A merchant increases his capital annually by one- 
 third of it, and at the end of each year sets aside $2700 for 
 expenses. At the end of four years, after deducting the 
 amount for expenses, he fiuds that his capital is reduced to 
 $2980. What was his capital at first? 
 
 85. A man owns a harness valued at $25, a horse, and a 
 carriage. The harness and carriage are together worth two- 
 thirds the value of the horse, and the horse and harness are 
 together worth $15 more than twice the value of the carriage. 
 Required the value of the horse, and of the carriage. 
 
 86. Two men, A and B, 107 miles apart, set out at the 
 same time and travel towards each other. A travels at the 
 rate of 13 miles in 5 hours, and B at the rate of 11 miles in 
 
 4 hours. How far will each have travelled when they meet? 
 
 87. A mixture is made of a pounds of coffee at m cents 
 a pound, h pounds at n cents, and c pounds at j? cents. 
 Required the cost per pound of the mixture. 
 
 88. A, B, and C together can do a piece of work in 6 
 days; B's work is one-half of A's, and C's work is two- 
 thirds of B's. How many da3S will it take each working 
 alone ? 
 
 89. A and B start in business, A putting in f as much 
 capital as B. The first year, A gains $150, and B loses \ 
 of his money. The next j^ear, A loses \ of his money, aud 
 B gains $300; and they have now equal amounts. How 
 much had each at first ? 
 
 90. At what time between 9 and 10 is the hour-hand of 
 a watch exactly one minute in advance of the minute-hand ? 
 
PROBLEMS. 131 
 
 91. A and B together can do a piece of work in 1^ days, 
 A and C in 1 J days, and B and C in 2| days. How many 
 days will it take each working alone? 
 
 92. A man buys two pieces of cloth, one of which con- 
 tains 3 yards more than the other. For the larger piece he 
 pays at the rate of $5 for G yaixls, and for the other at the 
 rate of $ 7 for 5 yards. He sells the whole at the rate of 3 
 yards for $4, and makes $8 by the transaction. How many 
 yards were there in each piece ? 
 
 93. A gentleman distributing some money among beggars, 
 found that in order to give them a cents each, he should 
 need b cents more. He therefore gave them c cents each, 
 and had d cents left. Required the number of beggars. 
 
 94. A man let a certain sum for 3 years at 5 per cent 
 compound interest ; that is, at the end of each year there 
 was added -^ to the sum due. At the end of the third 
 year there was due him $2315.25. Required the sum let. 
 
 95. A man starts in business with $4000, and adds to his 
 capital annually one-fourtli of it. At the end of each year 
 he sets aside a fixed sum for expenses. At the end of 
 three years, after deducting the fixed sum for expenses, his 
 capital is reduced to $2475. What are his annual expenses? 
 
 96. A man invests one-third of his money in 3J per cent 
 bonds, two-fifths in 4 per cent bonds, and the balance in 4^- 
 per cent bonds. His income from the investments is $595. 
 What is the amount of his property ? 
 
 97. At what time between 8 and 9 o'clock is the minute- 
 hand of a watch exactly 35 minutes in advance of the hour- 
 hand ? 
 
 98. A fox is pursued by a greyhound, and has a start of 
 60 of her own leaps. The fox makes 3 leaps while the 
 greyhound makes but 2 ; but the latter in 3 leaps goes as far 
 as the former in 7. How many leaps does each make before 
 the gre3'hound catches the fox ? 
 
132 ALGEBRA. 
 
 XIV. SIMPLE EQUATIONS. 
 
 CONTAINING TWO UNKNOWN QUANTITIES. 
 
 180. If we have a simple equation containing two un- 
 known quantities, as x-\-y = 12^ it is impossible to deter- 
 mine the values of x and y definitely ; because, if an}' value 
 be assumed for one of the quantities, we can find a corre- 
 sponding value for the other. 
 
 Thus, if a;= 9, then 9 4-2/ = 12, or 2/= 3 ; 
 
 if a; = 8, then 8 + ?/ = 12, or 2/ = 4 ; etc. 
 
 Hence, any of the pairs of values, 
 
 a;=9,2/ = 3; a; = 8,2/ = 4:; etc., 
 
 will satisfy the given equation. 
 
 Similarly, the equation a; — 2/ = 4 is satisfied by any of 
 the following pairs of values : 
 
 .T = 9,2/ = 5; a; = 8, 2/ = 4; etc. 
 
 Equations of this kind are called indeterminate. 
 
 But suppose we are required to find a pair of values 
 which will satisfy both a; + 2/ = 12 and a; — 2/ = 4 at the same 
 time. It is evident b}^ inspection that the values 
 
 aj = 8, 2/ = 4 
 
 satisfy both equations ; and no other pair of values can be 
 found which will satisfy both simultaneously. 
 
 181. Simultaneous Equations are such as are satisfied by 
 the same values of their unknown quantities. 
 
 Independent Equations are such as cannot be made to 
 assume the same form. 
 
SEVIPLE EQUATIONS. 133 
 
 Thus, a; 4- 2/ = 9 and x — y = \ are independent equations. 
 
 But a; + 2/ = 9 and 2a; + 2^=18 are not independent, 
 since the first equation may be obtained from the second by 
 dividing each term by 2. 
 
 182. It is evident from Art. 180 that two unknown quan- 
 tities require for their determination two independent, simul- 
 taneous equations. 
 
 Two such equations may be solved b\' combining them so 
 as to form a single equation containing but one unknown 
 quantity. This operation is called Elimination. 
 
 183. There are three principal methods of elimiixation : 
 
 1. By Addition or Subtraction. 
 
 2. By Substitution. 
 
 3. By Comparison. 
 
 ELIMINATION BY ADDITION OR SUBTRACTION. 
 
 184. 1. Solve the equations 
 
 ^ (5a;-32/=19 
 \lx + ^y= 2 
 
 (1) 
 (2) 
 
 Multiplying (1) by 4, 
 
 20 a;- 122/= 76 
 
 
 Multiplying (2) by 3, 
 
 21a; + 12i/= 6 
 
 
 Adding these equations, 
 
 41a; = 82 
 
 
 Whence, 
 
 x= 2 
 
 
 Substituting the value of a; in (1), 10— 3?/ = 19 
 
 -32/= 9 
 Whence, yz= — S 
 
 Ans. a;=2, 2/ = — 3. 
 
 This solution is an example of elimination by addition. 
 
134 ALGEBRA. 
 
 2. Solve the equations | 15a: + 8^/ = 1 (1) 
 
 {l0x-7y=:-2A (2) 
 
 Multiplying (1) by 2, 30a; + 16?/= 2 (3) 
 
 Multiplying (2) by 3, 30a;- 21 2/ = - 72 (4) 
 
 Subtracting (4) from (3), 3 7 2/= 74 
 
 2/= 2 
 
 Substituting this value in (2), lOo;— 14 = — 24 
 
 10a; = -10 
 
 x = - 1 
 
 ^ns. a; = — 1, 2/ = 2. 
 
 This solution is an example of elimination by suhir action, 
 
 RULE. 
 
 Multiply the given equations by such numbers as will make 
 the coefficients of one of the unknown quantities equal. Add 
 or subtract the resulting equations axxordinjg as the equal 
 coefficients have unlike or like signs. 
 
 Note. If the coeflScients which are to he made equal are prime to 
 each other, each may be used as the multiplier for the other equation. 
 If they are not prime, such multipliers should be used as will produce 
 their lowest common multiple. 
 
 Thus, in Ex. 1, to make the coefficients of tj equal, we multiply (1) 
 by 4, and (2) by 3. But in Ex. 2, to make the coefficients of x equal, 
 since the L.C.M. of 15 and 10 is 30, we multiply (1) by 2, and (2) by 3. 
 
 EXAMPLES. 
 
 Solve the following by the method of addition or subtrac- 
 tion : 
 
 3 |7a; + 22/ = 31. ^ 
 l3a;-42/ = 23. 
 
 4 I 3a; 4-72/ = 33. g 
 a; + 22/=10. 
 
 2a;- 
 
 -32/ = 
 
 4. 
 
 Qx- 
 
 ■ 2/ = 
 
 28. 
 
 7.y- 
 
 -5a; = 
 
 -11. 
 
 15a; 
 
 -142/ 
 
 = 82. 
 
SIMPLE EQUATIONS. 135 
 
 y (2x-3y = -24:. ^g ( 7a;- 11?/ = -58. 
 
 13x4-2?/== 3. ' (15ic-f- Sy= 2. 
 
 g I 9a;-132/= 76. jg (Il2/-18.T= 2. 
 
 Il5a:-|- 4^=101. * (24x- 6y = -22, 
 
 g I 24a; + 13?/ = -27. ^^ f 24a;- 18?/ = -43. 
 
 l36a; + ll?/ = -15. * (42a; + 30?/= 17. 
 
 10 |15y- 8^=12. 15 I lla;-12?/ = -32. 
 
 l25?/ + 12a;= 1. * lll?/-12a:= 14. 
 
 „ (5a;-7?/=15. ^g ( 9a;-ll?/= 24. 
 
 \3x-5y=\3. ' llOx-f 9?/ = -37. 
 
 ^y (12x-^2\y = -23. 
 1 15a; + 28?/ = -30. 
 
 ELIMINATION BY SUBSTITUTION. 
 
 i'J rf< 3 7/ := 
 2y — DX = 
 
 = -62 (1) 
 
 2y — ox= 44 (2) 
 
 Transposing 5 a; in (2) , 2 ?/ = 5 a; + 44 
 
 ^ 5 a; + 44 ,o\ 
 Or, y = ^ — (3) 
 
 Substituting this in (1), 7a;-3('5-^±M'\ = _- 62 
 
 ^ ^ loo; +132 ^o 
 
 Or, 7a; j = — 62 
 
 Clearing of fractions, 14a; — 15a; — 132 = — 124 
 
 -x= 8 
 
 "Whence, a; = — 8 
 
 _ 40+44 
 Substituting this value in (3) , y = — — = 2 
 
 i 
 
 Ans. x = — 8, 2/ = 2. 
 
136 ALGEBRA. 
 
 BULE. 
 
 Find the value of one of the unknown quantities in terms 
 of the other from one of the given equations^ and substitute 
 this value for that quantity in tJie other equation, 
 
 EXAMPLES. 
 Solve the following by the method of substitution : 
 
 72/ = -19. 
 5y = -U. 
 
 I x-\- 2/= 7. g ^5x + 
 
 X3x-i-2y=19, ' X4tx-\- 
 
 Ox- 
 
 Ay— 15a; = — 7 
 
 2 (Sx— y= 10. g riOa;- 7y= 9 
 
 1 x-\-4ty = — l, ' 1 Ay — 
 
 4. 
 
 ^Sx- 
 \2x- 
 
 -Ay=2, 
 -5y = 6. 
 
 10. 
 
 ( Qx-5y = -7, 
 {l0x-{-3y= 11. 
 
 5. 
 
 |7.- 
 (82/- 
 
 -2y= 8. 
 -6x = -d. 
 
 11. 
 
 ^dx + 2y=15. 
 \Ax + 7y= 3. 
 
 6. 
 
 1 9a; 
 ll5a? 
 
 -Ay = -A. 
 4-82/ = -3. 
 
 12. 
 
 (Sx^ 7y = -2S. 
 l5y-12a; = -12. 
 
 7. 
 
 (20;- 
 liy- 
 
 -7y= 8. 
 -9a; = 19. 
 
 13. 
 
 (7y-Sx=lS9, 
 \2x + 5y= 91. 
 
 
 
 ELIMINATION 
 
 BY COMPARISON. 
 
 186. 1. 
 
 Solve the equations \ \ 
 
 lx — Dy = — 16 
 ^x-\-7y= 5 
 
 Transposing —by in (1), 
 Or, 
 
 
 2x = 5y-16 
 • a.= 52/-16 
 
 2 
 Transposing 72/ in (2), 3x = 5 — 7y 
 
 Or, a==^::il2 
 
 (1) 
 (2) 
 
 (3) 
 
SIMPLE EQUATIONS. 137 
 
 Equating these values of a;, ^^-^ — = — — ^ 
 
 Clearing of fractions, 15 ?/ — 48 = 10 — 14^/ 
 
 29?/ = 58 
 y = 2 
 
 Substituting this value in (3) , x= — = — 3 
 
 Ans. x= — 3,y = 2. 
 
 BULB. 
 
 Find the value of the same unknovm quantity in terms of 
 the other from each of the given equations^ and place these 
 values equal to each other, 
 
 EXAMPLES. 
 
 Solve the following by the method of comparison : 
 
 c. [ X- 2/=-l. 3 f5a; + 62/= 24. 
 
 |3a; + 52/= 21. * 1 9i/-8x= -26. 
 
 „ |6aj + 5^ = -8. g {Ix- 82^ = -ll. 
 
 l4a;-f3?/ = -5. ' 1 x-Uy^ 12. 
 
 ^ (3.^'-5l/= 25. ^Q I bx-\2y-l. 
 
 1 7i/-2a; = -24. ' (10a;- 92/ = 4. 
 
 g (3a;-102/=-36. ^^ | 7?/- 12a;= 17. 
 
 (2aj- 9t/ = -31. ' l8a;-f-lly = 20. 
 
 g |3a;-52/ = 51. j2 ( 7a; + 32/ = 6. 
 
 l2a; + 72/= 3. * ( 11a; -f-9?/ = 8. 
 
 y r7a;H- ?/ = -3. ^g | 15a; + 6.y = - 7. 
 
 1 a;-f6iy= 23. * \^y-2\x= 18. 
 
138 
 
 ALGEBRA. 
 
 MISCELLANEOUS EXAMPLES. 
 
 187. Before applying either method of elimination, each of 
 the given equations should be reduced to its simplest form. 
 
 1. Solve the equations 
 
 7 3 
 
 = 
 
 x-i-3 2/ + 4 
 
 x(y-2)-y{x-5) = -lS 
 From (1), 
 
 72/+ 28 -3a?- 9 = 0, or 7y-3x = -19 
 
 From (2), 
 
 xy—2x — xy-\-5y = — 13, or 5?/ — 2aj= — 13 
 
 Multiplying (3) by 2, Uy-.Qx = - 38 
 
 Multiplying (4) by 3, 15 y — 6 a? = — 39 
 
 Subtracting (5) from (6), y = — 1 
 
 -5 — 2a?=-13 
 
 -2a? = - 8 
 
 a? = 4. 
 
 Ans. a? = 4, y = — 1, 
 
 Substituting in (4) , 
 
 2. 
 
 Solve the following : 
 'll2/ + 6a?=115. 
 2a? 11?/ 5 
 
 3. 
 
 (X 
 
 l3 
 
 4-3?/ 
 
 46. 
 
 ?/_ 5 
 4~6' 
 
 4. 
 
 1 + 3.= 
 
 66. 
 
 3 a? 5y 
 2 3 
 
 = — 
 
 8^6 
 
 -4. 
 
 4. 
 
 (1) 
 (2) 
 
 (3) 
 
 (4) 
 (5) 
 (6) 
 
 x_y _47 
 5 6~9b' 
 
 .2 a? — .05?/ =.25. 
 .03 a? 4- .3?/ = .96. 
 
 .5a?+ 2?/= .01. 
 .lla?+.32/ = — .009. 
 
( x-\-y x — y ^Q 
 
 2 3 * 
 
 8. i 
 
 I 3 "^ 4 
 
 SIMPLE EQUATIONS. 139 
 
 5 3 
 
 4y — 3a; _3a; - 
 
 9. \ 
 
 x{2y-Z) = 2y{x-^\). 
 10. i 3 . 5 
 
 a;— 1 2/ + 2 
 
 = 0. 
 
 11. 
 
 ^(.V-3)-2/(a;-h4) = 22. 
 
 (7/ + l)(a:-2)-(y + 3)(a;-4) = 6. 
 
 12. 
 
 a; — y 3 
 a; — y — 1 
 
 16. 
 
 f 2a;4-3y _ 1 
 .T4-2/ + 13~ 2" 
 
 5a; 7y— 2 _ 
 
 = 11, 
 
 13. 
 
 --12 = ^ + 8. 
 2 4 
 
 a; + y _ 2y — a; ^.^^ 
 5 4 
 
 16. 
 
 §£i±I_I=Ll^ = a; 
 6 10 
 
 6 8 ^* 
 
 14. 
 
 r 3a; + 2 ^y + 2 
 I 5 3 
 
 2y-t-l_a; — 6 
 
 17. 
 
 a; + 3y ^ 3 
 2a;-y 8* 
 
 7y-a; ^ 
 2+a; + 2y 
 
 17. 
 
 18. 
 
 r ''LZ_2 _ 2a; — y — 1 ^ 2y — 2 
 14 3 ~ 5 ' 
 
 I 2y + x-l ^ a; + y 
 19 4 * 
 
140 
 
 ALGEBRA. 
 
 19. 
 
 ^-1 ^-L.?l 
 
 4 3 2 5 
 
 13 
 4 
 
 14?/ — 3aj=ll. 
 
 20. i 
 
 2 5 ' 
 
 4 2 3 
 
 21. 
 
 _ 2a; + ?/ _ 17 _ 2?/ + a? 
 3 "12 4 ' 
 
 5 ^x — y 2w — a; 
 
 4 4-^3 
 
 ' 2^ _ 3^ _ x-[-2y _ 3 _ 5a; — C>y 
 3 5 4 4 ' 
 
 a; , 3aj — v c ■ ^ 
 
 2 ^ 5 15 
 
 23. 
 
 a;-2 
 
 3a; 
 
 2a; — 4?/ — 1 6a;— 1 
 3_5y_4a;-13 
 
 a; — 
 
 a; + 2 
 
 24. 
 
 4 a;2 + 4 a;?/ + 272 = (a; 4- ?/) (4 a; + 1 7) . 
 2/(a;-y) + 54_5.v + 27^ 
 
 a;-2/ 
 
 25. 
 
 a;2 _ 42/' -17 = (a; 4- 2y - 2) (a; - 2?/ +1) 
 xy — b 1 — 2a; _ 
 2/-2 2/-1 
 
SIMPLE EQUATIONS. 
 
 141 
 
 Note. In solving literal simultaneous equations, the method of 
 elimination by addition or subtraction is usually to be preferred. 
 
 ( ax-\-by = c 
 26. Solve the equations < , 
 
 I a'x -h b'y = c' 
 
 Multiplying (1) by b', ab'x -\- bb'y = b'c 
 
 Multiplying (2) by 6, a'bx-{-bb'y = bc' 
 
 Subtracting, 
 
 Whence, 
 
 (1) 
 
 (2) 
 
 ab'x — a'bx = b'c — be' 
 b'c - be' 
 
 x = 
 
 ab' - a'b 
 
 Multiplying (1) by a', aa'x -\-a'by = a'c 
 Multiplying (2) by a, aa'x 4- ab'y = ac' 
 Subtracting (3) from (4) , ab'y — a'by = ac' — a'c 
 
 (3) 
 
 (4) 
 
 Whence, 
 
 y = 
 
 ab' - a'b 
 
 Solve the following equations 
 
 27. 
 
 {2x-^y = a. 
 \ 3a; + 4?/ = 6. 
 
 33. 
 
 29. 
 
 ax-\-by = m. 
 ex ■i-dy = n. 
 
 ax — by = c. 
 X— y = d. 
 
 ^-1 = 
 a b 
 
 X y 
 c^d 
 
 { ax—by = 0. 
 \ mx-\-ny =p. 
 
 ( ax -^ by = m. 
 \ ex — dy = n. 
 
 32. j*«^' -»*2/ =i>- 
 ( m'x — n'y=p'. 
 
 30. 
 
 31. 
 
 34. 
 
 35. 
 
 36. 
 
 x-\-ay = a(a-^2b), 
 y-- = b, 
 
 
 
 iax-\-by = 2. 
 ab{ay — bx) = a- — b^ 
 
 ? + ?' = 2a6. 
 a b 
 
 x-\-y = ab{a + b), 
 
142 
 
 37. 
 
 38. 
 
 
 
 4LGEBRA. 
 
 
 mx -f ny 
 
 
 
 nx 
 
 -\-my 
 
 m^ + n^ 
 mn 
 
 
 \ (« 
 
 + h)x 
 
 -{a-h)y = 
 
 = 4:ab 
 
 [(« 
 
 -h)x. 
 
 -{a + h)y^ 
 
 = 0. 
 
 39. ^ 
 
 a; — 6 __ y — g _o? — \? 
 a h ah 
 
 X y _ a^ -f- m^ — n^ 
 
 40. ^^ m4-7?~ a{m-^n) 
 (m -{-ny(m — n)x = a?y. 
 
 41. 
 
 + 
 
 y _ 1 
 
 a + 6 a — h of 
 
 ^ [ y - 1 
 
 2/ 
 
 42. 
 
 a — 6 a + 6 a^ 
 
 a + 6 a — 6 
 x — y = 4:ab. 
 
 2 a. 
 
 Note. Certain fractional equations, in which the unknown quanti- 
 ties occur in the denominators, are readily solved without previously 
 clearing of fractions. 
 
 43. Solve the equations 
 
 1^-?= 8 
 
 X y 
 
 a; y 
 
 (1) 
 
 (2) 
 
 Multiplying (1) by 5, 
 
 50 45 
 
 40 
 
 Multiplying (2) by 3, ?^ 4- 15 = _ 3 
 
 Adding, 
 
 74 
 
 = 37 
 
SIMPLE EQUATIOXS. 143 
 
 
 
 
 37a; = 74 
 
 
 
 x= 2 
 
 Substituting in (1), 
 
 
 y = -3 
 
 Ans. x 
 
 = 2, 7/ = -3. 
 
 Solve the following 
 
 equations : 
 
 '3 1 5 
 Ix y 
 
 
 48. ^ ^ ' 
 
 45. - 
 
 ^2_3__7 
 
 X y 5 
 
 15 8__1_7^ 
 ^x y 3' 
 
 
 fa h 
 
 a; V 
 49. i I 
 
 U y 
 
 11 7 3 
 
 46. r' ' ' 
 
 
 ri._A=_3. 
 
 60. '^ '^ 
 
 5 1 _17 
 [sx Ay 6 
 
 47. ^ 
 
 f 3 5 1^ 
 
 -— = 16. 
 
 X 2y 
 
 -l4-i = _i5. 
 ^2x y 
 
 
 1 — = mw(m+n) 
 
 51. ^ "^ ^ 
 
 » + ™ = m» + „^ 
 U 2/ 
 
144 ALGEBRA. 
 
 XV. SIMPLE EQUATIONS. 
 
 CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 
 
 188. If there are three simple equations containing three 
 unknown quantities, we may combine two of them by the 
 methods of elimination explained in the last chapter, so as 
 to obtain an equation containing onl}- two unknown quanti- 
 ties. We may then combine the third equation with either 
 of the others, and obtain another equation containing the 
 same two unknown quantities. By solving the equations 
 thus obtained, we derive the values of two of the unknown 
 quantities. These values being substituted in either of the 
 given equations, the value of the third unknown quantity 
 may be determined. 
 
 A similar method may be used when the number of equa- 
 tions and of unknown quantities is greater than three. 
 
 The method of elimination by addition or subtraction is 
 usually the most convenient. 
 
 189. 1. Solve the equations 
 
 6ic— 4^/— 72; = 
 
 9aj- ly-Uz = 
 
 10a;— oy— ?>z = 
 
 Multiplying (1) by 3, \^x-\2y -2lz=: 51 
 Multiplying (2) by 2, 18a;- 14^/- 322;= 58 
 
 Subtracting, 2y-{-\lz = - 1 (4) 
 
 Multiplying (1) by 5, 30a; - 20?/ - 352; = 85 (5) 
 
 Multiplying (3) by 3, 30a;- 151/- ^z= 69 (6) 
 
 Subti-acting (5) from (6), 5^ + 262 = - 16 (7) 
 
 17 
 
 (1) 
 
 29 
 
 (2) 
 
 23 
 
 (3) 
 
SIMPLE EQUATIONS. 145 
 
 Multiplying (4) by 5, lOy + 552 = — 35 
 
 MultiplyiDg (7) by 2, I0y + 52z = - 32 
 
 Subtracting, 32 = — 3 
 
 2 = - 1 
 
 Substituting in (4), 2y — 11 = — 7 
 
 .•■?= 2 
 Substituting the values of y and 2 in (1), 
 
 6a;-8 + 7= 17 
 r.x= 3 
 Ans. it* = 3, 2/ = 2, 2 = — 1. 
 
 In certain cases the solution may be abridged by aid of 
 the artifice which is employed in the following example. 
 
 2. Solve the equations 
 
 Adding the given equations, 
 
 3?^ + 3.'c + 3?/ + 32 = 30 
 Whence, u-\-x-\-y-\-z = 10 (5) 
 
 Subtracting (2) from (5), w= 3 
 
 Subtracting (3) from (5), x= 2 
 
 Subtracting (4) from (5), y= 1 
 
 Subtracting (1) from (5), 2=4 
 
 EXAMPLES. 
 
 Solve the following equations : 
 
 x-\-y= 2. r2x — 5y= — '[0. 
 
 3. ]y-\-z = -l. 4. ]32^ + 42= 13. 
 
 z-{-x= 3. (22-5a;= 12. 
 
 'ti + x + y= 6 
 
 (1) 
 
 x+y+z= 7 
 
 (2) 
 
 y+z+u= 8 
 
 (3) 
 
 ^ z + u + x= 9 
 
 (4) 
 
146 
 
 5. 
 
 6. 
 
 AL 
 
 3a; — 2?/ = — 1. 
 5y + 4z = -6. 
 
 GEBRA 
 13. 
 
 r7aj + 4?/— 2 = — 50 
 }4.x — by — 3z= 20 
 
 x-y-3z = ll. 
 
 
 ( x--3y~4.z=z 30 
 
 2x- y = 5. 
 3x + 2y- 2 = 6. 
 x — 3y + 2z=\. 
 
 14. 
 
 r a;-62/4-42= 3. 
 j 4a; 4- 4?/ -32 =10. 
 (2a;4- 2/ + 62 = 46. 
 
 7. 
 
 8. 
 
 11. 
 
 12. 
 
 ^+ y+ ^— 53. 
 
 a; + 22/ + 32 =107. 
 a; + 32/ + 42 =137. 
 
 3a;— 2/-22 = -23. 
 6a; + 2?/ + 32= 15. 
 
 4a;+3?/— 2 = — 6. 
 
 a; + 2/-2 = 
 2/ + 2 — a; = 
 2 + a; — 2/ = 
 
 3. 
 
 1. 
 
 11. 
 
 x—2y-\-3z= 0. 
 
 10. Xy — 2z+3x = — 2h. 
 
 z — 2x-\-3y— 9. 
 
 5a;-32/+22 = 41. 
 2a;+ 2/— 2; = 17. 
 5a; + 42/ — 22 = 36. 
 
 2a;+ 2/+ 2! = -2. 
 a; + 22/+ 2= 0. 
 a;+ 2/+22 = — 4. 
 
 Ba?— 92/— 72 = — 36. 
 
 15. ^ 12a;- y~3z= 36. 
 6aj— 22/— 2= 10. 
 
 4aj-3y+22 = 40. 
 
 16. ^ 5a7+92/-72 = 47. 
 9a; + 82/-32 = 97. 
 
 
 -M-i=-« 
 
 17. < 
 
 l-M= " 
 
 
 j-|-|=— 
 
 
 '2?^-3a;= 1. 
 
 18. . 
 
 1 
 
 3a;-42/ = -l. 
 42/ — 52= 1. 
 
 I 
 
 52 -6?^=— 2. 
 
 r22/ + 2 + 2i^=-23. 
 2/+32 =-2. 
 19. \ 4a;+2 = 13. 
 
 J + 3w = -20. 
 
SIMPLE EQUATIONS. 
 
 147 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 X y 
 
 = i. 
 
 
 hi 
 
 3 
 ~2* 
 
 
 1 1 
 
 Jz-^x 
 
 = 2. 
 
 
 '3 2 
 
 X y 
 
 = -13. 
 
 
 '- + ' 
 
 y ^ 
 
 = 14. 
 
 
 3 2 
 
 Z X 
 
 = 18. 
 
 
 -ax + 
 
 'x'y = 2. 
 
 
 aV + 
 
 a^z= 2. 
 
 
 . ah 4- a^a; = a^ 
 
 + 1 
 
 ^u-z=22—x-2y. 
 4x — y = So — Sz. 
 4ii — 2y = l9 — Sx. 
 z = 39-2u — 4:y. 
 
 25. 
 
 26. 
 
 27. 
 
 2/-.-^^=l. 
 
 5 6 
 
 4 2 
 
 4. 
 
 a?/ -f- bx = c. 
 ex + «2; = 6. 
 bz -\-cy = a. 
 
 2-z-^^±y=o. 
 
 4 
 
 8-^±l^ = 3x. 
 3 
 
 25-12(a; + 2) = -y. 
 
 2x-{-y y-2z ^^ 
 4 3 
 
 28. ^ 
 
 X -f- 3 V X 
 
 = -2. 
 
 z-\-y z-\-x _ 3 
 3 ^ 2* 
 
 X y z 
 
 -7. 
 
 X y z 
 
 X y z 
 
 * r ax-f-.y— 2 = a^+a— 1. 
 29. ^ ay-f^;— x = a^— a+1. 
 Ca2!+x— y = a. 
 
 X — ay + a^2; = a^. 
 x — by-\-bh= W. 
 x — cy-\-ch= c^. 
 
 Add the equations together. 
 
148 ALGEBRA. 
 
 XVI. PROBLEMS. 
 
 LEADING TO SIMPLE EQUATIONS CONTAINING MORE 
 THAN ONE UNKNOWN QUANTITY. 
 
 190. In solving problems where more than one letter is 
 used to represent the unknown quantities, we must obtain 
 from the conditions of the problem as many independent 
 equations as there are unknown quantities to he determined. 
 
 1. Divide 81 into two parts such that f the greater shall 
 exceed -| the less by 7. 
 
 Let X = the greater part, 
 
 and y = the less. 
 
 , .r + y = 81 
 By the conditions, ) 3 x _ 5 y ^ 
 ( 5 ~ 9 
 
 Solving these equations, x = 4:[>, y = 36. 
 
 2. If 3 be added to both numerator and denominator of 
 a fraction, its value is | ; and if 2 be subtracted from both 
 numerator and denominator, its value is ^. Required the 
 fraction. 
 
 Let 
 
 X = the numerator. 
 
 id y = the denominator. 
 
 
 'x+3_2 
 
 By the conditions, - 
 
 y-\-3 3 
 
 x-2 1 
 
 [y-2 2 
 
 Solving these equations, x = 7, y = 12. 
 
 Therefore the fract 
 
 ion is — 
 12 
 
PROBLEMS. 149 
 
 PROBLEMS. 
 
 3. Divide 50 into two parts such that three-eighths of the 
 greater shall be equal to two-thirds of the less. 
 
 4. Find two numbers such that 7 times the greater ex- 
 ceeds I the less by 97, and 7 times the less exceeds \ the 
 greater by 47. 
 
 5. If one-fifth of A's age were added to two-thirds of 
 B's, the sum would be 19 J years ; and if two-fifths of B's 
 age were subtracted from seven -eighths of A's, the remain- 
 der would be 18 J years. Required their ages. 
 
 6. If 1 be added to the numerator of a certain fraction, 
 its value is ^ ; and if 1 be added to its denominator, its 
 value is J. Required the fraction. 
 
 7. A gentleman at the time of his marriage, found that 
 his wife's age was J of his own ; but after they had been 
 married 12 3*ears, her age was ^ of his. Required their 
 ages at the time of their marriage. 
 
 8. A and B engaged in trade, A with $240 and B with 
 $96. A lost twice as much as B ; and on settling their 
 accounts, it appeared that A had three times as much 
 remaining as B. How much did each lose? 
 
 9. Eight years ago, A was 4 times as old as B ; but in 12 
 years he will be only twice as old. Required their ages at 
 present. 
 
 10. If 5 be added to both terms of a fraction, its value 
 is ^ ; and if 3 be subtracted from both, its value is \. 
 Required the fraction. 
 
 11. A and B agreed to dig a well in 10 days ; but having 
 labored together 4 days, B agreed to finish the job, which 
 he did in 16 days. In how many days could each of them 
 alone dig the well? 
 
150 ALGEBRA. 
 
 12. If the greater of two numbers be divided by the less, 
 the quotient is 2 and the remainder 12 ; but if 4 times the 
 less be divided by the greater, the quotient is 1 and the 
 remainder 14. Required the numbers. 
 
 13. If the numerator of a fraction be doubled, and the 
 denominator increased by 7, its value is f ; and if the 
 denominator be doubled, and the numerator increased by 2, 
 the value is f . Required the fraction. 
 
 14. If a — 1 be subtracted from the numerator of a cer- 
 tain fraction, its value is a + 1 ; and if a be added to its 
 denominator, its value is a. Required the fraction. 
 
 15. A gentleman's two horses, with their harness, cost 
 $300. The value of the poorer horse, with the harness, 
 was $ 20 less than the value of the better horse ; and the 
 value of the better horse, with the harness, was twice that 
 of the poorer horse. What was the value of each? 
 
 16. A merchant has three kinds of sugar. He sells 3 
 lbs. of the first quality, 4 lbs. of the second, and 2 lbs. of 
 the third, for 60 cents ; or, 4 lbs. of the first qualit}', 1 lb. 
 of the second, and 5 lbs. of the third, for 59 cents ; or, 1 
 lb. of the first quality', 10 lbs. of the second, and 3 lbs. of 
 the third, for 90 cents. Required the price per pound of 
 each quality. 
 
 17. A sum of money was divided equally between a cer- 
 tain number of persons. Had there been 3 more, each 
 would have received $1 less; had there been 6 less, each 
 would have received $5 more. How man}' persons were 
 there, and how much did each receive? 
 
 Let X = the number of persons, 
 
 and y = what each received. 
 
 Then, xy = the sum divided. 
 
 By the conditions, 
 
 ((a; + 3)(r/-l) = r;/ 
 l(x-6){y + d) = xj/. 
 Solving these equations, x = 12, y = 6. 
 
PROBLEMS. 151 
 
 18. A boy spent his money for oranges. If he had got 
 five more for his money, they would have cost a half-cent 
 each less ; if three less, they would have cost a half-cent 
 each more. How much money did he spend, and how many 
 oranges did he get? 
 
 19. A merchant has two kinds of grain, worth 60 and 90 
 cents per bushel respectively. How many bushels of each 
 kind must he take to make a mixture of 40 bushels, worth 
 80 cents per bushel? 
 
 20.. My income and assessed taxes together amount to 
 $50. If the income tax were increased 50 per cent, and 
 the assessed tax diminished 25 per cent, they would 
 together amount to $52.50. Required the amount of each 
 tax. 
 
 21. A man purchased a certain number of eggs. If he 
 had bought 20 more for the same money, they would have 
 cost a cent apiece less; if 15 less, a cent apiece more. 
 How many eggs did he buy, and at what price? 
 
 22. If a certain lot of land were 8 feet longer and 2 feet 
 wider, it would contain 656 square feet more ; and if it were 
 2 feet longer and 8 feet wider, it would contain 776 square 
 feet more. Required its length and width. 
 
 23. If B gives A $5, they will have equal amounts ; but 
 if A gives B $15, B will have ^ as much as A. How much 
 money has each ? 
 
 24. Find three numbers such that the first with half the 
 other two, the second with one-third the other two, and the 
 third with one-fourth the other two, may each be equal to 
 34. 
 
 25. There are four numbers whose sum is 136. Twice 
 the first exceeds the second by 46, twice the second ex- 
 ceeds the third by 44, and twice the third exceeds the fourth 
 by 40. Required the numbers. 
 
152 ALGEBRA. 
 
 26. The sum of the digits of a number of three figures is 
 13. If the number, decreased by 8, be divided by the sum 
 of its second and third digits, the quotient is 25 ; and if 99 
 be added to the number, the digits will be inverted. Re- 
 quired the number. 
 
 Let X — the first digit, 
 
 y — the second, 
 and z = the third. 
 
 Then, 100 a: + 10 ?/ + 2 = the number, 
 and 100 z ■\- Id y -\- x = the number with its digits inverted. 
 
 By the conditions, 
 
 ^+?/ + ^=13, 
 1003:+ 10?/ + ;2-8 ^o^ 
 
 and 100a:+ 103/ + 24-99=100« + 10y + a:. 
 Solving these equations, a: = 2, 3/ = 8, 2 = 3. 
 Therefore the number is 283. 
 
 27. The sum of the dioits of a number of two fio^iires is 
 11 ; and if 27 be subtracted from the number, the digits will 
 be inverted. Required the number. 
 
 28. The Slim of the disfits of a number of three figures is 
 11, and the units' figure is twice the figure in the hundreds' 
 place. If 297 be added to the number, the digits will be 
 inverted. Required the number. 
 
 29. A and B can perform a piece of work in 6 days, A 
 and C in 8 days, and B and C in 12 days. In how many 
 days can each of them alone perform it? 
 
 30. If I were to make my field 5 rods longer and 4 rods 
 wider, its area would be increased by 240 square rods ; but 
 if I were to make its length 4 rods less, and its width 5 rods 
 less, its area would be diminished by 210 square rods. 
 Required its length, width, and area. 
 
 31. Find three numbers such that the sum of tiie first and 
 second is c, of the second and third is a, and of the third 
 and first is 6. « 
 
proble:ms. 153 
 
 32. There is a number of three figures, whose digits have 
 equal differences in their order. If the number be divided 
 by half the sum of its digits, the quotient is 41 ; and if 396 
 be added to the number, the digits will be inverted. Re- 
 quired the number. 
 
 33. A sum of money is divided equally between a certain 
 number of persous. Had there been m more, each would 
 have received a dollars less ; if n lessjPeach would have 
 received b dollars more. How many persons were there, 
 and how much did each receive ? 
 
 34. A gentleman left a sum of money to be divided 
 between his four sons, so that the share of the eldest should 
 be ^ the sum of the shares of the other three, of the second 
 I the sum of the other three, and of the third J the sum of 
 the other three. It was found that the share of the eldest 
 exceeded that of the youngest by $140. What was the 
 whole sum, and how much did each receive? 
 
 35. A grocer bought a certain number of eggs, part at 2 
 for 5 cents and the rest at 3 for 8 cents, and paid for the 
 whole $1.71. He sold them at 36 cents a dozen, and made 
 27 cents by the transaction. How many of each kind did 
 he buy ? 
 
 36. If a number of two figures be divided by the sum of its 
 digits, the quotient is 7 ; and if the digits be inverted, the 
 quotient of the resulting number, increased by 6, divided by 
 the sura of the digits, is 5. Required the number. 
 
 37. If 45 be added to a certain number of two figures, 
 the digits will be inverted ; and if the resulting number be 
 divided by the sum of its digits, the quotient is 7 and the 
 remainder 6. Required the number. 
 
 38. A and B can do a piece of work in m days, B and C 
 in n days, and C and A in p days. In what time can each 
 alone perform the work? 
 
154 ALGEBRA. 
 
 39. A crew can row 10 miles in 50 minutes down stream, 
 and 12 miles in an hour and a half against the stream. 
 Find the rate in miles per hour of the current, and of the 
 crew in still water. 
 
 Let X = the rate of the crew in still water, 
 
 and y = the rate of the current. 
 
 Then, x -\- j/ = the rate rowing down stream, 
 
 and X — y — the rate rowing up stream. 
 
 Since the distancP divided by the rate gives the time, we have, by 
 the conditions, 
 
 \x^y 6 
 ] 12- ^3 
 yx-y 2 
 
 Solving these equations, x = 10, ?/ = 2. 
 
 40. A crew can row a miles in h hours down stream, and 
 c miles in d hours against the stream. Find the rate 
 in miles per hour of the current, and of the crew in still 
 water. 
 
 41. A boatman can row down stream a distance of 20 
 miles, and back again, in 10 hours ; and he finds that he 
 can row 2 miles against the current in the same time that 
 he rows 3 miles with it. Required his time iu going and in 
 returning. 
 
 42. A number consists of three digits whose sum is 21. 
 The sum of the first digit and twice the second exceeds the 
 third by 8 ; and if 198 be added to the number, the digits 
 will be inverted. Required the number. 
 
 43. A merchant has two casks of wine. He pours from 
 the first cask into the second as much as the second con- 
 tained at first ; he then pours from the second into the first 
 as much as was left in the first ; and again from the first 
 into the second as much as was left in the second. There 
 are now 16 gallons in each cask. How many gallons did 
 each contain at first? 
 
PROBLEMS. 155 
 
 44. A number consists of two figures. If the digits be 
 inverted, the sum of the resulting number and the original 
 number is 121 ; and if the number be divided by the sum of 
 its digits, the quotient is 5 and the remainder 10. Required 
 the number. 
 
 45. A man has $30,000 invested at a certain rate of 
 interest, and owes $20,000, on which he pays interest at 
 another rate ; and tlie interest which he receives exceeds 
 that which he pays by $800. Another man has $35,000 
 invested at the second rate of interest, and owes $24,000, 
 on which he pays interest at the first rate ; and the interest 
 which he receives exceeds that which he pays by $310. 
 What are the two rates of interest? 
 
 46. A certain sum of money, at simple interest, amounted 
 in 2 years to $132, and in 5 years to $150. Required the 
 
 47. A certain sum of money, at simple interest, amounted 
 in m years to a dollars, and in n years to b dollars. Re- 
 quired the sum, and the rate of interest. 
 
 48. A train running from A to B meets with an accident 
 which causes its speed to be reduced to one-third of what it 
 was before, and it is in consequence 5 hours late. If the 
 accident had happened 60 miles nearer B, the train would 
 have been only 1 hour late. What was the rate of the train 
 before the accident? 
 
 Let 3 x = the rate of the train before the accident. 
 
 Then, x = its rate after the accident. 
 
 Let 1/ = the distance to B from the point of detention. 
 
 By the conditions, 
 
 Solving these equations, x = 10. 
 Hence the rate of tlic train before the accident was 30 miles an hour. 
 
156 ALGEBRA. 
 
 49. A man rows down a stream, whose rate is 3| miles 
 per hour, for a certain distance in 1 hour and 40 minutes. 
 In returning, it takes him 6 hours and 30 minutes to arrive 
 at a point 2 miles short of his starting-place. Find the 
 distance which he rowed down stream, and his rate of pulling. 
 
 50. If a certain number be divided by the sum of its two 
 digits, the quotient is 6 and the remainder 1. If the digits 
 be inverted, the quotient of the resulting number increased 
 by 8, divided by the sum of the digits, is 6. Required the 
 number. 
 
 51. A train running from A to B meets with an accident 
 which delays it 30 minutes ; after which it proceeds at three- 
 fifths its former rate and arrives at B 2 hours and 30 minutes 
 late. If the accident had occurred 30 miles nearer A, the 
 train would have been 3 hours late. What was the rate of 
 the train before the accident? 
 
 52. A, B, and C together have $24. A gives to B and 
 C as much as each of them has ; B gives to A and C as 
 much as each of them then has ; and C gives to A and B as 
 much as each of them then has. The}' have now equal 
 amounts. How much did each have at first? 
 
 53. A and B are building a fence 126 feet long. After 
 3 hours, A leaves off, and B finishes the work in 14 hours. 
 If 7 hours had occurred before A left off, B would have 
 finished the work in 4| hours. How many feet does each 
 build in one hour? 
 
 54. Divide 115 into three parts such that the first part 
 increased by 30, twice the second part, increased by 2, and 
 6 times the third part, increased by 4, may all be equal to 
 each other. 
 
 55. Four men. A, B, C, and D, play at cards, B having 
 $1 more than C. After A has won half of B's money, B 
 one-third of C's, and C one-fourth of D's, A, B, and C 
 have each $18. How much had each at first? 
 
PROBLEMS. 157 
 
 56. A gives to B and C as much as each of them has ; B 
 gives to A and C as much as each of them then has ; and C 
 gives to A and B as much as each of them then has. Each 
 has now $48. How much did each have at iBrst? 
 
 57. A, B, and C, were engaged to mow a field. The first 
 day, A worked 2 hours, B 3 hours, and C 5 hours, and 
 together they mowed 1 acre ; the second day, A worked 4 
 hours, B 9 hours, and C 6 hours, and all together mowed 2 
 acres; the third day, A worked 10 hours, B 12 hours,, and 
 C 5 hours, and all together mowed 3 acres. In what time 
 could each alone mow an acre ? 
 
 58. A man invests $3600, partly in S}r per cent bonds, 
 and partly in 4 per cent bonds. The income from the 3^ 
 per cent bonds exceeds the income from the 4 per cent 
 bonds by $6. How much has he in each kind of bond? 
 
 59. A and B run a race of 480 feet. The first heat, A 
 gives B a start of 48 feet, and beats him by 6 seconds ; the 
 second heat, A gives B a start of 1 44 feet, and is beaten by 
 2 seconds. How many feet can each run in a second ? 
 
 60. The fore-wheel of a carriage makes 4 revolutions 
 more than the hind-wheel in going 96 feet ; but if the cir- 
 cumference of the fore-wlieel were f as great, and of the 
 hind-wheel ^ as great, the fore-wheel would make only 2 
 revolutions more than the hind-wheel in going the same dis- 
 tance. Find the circumference of each wheel. 
 
 61. A and B together can do a piece of work in 44 days : 
 but if A had worked one-half as fast, and B twice as fast, 
 they would have finished it in 4^y days. In how many days 
 could each alone perform the work? 
 
 62. A and B run a race of 300 yards. The first heat, A 
 gives B a start of 40 yards, and beats him by 2 seconds ; the 
 second heat, A gives B a start of 16 seconds, and is beaten 
 by 36 yards. How many yards can each run in a second? 
 
158 ALGEBRA. 
 
 XVII. INVOLUTION. 
 
 191. Involution is the process of raising a quantity to 
 any required power. 
 
 This is effected, as is evident from Art. 13, by taking the 
 quantity as a factor a number of times equal to the exponent 
 of the required power. 
 
 192. If the quantity to be involved is positive, all its 
 powers will evidently be positive ; but if it is negative, all 
 its even powers will be positive, and all its odd powers nega- 
 tive. Thus, 
 
 (_a)' = (-a)x(-a) = + a^ 
 
 (— fl)3=:(_a) X ( — a) X ( — a) = — a^ 
 
 (— a)^ = (— a) X ( — a) X (— a) X (— a) = + a'*; etc. 
 
 Hence, the even powers of any quantity are positive; and 
 the ODD powers of a quantity have the same sign as the quan- 
 tity itself 
 
 INVOLUTION OF MONOMIALS. 
 
 193. 1. Find the value of {bahy. 
 
 (5 a-cY = 5 ah x 5 a^c x oahx 6 ah = 625 aV, Ans. 
 
 2. Find the value of (- 3m^)3. 
 
 (-3m^)3 = (-3m^) x (-3m^) x (-3m^) = -27m^, ^ws. 
 
 From the above examples we derive the following rule : 
 
 liaise the numerical coefficient to the required power^ and 
 multiply the exponent of each letter by the exponent of the 
 required power. 
 
 Give to every even power the positive sign, and to every odd 
 power the sign of the quantity itself. 
 
INVOLUTION. 159 
 
 EXAMPLES. 
 Write by inspection the values of the following : 
 
 3. (-a6V)^ 7. {-b'cy. 11. (Sa^b^c)''. 
 
 4. {-bd'by. 8. (a-6-c'")^ 12. {-Qx^-ify. 
 6. (x^y)"^. 9. (-5m^n)*. 13. (4a'"62«)5 
 
 6. (2?7inV)«. 10. (4a-6V)^ 14. {-l^fz'^y. 
 
 A fraction is raised to any power by raising both numera- 
 tor and denominator to the required power. 
 
 For example, (^-_j = ^^. 
 
 Write by inspection the values of the following : 
 
 "■(1ST "-(H- ^-(-SJ- 
 
 SQUARE OF A POLYNOMIAL. 
 194. We find by multiplication : 
 
 a + b-i-c 
 a-{-b-{-c 
 
 a^-\- ab-\- ac 
 
 ■i- ab +&-+ 6c 
 
 4- ac -^ bc + cP 
 
 a^ + 2ab-{-2ac-\-b^ + 2bc-{'€p 
 
 This result, for convenience of enunciation, may be written 
 as follows : 
 
 (a + 6 + c)2 = a^ + 62 _|. c2 + 2a6 + 2ac + 26c. 
 
160 ALGEBRA. 
 
 In a similar manner, we find : 
 (a + b-^c + dy = a^ + b'- + c'-\-dP 
 
 + 2ab + 2aG + 2ad + 2bc-^2bd-\-2cd; 
 and so on. 
 
 We have then the following rule for the square of any 
 polynomial : 
 
 Write the square of each term, together with twice its 
 product by each of the following terms, 
 
 EXAMPLES. 
 
 1. Square 2iB^ — 3i» — 5. 
 
 The squares of the terms are 4 a;*, 9cc^, and 25. Twice 
 the first term into eacli of the following terms gives the 
 results, — 120/* and — 20. t- ; and twice the second term into 
 the following term gives the result, 30a:. Hence, 
 
 {2sc^-3x-6y^4x^-j-da^-\-2o-12x^-20x'-\-^0x 
 = 4a;*-12a;'^-llar2 + 30a; + 25, Ans. 
 
 Square the following expressions : 
 
 2. a-b-{-c. 11. i^-2x + 5. 
 
 3. a + b-c. 12. 2ar' + 3j»2-f 1. 
 
 4. 2x'-\-x + l. 13. Sa'-2ab-6b\ 
 
 5. x^ — Sx-i-l. 14. 4 m^-j-mn^ — 3 7i\ 
 
 6. 0^2+ 4a: — 2. 15. a — b — c-\-d. 
 
 7. 2a:2_^_3 iq a — b-hc-d. 
 
 8. 3a2-5a + 4. 17. 1 + a: + a:^ -f a:^^ 
 
 9. 2a:2_|_5^_7^ 18. Sx^- 2^" ~x-^A. 
 10. x^2y-3z. 19. a:^- 4a:2_ 2a: — 3. 
 
INVOLUTION. 161 
 
 CUBE OF A BINOMIAL. 
 195. We find by multiplication : 
 
 (a-\-by=a'-h2ab -f 6^ 
 a -h6 
 
 a'b-h2ab^-i' 
 
 (a -^by = a^ + Sa-b -\- Sab^ + 1/' 
 
 {a-by = a^-2ab +6* 
 a -6 
 
 o.3~2a26 4- ab' 
 - a'b-\-2ab^-b^ 
 
 (a - by = a^-3a^b + 3ab^-b^ 
 
 That is, 
 
 The cube of the sum of two quantities is equal to the cube 
 of the first ^ plus three times the square of the first times the 
 second^ plus three times the first times the square of the sec- 
 ond, plus the cube of the second. 
 
 The cube of the difference of two quantities is equal to the 
 cube of the first, minus three times the square of the first times 
 the second, plus three times the first times tJie square of the 
 second, minus the cube of the second. 
 
 EXAMPLES. 
 
 1. Find the cube of a + 26. 
 
 (a + 26)3=a3-f-3a2(26)H-3a(26)2 + (26)3 
 = a^-\-6a'b + 12al/-\-Sb^, Ans. 
 
 2. Find the cube of 2a; — Si/*. 
 
 (2x-3fy=(2xy-s(2xy(3f) + s(2x)(3fy-{3y'y 
 
 = 8x'^-~36x'f-\-54:X7/-27f, Ans. 
 
162 ALGEBRA. 
 
 Find the cubes of the following : 
 
 3. 0^4-3. 7. Sm^-1. 11. 2i»3_3a,. 
 
 4. 2flj-l. 8. af + 4.. 12. Qx'-j-xy. 
 
 5. ah — cd. 9. a + 55. 13. 3m + 5n. 
 
 6. a + 4&. 10. 2x — oy. 14. Sa-?/ — 4a2. 
 
 The cube of a trinomial may be found by the above 
 method, if two of its terms be enclosed in a parenthesis and 
 regarded as a single term. 
 
 15. Find the cube oi o?—2x — l. 
 
 = (ic2 _ 2a;)3 _ 3(a^ _ 2iK)2 + 3(a^ _ 2a^) -1 
 = a^ -6a^ + 12a^ - 8if» - 3(i»^ - 4aT^ + 4ic2^ 
 
 + 3(a^-2aj)-l 
 ^ic^-Gar^-fOaj^ + ^ar^-Qo^-Gaj-l, Ans. 
 
 Find the cubes of the following : 
 
 16. x^-^x-l. 18. a-\-h-c. 20. a^ + Sa^+l. 
 
 17. a-& + l. 19. a^-2a; + 2. 21. 2a^-3ic-l. 
 
 ANY POWER OF A BINOMIAL. 
 196. By actual multiplication, we obtain : 
 •{a + hy^a^-\-2ah + 6^ 
 (a + 6)3 = a^ 4- 3a-6 + 3a62 -f h^ 
 (a + 6)^ = a^ + 4a36 + ea^ft^ _|_ 4a63 + ft^ ; etc. 
 
 {a-hY^a?-2ab +6^ 
 
 (a _ Z>) 3 = a=^ _ 3 a^^ _f- 3 a62 _ 53 
 
 (a - hY = a^ - 4a36 + 6 a-ft^ - 4a6-^ + h' ; etc. 
 
INVOLUTIOX. 163 
 
 In these results we observe the following laws : 
 
 I. The number of terms is one more than the exponent 
 of the binomial. 
 
 II. The exponent of a in the first term is the same as the 
 exponent of the binomial, and decreases by 1 in each suc- 
 ceeding term. 
 
 III. The exponent of h in the second term is 1, and 
 increases by 1 in each succeeding term. 
 
 rV. The coefficient of the first term is 1 ; and of the sec- 
 ond term, is the exponent of the binomial. 
 
 V. If the coefficient of any term be multiplied by the 
 exponent of a in that term, and the result divided by the 
 exponent of b increased by 1, the quotient will be the coeffi- 
 cient of the next term. 
 
 VI. If the second term of the binomial is negative, the 
 terms in the result are alternately positive and negative. 
 
 By aid of the above laws, any power of a binomial may 
 be written by inspection. 
 
 EXAMPLES. 
 1. Expand (a-\-xy. 
 
 The exponent of a in the first term is 5, and decreases by 
 1 in each succeeding term. 
 
 The exponent of x in the second term is 1, and increases 
 by 1 in each succeeding term. 
 
 The coefficient of the first term is 1 ; of the second term, 
 5 ; multiplying the coefficient of the second term by 4, the 
 exponent of a in that term, and dividing the result by the 
 exponent of x increased by 1, or 2, we have 10 for the 
 coefficient of the third term ; and so on. Hence, 
 
 (a ■i-xy = a^-j-6 a*x + 10 aV -f 10 aV -|- 5 ax* + ar', Ans. 
 
164 ALGEBRA. 
 
 Note. The coefficients of terms equally distant from the beginning 
 and end of the expansion are equal. Thus the coefficients of the lat- 
 ter half of an expansion may be written out from the first half. 
 
 2. Expand (l-xy. 
 
 + 15.1-.a;'-C.l 'a^-{-a^ 
 = 1 — 6x -h Icfx^ — 20 a^ -\- 15 X* — Gaf' -\- of , A71S. 
 
 Note. If the first term of the binomial is numerical, it is con- 
 venient to write the exponents at first without reduction. The result 
 should afterwards be reduced to its simplest form. 
 
 Expand the following : 
 
 3. (a -by. 7. (1-a;)^ 11. {x-4.y. 
 
 4. {a-\-by. 8. (x + yy. 12. (a-3)^ 
 
 5. {a -by. 9. (m-ny. 13. (a-f2)^ 
 
 6. {x-iy. 10. (2 + 0^)^ 14. (x-2y. 
 
 15. Expand (3 m — w^)^. 
 {3m-7iy = l{Sm)-{n')Y 
 
 = (3m)^-4(3m)3(n2) + 6(3m)2(n2)2 
 
 -4.(3m)(ny-{-{n'y 
 = 81 m* - 108mV + 54mV — 12mn« + n\ Ans. 
 
 Note. If either term of the binomial has a coefficient or exponent 
 other than unity, it should be enclosed in a parenthesis before apply- 
 ing the laws. 
 
 Expand the following : 
 
 16. (a-Sxy. 18. (a' + bcy. 20. (2 a' -{-by. 
 
 17. (3 + 26)*. 19. {:>f-4y. 21. {2m^-3ny. 
 
EVOLUTION. 165 
 
 XVIII. EVOLUTION. 
 
 197. If a quantity be resolved into any number of equal 
 factors, one of these factors is called a Eoot of the quantity. 
 
 198. Evolution is the process of finding any required 
 root of a quantity. This is effected, as is evident from the 
 preceding article, by finding a quantity which, when raised 
 to the proposed power, will produce the given quantit}-. 
 
 199. The Radical Sig^, V' when prefixed to a quantity, 
 indicates that some root of the quantity is to be found. 
 
 Thus, y'a indicates the second or square root of a ; 
 ■y/a indicates the third or cube root of a ; 
 ■^a indicates the fourth root of a ; and so on. 
 
 The index of the root is the figure written over the radical 
 sign. When no index is written, the square root is under- 
 stood. 
 
 EVOLUTION OF MONOMIALS. 
 
 200. Required the cube root of a""'6V. 
 
 By Art. 198, we are to find a quantity which, when raised 
 to the third power, will produce a^6V. That quantity is 
 evidently a6V. Hence, 
 
 That is, any root of a monomial is obtained by dividing the 
 exponent of each factor by the index of the required root. 
 
 201. From the relation of a root to its corresponding 
 power, it follows from Art. 192 that : 
 
 1. The odd roots of a quantity have the same si(jn as the 
 quantity itself. 
 
 Thus, \a^ = a, and V— ci* = — a. 
 
166 ALGEBRA. 
 
 2. The even roots of a positive quantity are either positive 
 or negatice. 
 
 For the even powers of either a positive or a negative 
 quantity ai-e positive. 
 
 Thus, -^a^ = a OY — a ; that is, -^a* =±a. 
 
 Note. The sign ±, called the double sign, is prefixed to a quantity 
 when we wish to indicate that it is either + or — . 
 
 3. The even roots of a negative quantity are impossible. 
 
 For no quantity when raised to an even power can pro- 
 duce a negative result. Such roots are called imaginary 
 quantities. 
 
 202. From Arts. 200 and 201 we derive the following 
 rule : 
 
 Extract the required root of the numerical coefficient^ and 
 divide the exponent of each letter by the index of the root. 
 
 Give to every even root of a positive quantity the sign ± , 
 and to every odd root of any quantity the sign of the quantity 
 itself. ' 
 
 Note. Any root of a fraction may be found by taking the required 
 root of each of its terms. 
 
 EXAMPLES. 
 
 1 . Find the square root of 9 a^lrc^. 
 
 By the rule, V9 a^W- = ± 3 a-b(?, Ans. 
 2. Find the fifth root of - 32o^V'". 
 
 . V-32a^V'« = -2a2ic™, Ans. 
 Find the values of the following ; 
 3. V-l'2oxY. 7. -^-Sa^^V. 11. -v/Slm^^o. 
 
 4. V49a^62g^. 8. Vl2rS^. 12. V - 243 c^^cZ^^. 
 
 5. v^mVp'"- 9- ^V^r^. 13. -s/U^}Wc\ 
 
 6. VUa^\ 10. VsT^v^. 14. -v/^^H^y^8. 
 
EVOLUTION. 167 
 
 15. ^\^-^^, 17. xl-^- 19 
 
 \ f^ 
 
 ' \l6m« 
 
 3 
 
 64 mV' 
 
 • v 
 
 125 
 
 . ± 
 
 a^'" 
 
 16 
 
 SQUARE ROOT OF POLYNOMIALS. 
 
 203. Since (a-\-by = a^-\-2ab-\-b-, we know that the 
 square root of a^ + 2 a6 -f 6^ is a -f- 6. 
 
 It is required to find a process by which, when the quan- 
 tity a^H-2a6 + 6^ is given, its square root, a-^b, ma}' be 
 determined. 
 
 «2 
 
 2a-\-b 
 
 a 4-b '^^^ square root of the first term is a, 
 which is the first term of the root. Sub- 
 tracting its square from the given ex- 
 2(lb -\-b' pression, the remainder is 2ab -\- h^, or 
 
 2ab-\-b^ (2 a + 6)6. Dividing the first term of 
 
 " this remainder by 2 a, or twice the first 
 
 term of the root, we obtain 6, the second term. This being added to 
 2 a, gives the complete divisor 2 a + 6 ; which, when multiplied by 6, 
 and the product, 2 aft + b^, subtracted from the remainder, completes 
 the operation. 
 
 From the above process we derive the following rule : 
 
 Arrange the terms according to the powers of some letter. 
 
 Find the square root of the first term^ uorite it as the first 
 term of the root, and subtract its square from the given 
 expression. 
 
 Divide the first term of the remainder by twice the first term 
 of the root, and add the quotient to the root and also to the 
 divisor. 
 
 Multiply the complete divisor by the term of the root last 
 obtained, and subtract the product from the remainder. 
 
 If other terms remain, proceed as before, doubling the 
 part of the root already found for the next trial-divisor. 
 
168 
 
 ALGEBRA, 
 
 EXAMPLES. 
 
 204. 1. Find the square root of 9x^ — SOa^x^ + 25 a^. 
 
 9x*-S0a^af + 26a^ 3a^-5a^ Ans. 
 9x^ 
 
 6aj2-5a-^ 
 
 S0a^x^-^2oa^ 
 30aV + 25a« 
 
 The square root of the first term is 3^2, which is the first term of the 
 root. Subtracting 9x^ from the given expression, we have — SOa^x'^ as 
 the first term of the remainder. Dividing this hy twice the first term 
 of the root, Qx^, we obtain the second term of the root, — Sa^, which, 
 added to Qx^, completes the divisor 6x2 — 5 o^. Multiplying this divi- 
 sor by — 5a^, and subtracting the product from the remainder, there is 
 no remainder. Hence, 3 ar^ — 5 a^ is the required square root. 
 
 2. Find the square root of 
 
 12x^ - Ux^ -}- 1 ~ 8x^-^ da^ + Ax. 
 Arranging according to the descending powers of x, 
 
 da^ + Ux^-Sx^-Ux^ + ix+l 
 9x^ 
 
 6af'+2a^ 
 
 o 
 
 3i«3+2a^-2a;-l, 
 Ans. 
 
 12ic^ 
 12.^'^+ 4:X* 
 
 6a^+4:x'-2x 
 
 -Ux" 
 
 6a^ + 4ar^ — 4a;— 1 
 
 
 It will be observed that each trial-divisor is equal to the 
 preceding complete divisor, with its last term doubled. 
 
 Note. Since every square root has the double sign (Art. 201), the 
 result may be written in a different form by changing the sign of each 
 term. Thus, in Example 2, another form of the answer is 
 
 - 3x3-20:2+ 2x+l. 
 
EVOLUTION. 169 
 
 Find the square roots of the following : 
 3. a*-4a^ + 6a2_4a-f 1. 
 
 6. 19a:^ + Go(^-\-26+x*-\-SOx. 
 
 7. 4:0x-{-25-Ux^-\-dx'-24.x'. 
 
 8. m2 + 2m-l--4-— / 
 
 9. Aa' 4- G46^ - 20a36 - 80«6'^ + oTa^fe^. 
 
 10. 28a;3 + 4a;*-14a;+l-f45ar. 
 
 11. a2 + 62 4.c2_2a6-2ac + 26c. 
 
 12. a2^4^2_^922_4a^4.6a;2_i22/2;. 
 
 13. dxf^-{-S0a^ + 25x*-4:2x^-10x'-\-4d. 
 
 14. 16c«-40c*-24(r^ + 25c2 + 30o+9. 
 
 15. 9 + a« + 30a-4a^ + 13a-+ 14a*-14a'. 
 
 16. 4a;^-4ar'y-3a;y-6af^2/^+5a^y*4-4a?y*-f 4a^. 
 
 17. 25a;*-44ic3_40a;4-4a;« + 25H-46ar'-12a?5. 
 
 18. ^_2^ + t£!^_a63 + ^. 
 
 9 3 3 4 
 
 19. da^-12x^y-^10xY-16x^y^-\-dx^y*-4:xf-\-if. 
 
 Find to four terms the approximate square roots of the 
 following : 
 
 20. l + x. 22. a--Aab + b\ 
 
 21. l-2a. 23. 4:x- + 2y, 
 
170 ALGEBRA. 
 
 SQUARE ROOT OF NUMBERS. 
 
 205. The method of Art. 204 may be used to extract the 
 square roots of arithmetical numbers. 
 
 The square root of 100 is 10; of 10,000 is 100; etc. 
 Hence, the square root of a number less than 100 is less 
 than 10 ; the square root of a number between 10,000 and 
 100 is between 100 and 10 ; and so on. 
 
 That is, the integral part of the square root of a number of 
 one or two figures, contains one figure ; of a number of three 
 or four figures, contains two figures ; and so on. Hence, 
 
 Jf a point he placed over every second figure in any integral 
 number^ beginning with the units* place ^ the number of points 
 shows the number of figures in the integral part of its square 
 root. 
 
 206. Let it be required to find the square root of 4624. 
 
 4624 
 3600 
 
 gQ _j_ g Pointing the number according to 
 _ I T the rule of Art. 205, we see that there 
 are two figures in the integral part of 
 120 + 8 1024 the square root. 
 
 = 2a -j- b 1024 Let a denote the value of the num- 
 
 ber in the tens' place in the root, and 
 b the number in the units' place. Then a must be the greatest multi- 
 ple of 10 whose square is less than 4624 ; this we find to be 60. Sub- 
 tracting a2, that is, the square of 60 or 3600, from the given number, 
 the remainder is 1024. Dividing the remainder by 2 a or 120, we have 
 8 as the value of b. Adding this to 120, multiplying the result by 8, 
 and subtracting the product, 1024, there is no remainder. Hence, 
 60 + 8 or 68 is the required square root. 
 
 The ciphers being omitted for the sake of brevit}^, the work 
 will stand as follows : 
 
 4624 
 36 
 
 128 1024 
 1024 
 
 68 
 
EVOLUTION 171 
 
 From the above process we derive the following rule : 
 
 Separate the number into periods by pointing every second 
 figure, beginning with the units* place. 
 
 Find the greatest square in the left-hand period, and write 
 Us square root as the first figure of the root; subtract its 
 square from the number, and to the result bring down the next 
 period. 
 
 Divide this remainder, omitting the last figure, by twice the 
 part of the root already found, and annex the quotient to the 
 root and also to the divisor. 
 
 Multiply the complete divisor by the figure of the root last 
 obtained, and subtract the product from the remainder. 
 
 If other periods remain, proceed as before, doubling the 
 part of the root already found for the next trial-divisor. 
 
 Note 1. It should be observed that decimals require to be pointed 
 to the right. 
 
 Note 2. As the trial-divisor is an incomplete divisor, it is sometimes 
 found that after completion it gives a product greater than the remain- 
 der. In such a case, the last root-figure is too large, and one less must 
 be substituted for it. 
 
 Note 3. If any root-figure is 0, annex to the trial-divisor, and 
 bring down to the remainder the next period. 
 
 EXAMPLES. 
 ' 207. 1. Find the square root of 49.449024. 
 
 49.449024 
 
 7.032, 
 
 49 
 
 
 1403 
 
 4490 
 4209 
 
 
 14062 
 
 28124 
 28124 
 
 
 Since the second root-figure is 0, we annex to the trial- 
 divisor 14, and bring down to the remainder the next period, 
 90. 
 
172 
 
 ALGEBRA. 
 
 Extract the square roots of the following : 
 
 2. 45796. 6. .247009. 10. 446.0544. 
 
 3. 273529. 7. .081796. 11. .0022448644. 
 
 4. 654481. 8. .521284. 12. 811440.64. 
 
 5. 33.1776. 9. 1.170724. 13. .68112009. 
 
 If there is a final remainder, the given number has no 
 exact square root ; but we may continue the operation by 
 annexing periods of ciphers, and thus obtain an approximate 
 value of the square root, correct to any desired number of 
 decimal places. 
 
 14. Extract the square root of 12 to five figures. 
 
 3.4641..., Ans. 
 
 12.06060606 
 
 9 
 
 64 
 
 300 
 256 
 
 686 
 
 4400 
 4116 
 
 6924 
 
 [ 28400 
 27696 
 
 69281 
 
 70400 
 69281 
 
 1119 
 
 Extract the square roots of the following to five figures : 
 
 15. 2. 18. 11. 21. .7. 24. .001. 
 
 16. 3. 19. 31. 22. .08. 25. .00625. 
 
 17. 5. 20. 17.3. 23. .144. 26. 2.08627. 
 
 The square root of a fraction may be obtained by taking 
 the square roots of its terms. 
 
EVOLUTION 173 
 
 If the denominator is not a perfect square, it is better to 
 reduce the fraction to an equivalent fraction whose denomi- 
 nator is a perfect square. 
 
 3 
 
 Thus, to obtain the square root of -, we should proceed 
 
 as follows : 
 
 J5 =JA=V6^ 2^4949^ ^gj237... 
 \8 \l6 4 4 
 
 Extract the square roots of the following to five figures : 
 
 -!■ 
 
 »f 
 
 -1 
 
 ..u. 
 
 35. 
 
 7 
 18 
 
 -*• 
 
 30. i. 
 
 5 
 
 32. i 
 
 »-i 
 
 36. 
 
 13 
 72 
 
 CUBE ROOT OF POLYNOMIALS. 
 
 208. Since (a -^by = a^-\-3a^b + S ab^ + 6^ we know that 
 the cube root of a^ -f 3 a^6 + 3 a6- + 6^ is a-\-b. 
 
 It is required to find a process by which, when the quan- 
 tity a^ -f 3 a^6 -t- 3 ab^ -}- ^ is given, its cube root, a-\-b, may 
 be determined. 
 
 /»3 
 
 Sa' + Sab-^-b- 
 
 a-\-b 
 
 3a'b + Sab^-{-b^ 
 3a-b-hSab- + b^ 
 
 The cube root of the first term is a, which is the first term of the 
 root. Subtracting its cube from tlie given expression, the remainder 
 is 3a26 4- 3a62 -f lr\ or (3a^ + 3«6 + 6^) ft. Dividing the first terra of 
 tliis remainder by 3 a', or three times the square of the first term of 
 the root, we obtain b, the second term. 
 
 Adding to the trial-divisor Sab, that is, three times the product of 
 the first term of the root by the second, and b'\ that is, the square of the 
 last term of the root, completes the divisor, Sa'^ + 3a6 + b^. This being 
 multiplied by 6, and the product, Sa^b ■{■ Sab'^ -{■ b^, subtracted from 
 the remainder, completes the operation. 
 
174 ALGEBRA. 
 
 From the above process we derive the following rule : 
 
 Arrange the terms according to the powers of some letter. 
 
 Find the cube root of the first term^ write it as the first term 
 of the root^ and subtract its cube from the given expression. 
 
 Divide the first term of the remainder by three times the 
 square of the first term of the root, and write the quotient as 
 the next term of the root. 
 
 Add to the trial-divisor three times the product of the first 
 term of the root by the second, and the square of the second 
 term. 
 
 Multiply the complete divisor by the term of the root last 
 obtained, and subtract the product from the remaiyider. 
 
 If other terms remain, proceed as before, taking three 
 times the square of the root already found for the next trial- 
 divisor. 
 
 EXAMPLES. 
 
 209. 1. Find the cube root of 8ic«- 36a;V + 54^2/- 
 27^*. 
 
 8a;« 
 
 Ux'^-l^x'y+^y- 
 
 2a^ — 3?/, Ans. 
 
 -3Qx'y + 54:Qfy^-27f 
 -S6x^y-\-54:afy^-27f 
 
 The cube root of the first term is 2 x"^, which is the first term of the 
 root. Subtracting 8x^ from the given expression, we have —S6x*y as 
 the first term of the remainder. Dividing this by three times the 
 square of the first term of the root, 12a:*, we obtain —Sy as the second 
 term of the root. Adding to the trial-divisor three times the product 
 of the first term of the root by the second, — 18 x'^^/' ^^^ ^^^^ squal-e of 
 the second tdrm, 9 if, completes the divisor, 12^:* — ISar^^y + Oy^. 
 Multiplying this by —3?/, and subtracting the product from the re- 
 mainder, there is no remainder. Hence, 2 x^ — 01/ is the required cube 
 root. 
 
 2. Find the cube root of 40 0:^-60^ -96 a; -(-»«- 64. 
 
EVOLUTIOX. 
 
 175 
 
 Ari'anging according to the descending powers of x, 
 
 -4, 
 Ans. 
 
 x^ 
 
 a^-2 
 
 Sx'-Ga^+ix" 
 
 -Ga^-\-l2x'-8x^ 
 
 
 3x^-12x^-\-12a^ 
 
 -12a^+24a;4-16 
 
 -12a;^+48ar'-96a;-64 
 
 3a;*-12ar'^ +24a;+16 
 
 -12x*-{'^8a^-dG 
 
 X-Q4: 
 
 The second complete divisor is foi-med as follows : 
 
 The trial-divisor is 3 times the square of the root already found ; 
 that is, 3 {x"^ — 2a:)-, or 3x* - 12 r' + 12x2. Three times the product of 
 the root already found by the last term of the root is 3{— 4)(j-2 — 2x), 
 or — 12 a:'' + 24 a:; and the square of the last root-term is 16. Adding 
 these, we have for the complete divisor 3x* — 120^* + 24 a: + 16. 
 
 Find the cube roots of the following : 
 
 3. l-(^y^l2y^-8f. 
 
 4. 27iB«-f 27a;*-f 9»2+l. 
 
 6. 54a/-f 272/^-f36flr^y4-8ar'. 
 
 6. 64a3_i44a2iC2/ + 108aary-27a^2^. 
 
 7. a;^ + 6ar^-40ar*'-f-96a;-G4. 
 
 8. f-l+bf-^^f-^y. 
 
 9. lbx^-Gx-Go^-\-\bx^-\-l+o^-20a?, 
 
 10. 9a^-21a^-36a^ + 8a:«-9a; + 42a^-l. 
 
 11. 8a«-12a*-54a^-f 59a''-f 135a2-75a-12o. 
 
 12. 30ar^-12ar^-12a; + 8~25a;»+8a^ + 30ic^ 
 
 13. ^x^^Zx'y - 3a;y -llit^f + &a?y^ -irUxi^ - 8f. 
 
 14. 27a« - 54a*6 + 9a*b^ + 28a^b^ -dd'b* -Bab"- 6«. 
 
176 ALGEBRA. 
 
 CUBE ROOT OF NUMBERS. 
 
 210. The method of Art. 209 may be used to extract the 
 cube roots of arithmetical numbers. 
 
 The cube root of 1000 is 10; of 1,000,000 is 100; etc. 
 Hence, the cube root of a number less than 1000 is less than 
 10 ; the cube root of a number between 1,000,000 and 1000 
 is between 100 and 10 ; and so on. 
 
 That is, the integral part of the cube root of a number of 
 one, two, or three figures, contains one figure ; of a number 
 of four, five, or six figures, contains two figures ; and so on. 
 Hence, 
 
 If a point he placed over every third figure in any integral 
 number^ beginning with the units* place^ the number of 2wints 
 shows the number of figures in the integral part of its cube 
 root. 
 
 211. Let it be required to find the cube root of 157464. 
 
 157464 50+4 Pointing the number according 
 
 ^3__ 125000 = a + 6 *° *^^^ ^^^^ ^^ ^^** ^^^' ^^ ®^^ ^^** 
 
 ' there are two figures in the integral 
 
 oz4o4: p^j.^ q£ ^j^g cube root. 
 
 Let a denote the value of the 
 
 number in the tens' place in the 
 
 32464 root, and h the number in the units' 
 
 place. Then a must be the greatest 
 
 multiple of 10 whose cube is less than 157464 ; this we find to be 50. 
 Subtracting a^, that is, the cube of 50 or 125000, from the given num- 
 ber, the remainder is 32464. Dividing this remainder by Sa^, that is, 
 3 times the square of 50 or 7500, we obtain 4 as the value of h. Adding 
 to the trial-divisor 3 ah, that is, 3 times the product of 50 and 4, or 600, 
 and 6^, or 16, we have the complete divisor 8116. Multiplying this by 
 4, and subtracting the product, 32464, there is no remainder. Hence, 
 50 -f 4 or 54 is the required cube root. 
 
 The ciphers being omitted for the sake of brevity, the work 
 will stand as follows : 
 
 3a2 =7500 
 
 3a6= 600 
 
 6^= 16 
 
 8116 
 
EVOLUTION. 
 
 157464 
 
 54 
 
 125 
 
 
 7500 
 
 32464 
 
 
 600 
 
 
 16 
 
 
 8116 
 
 324G4 
 
 
 177 
 
 From the above process, we derive the following rule : 
 
 Separate the number into periods by pointing eveiy third 
 figure^ beginning with the units' jjlace. 
 
 Find the greatest cube in the left-hand period^ and wnte its 
 cube root as the first figure of the root; subtract its cube from 
 the number^ and to the result bring down the next period. 
 
 Divide this remainder by three times the square of the root 
 already found, with two ciphers annexed, and write the quotient 
 as the next figure of the root. 
 
 Add to the trial-divisor three times the product of the last 
 root-figure and the part of the root previously found, with one 
 cipher annexed, and the square of the last root-figure. 
 
 Multiply the complete divisor by the figure of the root last 
 obtained, and subtract tJie product from the remainder. 
 
 If other periods remain, proceed as before, taking three 
 times the square of the root already found for the next 
 trial-divisor. 
 
 The notes to Art. 206 apply with equal force to examples 
 in cube root, except that in Note 3 two ciphers should be 
 annexed to the trial-divisor. 
 
 212. In the illustration of Art. 208, if there had been 
 more terms in the given quantity, the next trial-divisor 
 would have been three times the square of a + 6 ; that is, 
 3 a^ -h 6 a6 -f 3 6-. We obsei-ve that this is obtained from the 
 preceding complete divisor, 3 a^ + 3 a6 + b^, by adding to it 
 its second term, 3 ab, and twice its third term, 26^. We may 
 
178 
 
 ALGEBRA. 
 
 then use the following rule for forming the successive trial- 
 divisors in the cube root of numbers : 
 
 To the preceding complete divisor^ add its second term and 
 twice its third term; and aymex two ciphers to the result. 
 
 EXAMPLES. 
 213. 1. Find the cube root of 8.144865728. 
 
 8.144865728 
 
 8 
 
 120000 
 
 144805 
 
 600 
 
 
 1 
 
 
 120601 
 
 120601 
 
 600 
 
 24264728 
 
 2 
 
 
 12120300 
 
 
 12060 
 
 
 4 
 
 
 121323C 
 
 4 
 
 24264728 
 
 2.012, Ans. 
 
 Since the second root-figure is 0, we annex two ciphers to the trial- 
 divisor 1200, and bring down to the remainder the next period, 865. 
 
 The second trial-divisor is formed by the rule of Art. 212. The pre- 
 ceding complete divisor is 120601 ; adding its second term, 600, and 
 twice its third term, 2, we have 121203; annexing two ciphers to this, we 
 obtain the result 12120300. 
 
 Extract the cube roots of the following : 
 
 2. 29791. 7. .000941192. 12. 116.930169. 
 
 3. 97.336. 8. 8.242408. 13. .031855013. 
 
 4. .681472. 9. 51478848. 14. .724150792. 
 
 5. 1860867. 10. 10077.696. 15. 1039509.197. 
 
 6. 1.481544. 11. .517781627. 16. .000152273304. 
 
EVOLUTION. 179 
 
 Extract the cube roots of the following to four figures.: 
 
 17. 2. 19. 7.2. 21. -' 23. — 
 
 8 27 
 
 18. 6. 20. .03. 22. -• 24. -. 
 
 4 3 
 
 214. When the index of the required root is the product 
 of two or more numbers, we may obtain the result by suc- 
 cessive extractions of the simpler roots. 
 
 For, by Art. 198, ("'-^a)"'" = a. 
 
 Taking the »th root of both members, 
 
 ("-/«,)"•= -;ya. (1) 
 
 Taking the with root of both members of (1), 
 
 That is, 
 
 Tlie mnth root of a quantity is equal to the mfli root of 
 the nth root of that quantity. 
 
 For example, the fourth root is the square root of the 
 square root ; the sixth root is the cube root of the square 
 root ; etc. 
 
 EXAMPLES. 
 Find the fourth roots of the following : 
 
 1. Ux*-96a^y-{-21Gx'y--2l6xf + 8\y*. 
 
 2. a^-4af-hl0a,-«-16ar'+19a^-16ar»-}-10a^-4a;-hl. 
 
 3. a^-Sx^-\-Uoi^-^lQx^—56x^-32a^-{-64:a^-\-6ix-\-16. 
 
 Find the sixth roots of the following : 
 
 4. a^-6a^« + 15a«-20a« + 15a*-6a2-|.i. 
 
 5. 64a^ + 192ar'-f 240a;^ 4- 160x'3 + G0ar + 12x4-1. 
 
180 ALGEBRA. 
 
 XIX. THE THEORY OF EXPONENTS. 
 
 215. In the preceding chapters we have considered an 
 exponent only as a positive whole number. It is, however, 
 found convenient to employ fractional and negative expo- 
 nents ; and we proceed to define them, and to prove the 
 rules for their use. 
 
 216. In Art. 13 we defined a positive integral exponent as 
 indicating how many times a quantity was taken as a factor ; 
 thus, 
 
 a"* signifies a x ax ax ••• to m factors. 
 
 We have also found the following rules to hold when m 
 and n are positive integers : 
 
 I. frxa" = a'"+". (Art. 79.) 
 
 II. (a'^)" = a"'". (Art. 193.) 
 
 217. The definition of Art. 13 has no meaning unless the 
 exponent is a positive integer, and we must therefore adopt 
 new definitions for fractional and negative exponents. It 
 is convenient to have all forms of exponents subject to the 
 same laws in regard to multiplication, division, etc., and we 
 shall therefore assume Rule I. to hold for all values of m 
 and 71, and find what meanings must be attached to fractional 
 and negative exponents in consequencco 
 
 218. Required the meaning of a^. 
 
 Since Rule I. is to hold universally, we must have 
 a3 X a^ X as = a3 ^ 3 ^ 3 = ^s^ 
 
 That is, a^ is such a quantity that when raised to the third 
 power the result is a^. Hence (Art. 198), a^ must be the 
 cube root of a^ ; or, a^ = -^a^. 
 
THEORY OF EXPONENTS. 181 
 
 We will now consider the general case : 
 
 p 
 Required the meaning of a«, where p and q are positive 
 
 integers. 
 
 p p p 
 By Rule I., a« X a^ X a» X ••• to 7 factors 
 
 p 
 That is, a» is such a quantity that when raised to the gth 
 
 p 
 power the result is a^. Therefore a» must be the gth root of 
 
 a^', or, 
 
 Hence, in a fractional exponent, the numerator denotes a 
 power and the denominator a root. 
 
 For example, a* = -^a^ ; c- = ^c* ; x^ = -^x ; etc. 
 
 EXAMPLES. 
 219. Express the following with radical signs : 
 
 1. a*. 3. 2c*. 6. x^y^. 7. 4a"6". 9. by^z^. 
 
 2. h\ 4. 3ami 6. m^n^. 8. 2c' d-. 10. oft^c^di 
 
 Express the following with fractional exponents : 
 
 11. ■^:x^. 13. V^- 15. 3Vm^ 17. ^a*</5. 
 
 12. ^y\ 14. ^c. 16. 4.ya\ 18. V^v^2^- 
 
 19. SV^^'a/^'- ' 20. 2a^xyy'^. 
 
 The value of a numerical quantity affected with a fractional 
 exponent may be found by first extracting the root indi- 
 cated by the denominator, and then raising the i*esult to the 
 power indicated by the numerator. 
 
 Thus, (-8)^ = (•^/:r8)2 = (_ 2)2= 4. 
 
182 ALGEBRA. 
 
 Find the values of the following : 
 
 21. 92. 23. 36i 25. (-27)i 27. 64^ 
 
 22. 21K 24. 16^. 26. (-32)i 28. (-216)*. 
 
 220. Required the meaning of oP, 
 
 Since Rule I. is to hold universally, we must have 
 
 or X a" = a"*+^ = a'". 
 Therefore a^ must be equal to 1 . 
 That is, any quantity whose exponent is is equal to 1. 
 
 221. We pass next to the case of negative exponents. 
 Required the meaning of a ^. 
 
 By Rule I., « '^ x a'' = a-3+3 = a«= 1. (Art. 220.) 
 
 Hence, a"^=— • 
 
 a^ 
 
 We will now consider the general case : 
 
 Required the meaning ofa"^, n being integral or fractional. 
 
 By Rule I., a"" x a'' = a-^ + '* = «''= 1. 
 
 Hence, a""* = — 
 
 For example, 
 
 a-2 = i; «'~* = -^; 3a;-V^ = A; etc. 
 a^ (T' xy^ 
 
 222. In connection with Art. 221, the following principle 
 may be noticed : 
 
 Any factor of the numerator of a fraction may he transferred 
 to the denominator^ or any factor of the denominator to the 
 numerator^ if the sign of its exponent he cimnged. 
 
THEORY OF EXPONENTS. 183 
 
 Thus, the fraction ^ can be written in any of the forms 
 
 
 , etc. 
 
 a-^cd*' d^ ' a ^6 «cci* 
 
 EXAMPLES. 
 223. Write the following with positive exponents : 
 1. xY"- 5. a-^h-\ 9. ba^h-^c. 
 
 2. X 'y^. 
 
 6. 3a*6-i 
 
 
 10. 2m-^n-\ 
 
 3. m'^n^. 
 
 7. 2a;-'3,-?. 
 
 
 11. 3aj-V^- 
 
 4. 4a^-i 
 
 8. a-^6-V. 
 
 
 12. o-*6-^c-i 
 
 Transfer the literal factors from 
 
 the denominators to the 
 
 numerators in 
 
 the following : 
 
 
 
 13. 1. 
 
 X 
 
 2^ 
 
 
 19. ^«^ 
 
 2 6c« 
 
 14- ^- 
 
 17. i^• 
 
 
 20 "^ . 
 
 ' 2x^y^ 
 
 u. A, 
 
 18. «^:. 
 
 
 21 ^^ 
 
 2 
 
 Transfer the literal factors from the 
 
 numerators to the 
 
 denominators 
 
 in the following : 
 
 
 
 22. f. 
 
 25. 2«-l 
 5 
 
 
 28. wr^ni 
 
 ^■t- 
 
 26. 3ai 
 
 
 20. -'fl 
 
 2;- 
 
 24. -'. 
 
 giy 5a-2c^ 
 
 
 30. ''«-'^"-. 
 
 fti 3c» 
 
184 ALGEBRA. 
 
 224. Since the definitions of fractional and negative expo- 
 nents were obtained on the supposition that Rule 1., Art. 
 216, was to hold universally, we have for any values of m 
 and w, 
 
 a*" X a** = a'*+*. 
 
 For example. 
 
 
 EXAMPLES. 
 Find the values of the following : 
 
 1. a^xa-\ 6. Saxa'K 11. 2c"^x3a-^c^. 
 
 2. d'xa-^ 7. 5c-»x3c"i 12. 2a-^b^xab~\ 
 
 3. a;-^ X a;-^ 8. a' x i/a\ 13. aj^^ X ^^. 
 
 4. ?i^xw~*. 9. a;-' X ^a;-'. 14. -.j/ajx 5^^?"^. 
 
 5. 2a;^xa;"l 10. m^x-^- 15. -i- x 
 
 ■V^ ah-' a-%^ 
 
 16. Multiply a-|-2a^-3a* by 2-4a~» -6a~^. 
 
 a-f-2a^ — 3a^ 
 2-4a~^-6a"^ 
 
 2a + 4a^ -6a^ 
 
 -4a^ _8a^4-12 
 
 _6a*-12 + 18a~* 
 
 2a -20a* +18a~*, ^w5. 
 
 Note. In examples like the above, it should be borne in mind that 
 any quantity whose exponent is is equal to 1. (Art. 220.) 
 
THEORY OF EXPONENTS. 185 
 
 Multiply the following : 
 
 17. a--2-ha-hja'-\-2-{-a-\ 
 
 18. a^ -f a^x^+ x^ by a^ — x^, 
 
 19. £C~* _ a;-^ + a;"^ - 1 by x~^' + 1. 
 
 20. x-^-2x-^-\-l-2xby x-^-\-2x-\ 
 
 21. 3a-i - a-^b-^ + «-'&"' by Ga^d^ _,. 2^25 -|- 2a. 
 
 22. 2a;^-3a;^-4 + a;"Hy 3a;^4-«-2a;i 
 
 23. x-^y'--x-^y-2x-^'hy 2o(?y-^-\-2a?y-^ -A^a^y-^, 
 
 24. a^aj"^ + 2 + a~^a;^ by 2a~^a;* - 4a"^a;^ + 2a-»a;i 
 
 25. 3a^6-i + a^-2a-Hby 6aU-^-2a-*-3a-U. 
 
 225. The rule of Art. 89 for the division of exponents 
 holds universally ; for, it follows from Art. 222 that 
 
 (Art. 224.) 
 
 
 or _ 
 a- 
 
 ■.a''xa- 
 
 " = a"-". 
 
 
 example 
 
 a 
 
 = a-i-' 
 
 = a-i; 
 
 
 
 
 = «*+» 
 
 = ai; 
 
 
 
 
 = a-»+» 
 
 = a-V; 
 
 etc, 
 
 EXAMPLES. 
 Divide the following : 
 
 1. a^ by a-K 4. a"^ by oTK 7. a«^ by ^. 
 
 -y/ar 
 
 2. abyal 6. 3c-i by </c*. 8. 15a by 3a-^ ^6. 
 
 3. a' by a*. 6. m^ by -^m-^. 9. Ga-^/by 3a^"^. 
 
186 ALGEBRA. 
 
 10. Divide 2a3 _ 20 -f 18a"^ by a + 2a^ - 3a*. 
 
 a 4- 2a* -3a* 
 
 2a^-20 + 18a" 
 2a* 4- 4a* -6 
 
 2a *--4a ^ — 6a\ Ans. 
 
 -4a* -14 + 18a"* • 
 -.4a*- 8 + 12a"* 
 
 -6-12a"* + 18a"* 
 -6-12a~* + 18a"* 
 
 Note. It is important to arrange the dividend and divisor in the same 
 order of powers, and to keep this order throughout the work. 
 
 Divide the following : 
 
 11. a-6bya*-6*. 12. a ^-f- 1 by a~*+ 1. 
 
 13. x-"-5x-^-4:6-4:0xhy x-^-\-4. 
 
 14. x~^-lhy x~^~x~^^-\-x~^ — l. 
 
 15. m — 3m^7i* -|-3m**i3 — n by m* — n*. 
 
 16. x-^y-^ — Sx'^y-'^ + x~'^y~^ by x-^y-^ -f- x-^y-'^ — x-*y-\ 
 
 17. a + a*6^ + 6by a*-a46f + 6*. 
 
 18. m~ ^ -f- nf"^n~^ -f- n~"* by 7n~^ + m~^n~^ -\- m~^n~^. 
 
 226. We will now prove that Rule II., Art. 216, holds 
 for all values of m and n. 
 
 We will consider three cases, in each of which m may 
 have any value, positive or negative, integral or fractional. 
 
 Case I. Let n be a positive integer. 
 Then, from the definition of a positive integral exponent, 
 (a'")"= a"'xa'^X a"^ X ••• to n factors 
 
 __ ^w -f ?» -f m +••• to n terms __ ^mn 
 
THEORY OF EXPONENTS. 18^ 
 
 Case II. Let nhe a. positive fraction, which we will de- 
 
 note by ?.. 
 
 Then, (a"*)«= V{ary, by the definition of Art. 218, 
 
 = Va^, by Case I., 
 = aT, by Art. 218, 
 
 a"*^?. 
 
 Case III. Let n be a negative quantity-, which we will 
 denote by — s. 
 
 Then, (oT) ' = — i— , by the definition of Art. 221, 
 
 "^ ^ {ary ^ 
 
 = — , bv Cases I. or II., 
 
 
 = «-»- = 
 
 :a"'<-*). 
 
 
 We have therefore for all 
 
 values of 
 
 m and n, 
 
 For example, 
 
 
 
 = a-*; 
 
 
 (a^)-3 = 
 
 = a2^-» 
 
 = a-«; 
 
 (a-3)~^ = a"^^~^ =a; etc. 
 
 EXAMPLES. 
 227. Find the values of the following : 
 
 1. (a2)-3. 5. (a;-4)-2, 9. (^m«)t. 13. /"-LV. 
 
 2. (ci-2)2. 6. (a-i)^. 10. {Vy-'y- 14. /'-^3¥. 
 
 3. {a?)K 7. (a*)i 11. (i)^- 15. ^[(0."^)^. 
 
 4. (c-^)"^. 8. {-yjxy^. 12. (a;l)^l 16. (a^-^)Si^. 
 
188 ALGEBRA. 
 
 228. To prove that {aby = a^b"", for any value of n. 
 
 In Art. 193 we showed the truth of the theorem for a 
 positive integral value of 7i. 
 
 Case I. Let n be a positive fraction, which we will denote 
 
 q 
 
 By Art. 226, [(aft)*]* = {aby. 
 
 By Art. 193, [ah^^ =aPbP = {aby. 
 
 Therefore, [(ab)^^ =[ah^y. 
 
 Taking the gth root of both members, 
 
 p p p 
 (^aby —a^b9. 
 
 Case II. Let n be a negative quantity, which we will 
 denote by — s. 
 
 Then, (aby = — ?— - = — , by Art. 193, or Case I. 
 
 (aby a'b' 
 
 = a-'b-\ 
 
 MISCELLANEOUS EXAMPLES. 
 
 229. Square the following (Art. 95) : 
 
 1. a^-b^. 2. a~^ + 2a. 3. x'^y^-Sx'y-^ 
 
 Extract the square roots of the following : 
 
 4. a-^xK 5. 9mnK 6. ^-^. 7. ^. 
 
 8. dx-*-12x~^~2x-^ + 4x''^ + l. 
 
 9. 4a;^ + 4a;*-15a^-8flj^ + 16cci 
 
 10. a^b~^ -4.ah~^' + 6 - 4a"' 6* + a-^b^. 
 
THEOKY OF EXPONENTS. 189 
 
 Extract the cube roots of the following : 
 11. ah\ 12. -Hx 'y^. 13. ^Irnhf', 14. ^. 
 
 15. 82/ 2_i22/"'V- + 6y-^-2/-i. 
 
 16. x^ -9a;^ + 33a;' — 63a; + 66a;' -36a;^ + 8a;i 
 
 Reduce the following to their simplest forms : 
 17. a'-y-^^'a-'^-^^-^'a'. 20. [.T^'-^^a^'-"^]^'. 
 
 19. (a;")-*-j- (a;-'*)-\ 22. [(a;«-6)' «]«+*. 
 
 23 ^- (<^- - a^- ) - a;^ (a" ^ - a;" ^ ) 
 2(a*+a;^)(a^-a;^) 
 
 24. ^-^ 25. (^-«-^-)'+(^-+a^)^ 
 
 «}&-! + a"^ 6' 1 - a^-x^ + a*(a^ + a;^) 
 
 26. (4.'c3-3.T)(.T2+l)-^ + 3a;(a^+l)l 
 
 27 (l-3a;4-a^)--a;(a;-3)(l-3a; + a^)"^ 
 
 1 — 3 a? 4- a:- 
 
 _ 1 _ 1 
 
 28 ^[a? " + (^4-a?) -] , m + 2x 
 
 2 [a)^2i + (^i _|_ a;) ^] 2 x^ {m + a;) ^ 
 29. ^+[l+(l+x^ . 
 
190 ALGEBRA. 
 
 XX. RADICALS. 
 
 230. A Radical is a root of a quantity indicated by a rad- 
 ical sign ; as, Va, or -vx + 1. 
 
 If the indicated root can be exactly obtained, it is called a 
 rational quantity ; if it cannot be exactly obtained, it is 
 called an irrational or surd quantity. 
 
 231. The degree of a radical is denoted by the index of 
 
 3 / 
 
 the radical sign ; thus, -vx -f 1 is of the third degree. 
 
 232. Similar Badicals are those of the same degree, and 
 with the same quantity under the radical sign ; as, 2-\/ax 
 and 3 Va.x'. 
 
 233. Most problems in radicals depend for their solution 
 
 on the following important principle (Art. 228) : 
 
 111 
 For any value of ?i, (ab) « = a« x 6". 
 
 That is, ^ab = Va X Vft. 
 
 TO REDUCE A RADICAL TO ITS SIMPLEST FORM. 
 
 234. A radical is in its simplest form when the quantity 
 under the radical sign is not a perfect power of the degree 
 denoted by any factor of the index of the radical, and has 
 no factor which is a perfect power of the same degree as the 
 radical. 
 
 Case I. 
 
 235. When the quantity under the radical sign is a perfect 
 jjower of the degree denoted by a factor of the index. 
 
 1. Reduce V8 to its simplest form. 
 
 i/8 = ■\/2' = 2^ = 2* = V2, Ans, 
 
RADICALS. 191 
 
 EXAMPLES. 
 Reduce the following to their simplest forms : 
 2. ^^25. 5. ^27. 8. ^64. 11. <^49mV. 
 
 3. </9. 
 
 6. VlOO. 
 
 9. \/25aJ*. 
 
 12. ^125 a«6^ 
 
 4. </S. 
 
 7. ^'81. 
 
 10. y/S2a\ 
 Case IF. 
 
 13. "Va"6^. 
 
 236. TFZien ^7ie quantity under the radical sign has a factor 
 which is a perfect power of the same degree as the radical. 
 
 1. Reduce V54 to its simplest form. 
 
 ^/54 = ^27x2 = ^27 x v'2 (Art. 233) = 3 \/2, Ans, 
 
 2. Reduce Vl8a-6^ — 27a"b^ to its simplest form. 
 
 Vl8a-V-27a^6*= V9a-6^(26-3a) 
 
 = V9a¥x V26-3a 
 
 = 3a?;2V26-3a, ^»s. 
 
 RULE. 
 
 Resolve the quantity under the radical sign into two factors, 
 one of which is the highest perfect power of the same degree as 
 the radical. Extract the required root of this factor, and prC' 
 Jix the result to the indicated root of the other. 
 
 Note. If the highest perfect power in the numerical portion of the 
 quantity cannot be determined by inspection, it may be found by re- 
 solving the number into its prime factors. 
 
 Thus, \/l944 = \/Wx^^ = v/22 x 3* x V^l<3 
 
 = 2 x 3'-^ X V6 = 18 Ve. 
 
192 ALGEBKA. 
 
 EXAMPLES. 
 Reduce the following to their simplest forms. 
 
 3. VoO. 6. ■v^320. 9. -y/sT^f. 
 
 4. 3V24. 7. 2^80. 10. 7V63a^6V. 
 
 6. V72. 8. V98a=^6^. 11. V250a^//. 
 
 12. V25ar^/-50a;y. 14. V (ar^ - 2/^) (a; + 2/) 
 
 13. V54a^6^ + 135a35^ • 15. Vax-^ - 6 aa; + 9 a. 
 
 16. V20^''+60a; + 45. 
 
 17. V3 m^ - 54 m^^i + 243 mn\ 
 
 If the quantity under the radical sign is a fraction, multiply 
 both terms hy such a quantity as will make the denominator- a 
 perfect power of the same degree as the radical. Then pro- 
 ceed as before. 
 
 18. Reduce \/ — -to its simplest form. 
 
 \8a« ^ 
 
 .\8a3 A(l6a^ \16a* 4a2 
 
 Reduce the following to their simplest forms ; 
 
 19. \h 22. -M. 25. A^li. 
 
 20. j5. 23. S^. 26. ^"^'^ 
 
 I _ 
 
 \6 ■ \ 4 ■ \10cd 
 
 21. \L. 24. A^. 27. J^. 
 
 28. aP^^. 29. ^^,# 
 
RADICALS. 193 
 
 237. Conversely^ the coefficient of a radical may be intro- 
 duced under the radical sign by raising it to the power 
 denoted by the index. 
 
 I. Introduce the coefficient of 2aV3a^ under the radical 
 sign. 
 
 2a-\/3a^=^8a3x3iB2= y/24.a?x', Ans. 
 
 Note. A rational quantity may be expressed in the form of a radical 
 by raising it to the power denoted by the index, and writing the result 
 under the corresponding radical sign. 
 
 EXAMPLES. 
 
 Introduce the coefficients of the following under the radical 
 signs : 
 
 2. 3V5. 4. 3</2. 6. WVab. 8. 5a\/2^. 
 
 3. 2\/7. 5. 4^0. 7. o^h^atiK 9. 3mnH'|^. 
 
 \ 27 
 
 10. (a:-l)J^+I. 12. 1±^JIH«. 
 
 II. (l+a;)J_^-l. 13. ?^^^^x| ji-^,-1. 
 
 ADDITION AND SUBTRACTION OF RADICALS. 
 
 238. The sum or difference of two similar radicals (Art. 
 232) may be found by prefixing the sum or difference of 
 their coefficients to their common radical part. 
 
 1. Find the sum of V20 and V45. 
 
 By Art 236, V20 = V4 x 5 = 2 V5, 
 \/45 = Vir3r5 = 3V5. 
 Hence, V20 + V45 = 2V5 + 3V5 = 5V5, Ans, 
 
194 ALGEBRA. 
 
 2. Simplify ^l+^f-^l 
 
 RULE. 
 
 Reduce each radical to Us simplest form. Unite the similar 
 radicals, and indicate the addition or subtraction of the 
 dissimilar. 
 
 EXAMPLES. 
 
 Simplify the following : 
 
 
 3. V27 + V12. 
 
 9. V4:a26 + V9R 
 
 4. V96 4-V54. 
 
 10. v7o + V48 - V245, 
 
 5. ■y/180--^4.5. 
 
 11. -^16 + ^54+^128. 
 
 6. ^162 --^48. 
 
 «• ^i-VIWs- 
 
 7. V128 + V98 + V50. 
 
 - VI-^sW^- 
 
 « 116 13 
 
 14 3(1 J. 3_L J- s 2 
 
 15. 7V27-V75-24^±-27^±. 
 
 16. V27^2^V75a^+(a-36)V3a. 
 
 17. V9a' + 18a^6 - V4a6« + Sb'. 
 
RADICALS. 19;: 
 
 18. ^24 + 5^54-^250-^192. 
 
 19. ^28a^x — 2Sax-\-lx— ^/ 7 a-x -^ ^2 ax -\- 63 x. 
 
 20. ^J^ + 2,J^±J'-i^V^^. 
 Mx-}-y Mx — y ar — 2/ 
 
 TO REDUCE RADICALS OF DIFFERENT DEGREES TO 
 EQUIVALENT RADICALS OF THE SAME DEGREE. 
 
 239. 1. Reduce ^2, ^3, and ^5 to equivalent radicals 
 of the same degree. 
 
 By Art 218, V2 = 2^ = 2^"^ = ^^^ = ^64. 
 ^3 = 3 ' = 3T^ = ^3* = ^81 . 
 ^5 = 5^ = 5^^ = ^5^ = 1^125. 
 
 RULE. 
 
 Wnte the radicals with fractional exponents, and reduce 
 these exponents to a common denominator. 
 
 Note. The relative magnitude of radicals may be compared by re- 
 ducing tliem to the same degree. Thus, in Ex. 1, 1^125 is greater than 
 1^/81, and ?^81 than 5^64. Hence, ^6 is greater than ^3, and ^3 
 than v'2. 
 
 EXAMPLES. 
 
 Reduce the following to equivalent radicals of the same 
 degree : 
 
 2. ^2 and -^3. 5. v^, \/36, and VTc. 
 
 3. V^' \/^' and -^3. 6. Vxy, ^i/yz, and Vzx. 
 
 4. ^5, -^6, and -^7. 7. ^a + 6 and -^a-b. 
 
196 ALGEBRA. 
 
 8. Which is the greater, -^2 or ^3? 
 
 9. Which is the greater, -^^3 or -^5 ? 
 
 10. Arrange in order of magnitude ^3, ^4, and -y/1, 
 
 MULTIPLICATION OF RADICALS. 
 240. 1. Multiply V6 by Vl5. 
 By Art. 233, 
 
 V6 X Vl5 = V6 X 15 = V90 = 3 VlO, Ans. 
 
 2. Multiply V2a by ^/So^. 
 
 Reducing to equivalent radicals of the same degree, 
 
 \/2^ =(2a)^ =(2a)^ = v" (2^^ = -^ST? 
 
 ^2/3^ = (3 a^) ^ = (3 a^) * = ^^^3^ = -^9^ 
 
 Hence, V2a x V"^? = V 80^ x -^¥a^ = Vl2a^ 
 = a^l2a^ Ans. 
 
 RULE. 
 
 Reduce the radicals to equivalent radicals of the same degree. 
 Multiply together the quantities under the radical signs, and 
 write the product under the common radical sign. 
 
 Note. The result should be reduced to its simplest form. 
 
 EXAMPLES. 
 Multiply the following : 
 
 3. V6 and V42. 7. 5^12 and ?^2. 
 
 4: O 
 
 4. 5V10and3V15. 8. ^2 and ^3. 
 
 5. 2 V3^ and 5 Vl5aj. 9. Va^ and V6^. 
 0. Va-6 and S/a6c^. 10. •\/4a^ and V2a. 
 
RADICALS. 197 
 
 11. 4^/3 and 3 V2. 13. V^, ^2, and «/l. 
 
 12. -i/xy, -y/yz^ and -Vzx. 14. ^/2x^ -v/Sic, and ■\\-r-i,- 
 
 15. Multiply 2v3 + 3V2l>y 3V3- V2- 
 
 2V3+3V2 
 3V3- V2 
 
 18 + 9V6 
 
 - 2 V6 - 6 
 
 18H-7V6-6 = 12-|-7V6, Arts. 
 
 Note. It should be remembered that to multiply a radical of the 
 second degree by itself simply removes the radical sign; thus, 
 V3xV3 = 3. 
 
 16. Multiply 3 V^+1 + 4a; by 2 VV+1 - x. 
 
 3V^TTH-4; 
 
 2V^+1- X 
 
 Q>{x' + l) +8a;va:'+l 
 
 -3a;Var'-hl-4a^ 
 
 6ar^ + 6 + 5a; Var^ + 1 - 4a;2 
 
 =^2Qi? + Q>-{-bxV^~+\, Ans. 
 Multiply the following : 
 
 17. V^ — 2andV^ + 3• 
 18. V^-^V^ and 2V5H-V2- 
 
 19. V^ - 4 V3 and 2 V^ + V^- 
 
 20. 2Va-3V^ and 4V«+V^- 
 
 21. ^a;— V2/+V^ and V^ + V2/ — V^^- 
 
198 ALGEBRA. 
 
 22. Vx -j- 1 - 2 Vaj and 2 Vi» -I- 1 + Vx. 
 
 23. V2 - V'3 + V5 and V2 + V^ + V^- 
 
 24. 3V5-2V6 + V7a»(l 6V5 + 4V6-2V7. 
 
 25. 8V3 + 10V2-3V5 and4V3-5V2- V^- 
 Expand the following (Art. 95) : 
 
 26. (2V3-3)^ 28. (Vni^ + a)^. 
 
 27. (3V8 + 5V3)'. 29. {V^H^ - V^^^^^y. 
 
 30. (Va;--f l+a;)(Va^ + l-x). 
 
 31. (Vi»4-1 -{- ^x - 1) (-Vx -{- 1 — -y/x - 1) , 
 
 32. (3V2a;H-5 + 2V3a;-l)(3V2a; + 5-2V3a;-l). 
 
 DIVISION OF RADICALS. 
 241. By Art. 233, ^'^= Va X V6. 
 
 Whence, ^=Vb. 
 
 RULE. 
 
 Reduce the radicals to equivalent radicals of the same de- 
 gree. Divide the quantities under the radical sign, and write 
 the quotient under the common radical sign. 
 
 EXAMPLES. 
 1. Divide Vlo by Vo. 
 
 Reducing to equivalent radicals of the same degree, we 
 have 
 
 V5 -^125 '\12.5 "V5' 
 
RADICALS. 199 
 
 Divide the following : 
 
 2. V108 by V6. 7. ^2 by -^3. 
 
 3. ^M? by V2^. 8. ^12 by ^2. 
 
 4. -W^' by </S^. 9. A^I^ by ■\/2^. 
 
 5. V6 by ^3. 10. a/3^ by -^^6^-. 
 
 6. ^18 by VC. 11. V 12^W by </2¥^\ 
 
 INVOLUTION AND EVOLUTION OF RADICALS. 
 
 242. Any power or root of a radical may be found by 
 using fractional exponents. 
 
 1. Raise ^12 to the third power. 
 
 (^12)3 = (12^)3= 12^ = 12^ = V12 = 2V3, Ans. 
 
 2. Raise ^2 to the fourth power. 
 
 (^2)^ =(2^)^ =2^=^/2^ = -^16 = 2^2, Ans. 
 
 Note 1. The following rule for the involution of radicals is evi- 
 dent from the above : 
 
 If possible, divide the index by the exponent of the required power; 
 otherwise, raise the quantity under the radical sign to the required power. 
 
 EXAMPLES. 
 Find the values of the following : 
 
 3. (-v/5)«. 6. (-^18)^ 9. (v'32)3. 
 
 4. (</7)2. 7. (^/^^&)^ 10. {Sa</'b^y. 
 
 5. {</^xy. 8. (4V3^)3. 11. (3^/24^¥)2. 
 
200 ALGEBRA. 
 
 12. Extract the cube root of V27a^. 
 
 3 
 
 = (3aj)* = V3a^, A7is. 
 13. Extract the square root of -^6. 
 
 V(-^6) = (6^)^ = 6^ = ^6, Ans. 
 
 Note 2. The following rule for the evolution of radicals is evident 
 from the above : 
 
 If possible, extract the required root of the quantity under the radical 
 sign; otherwise, multiphj the index of the radical by the index of the 
 required root. 
 
 If the radical has a coefficient which is not a perfect power of the 
 same degree as the required root, it should be introduced under the 
 radical sign before applying the rule. Thus, 
 
 Find the values of the following : 
 14. V(V2). 17. ^'(-v/27^). 20. -v/CSVS). 
 
 15. V(Vl25). 18. V(Va + 6). 21. V(Va^2/^). 
 
 16. V(V32). 19. V(W-2a^+l). 22. V(4V2). 
 
 TO REDUCE A FRACTION HAVING AN IRRATIONAL 
 
 DENOMINATOR TO AN EQUIVALENT FRACTION 
 
 WHOSE DENOMINATOR IS RATIONAL. 
 
 Case I. 
 
 243. When the denominator is a monomial. 
 
 The reduction is effected by multiplying both terms by a 
 radical of the same degree as the denominator, having such 
 a quantity under the radical sign as will make the denomi- 
 nator of the resulting fraction rational. 
 
RADICALS. 201 
 
 5 
 1. Reduce — ; to an equivalent fraction with a rational 
 
 denominator. 
 
 Multiplying both terms by V9a, 
 
 5 oV9a 5V9a SVOo^ ^ 
 
 -, Ans, 
 
 -v^Sa^ -^Sa^-v/ya ^/27d' 3a 
 
 EXAMPLES. 
 
 Reduce the following to equivalent fractions with rational 
 denominators : 
 
 2. A. 4. -^. 6. ^ 
 
 V2 Voa^ Vl6ar* 
 
 -L. 5.-4=. 7. -1^ 
 
 ■v/4 ■v/9a^ •v/27a2 
 
 Case II. 
 
 244. Wlien the denominator is a binomial, containing rad- 
 icals of the second degree only. 
 
 ■y/^ ■y/2 
 
 1. Reduce to an equivalent fraction with a 
 
 rational denominator. 
 
 Multipl3'ing both terms by Vo — V2, 
 
 V5-V2 ^ (V5_V2)2 ^ 5-2VlO + 2 
 
 V^+V^ (V^+V^)(V^-V2) 5-2 
 
 7-2V1O . 
 
 = , A71S. 
 
 \ -|_-v/l X 
 
 2. Reduce -—^ — ' to an equivalent fraction witli a 
 l-Vl-x 
 rational denominator. 
 
202 ALGEBRA. 
 
 Multiplying both terms by 1 -|-Vl —x^ 
 
 1+Vl-a? ^ (1-fVl-a;)' 
 
 1-Vl-a; (1 - Vr^) (1 + Vl - a;) 
 
 l + 2Vl-a;-f-l-a; 
 
 2-.+ 2Vl-.^ ^^^^^ 
 
 RULE. 
 
 Multiply both terms of the fraction by the denominator with 
 the sign between its terms changed. 
 
 EXAMPLES. 
 
 Reduce the following to equivalent fractions with rational 
 denominators : 
 
 3. _1_. 7. 2V54-V2 
 
 3+V2 ' V5-3V'2 
 
 4 ^ — V3 g g — Va ;_ 
 
 2-V3 * a+Va; 
 
 g V2-V3 g -Va-hl-2 
 V2+V3 ' Vo+T-l 
 
 11. 
 
 a_Va'-l 
 
 a + Va^-l 
 
 12 
 
 a? -)- Var' — 4 
 
 
 a;_Va^-4 
 
 13. 
 
 Va+a;+Va— a: 
 
 Va-f aj— Va— a; 
 
 14. 
 
 Va2_i_Va^4-l 
 
 Q Va+V?> iQ Va.-4-2— Va; 
 
 Va— V6 Va; + 2-j-Va; Va" — 1 + Va-+I 
 
 245. The approximate value of a fraction, whose denomi- 
 nator is irrational, may be most conveniently found by 
 reducing it to an equivalent fraction with a rational denomi- 
 nator. 
 
RADICALS. 203 
 
 1. Find the approximate value of to three places 
 
 of decimals. ~ ^ 
 
 1 2+V2 
 
 2-V2 (2-v2)(2+V2) 
 
 ^ 2+V2 
 4-2 
 
 2 + 1.414... , ^^7 J 
 
 = — - — 2 = l./07..., Ans. 
 
 2 
 
 EXAMPLES. 
 
 Find the approximate values of the following to three 
 decimal places : 
 
 g 3 3.^. 4 V3-V2 g 2V5-V3 
 
 y/2-1 sy9 V3 + V2 3Vo + 2v3 
 
 IMAGINARY QUANTITIES. 
 
 246. An Imaginary Quantity is an indicated even root of 
 a negative quantity (Art. 201) ; as, V — 4, or V — a^. 
 
 In contradistinction, all other quantities, rational or irra- 
 tional, are called real quantities. 
 
 247. Every imaginary square root can be expressed as 
 the product of a real quantity multiplied b}' V— 1. Thus, 
 
 V — a^ = Va- X (— 1) = Vo^'x V— 1 =aV— 1 ; 
 
 V— 5 =V5 x(-l) = V5V-l; etc. 
 
 248. Let it be required to find the powers of V— 1. 
 
 By Art. 198, V— 1 signifies a quantity which, when 
 multiplied by itself, will produce — 1 ; that is. 
 
204 
 
 ALGEBKA. 
 
 Therefore, 
 
 (V^y=(V-irx (V^)^=(-l) X (-i) = l ; 
 (V^)'=(V^)^X V^ = 1 xV^=V^=o:; etc. 
 
 Thus the first four powers of V — 1 are V — 1, — 1 , — V — 1, 
 and 1 ; and for higher powers these terms recur in the same 
 order. 
 
 MULTIPLICATION OF IMAGINARY QUANTITIES. 
 
 249. The product of two or more imaginary square roots 
 may be found by aid of the principles of Arts. 247 and 248. 
 
 1. Multiply V^^ by V^^. 
 By Art. 247, 
 
 V^xV^3 = V2V^=TxV3V^ 
 
 = V2V3(V^^)^ 
 
 = V6x(-l) (Art. 248) = -V6, A71S. 
 
 2. Multiply V— a-, V— &-, and V— c^. 
 
 V — a^ X V^^^ xV— c^ = aV — 1x6 V— 1 X cV— 1 
 
 = a&c(V — 1)^=— a&cV— 1, Ans. 
 
 RULE. 
 
 Reduce each imaginary quantity to the form of a real quan- 
 tity multiplied by V— 1. Form the product of the real quan- 
 tities, and multiply the result by the required power of V— 1 . 
 
RADICALS. 
 
 205 
 
 EXAMPLES. 
 
 Multiply the following 
 
 3. 4V^^ aiid2V^^. 
 
 4. \A^ and V^^. 
 
 6. V^, V-4, and V^^. 
 
 7. l_2V^and 3-f-V^^. 
 
 5. -3V^eand4V^r5. 8. 4-f V-7 and 8-2 V-7. 
 9. 2V^-3V^and4V^^ + 6V=^. 
 
 10. V^, V-i>, V-16, and V-25. 
 
 Expand the following : 
 
 11. (2-V^=^)^ 
 
 12. (V^r3 + 2V^^)^ 
 
 15. {x^ —x-\-y 
 
 13. (l + V3T)(i_V^). 
 
 14. (a-fV^^)(a-V^^). 
 2/)(a;V^a-yV^). 
 
 
 16. 
 
 (1+V- 
 
 -1)^+(1-- 
 
 V-i)^ 
 
 17. 
 
 Divide V— a; 
 
 't>y V-2/. 
 
 
 
 
 V-x 
 
 
 ^x 
 
 
 V-2/ 
 
 V;/ 
 
 Note. The rule of Art. 241 would have given the same result; 
 hence, that rule applies to the division of all radicals, whether real or 
 imaginary. 
 
 Divide the following 
 
 18. V"^ by V^. 
 
 19. V-24 bv V^^. 
 
 20. V^^^l by V^, 
 
 21. -V^^^ bv \/^^. 
 
206 ALGEBRA. 
 
 PROPERTIES OF QUADRATIC SURDS. 
 
 250. A Quadratic Surd is the indicated square root of an 
 imperfect square ; as, -^3, or -^7. 
 
 251. A quadratic surd cannot be equal to a rational quan- 
 tity plus a quadratic surd. 
 
 For, if possible, let ^a = h + V^* 
 
 Squaring the equation, a = 6^ + 2 6 ^c + c. 
 
 Or, 2b^c=:a-b^ — c. 
 
 iTTi / a — b^ — c 
 Whence, V^' = ^T 
 
 ii 
 
 That is, a surd equal to a rational quantity, which is 
 impossible. Hence -^ya cannot be equal to 6 + y/c. 
 
 252. To prove that if a-^- ^b = c -\--^d, then a = c, and 
 ^b = ^d. 
 
 If a is not equal to c, let a = c + x. Substituting, we have 
 
 c + x-\-^b = c + ^d. 
 
 Or, ic+V6 = v(^, 
 
 which is impossible by Art. 251. Hence a = c, and conse- 
 quently -\/b = -y/d. 
 
 253. To prove that if Va -\- y/b = wx -f- Vi/, then Va — V6 
 
 Squaring the equation Va + Vb = Va? + Vy, 
 
 we have a + V6 = x -{-2 ^xy -f- ?/. 
 
 Whence, by Art. 252, a = a; + ?/, (1) 
 
 and • V6 = 2V^. (2) 
 
 Subtracting (2) from (1) , a— V6 = aj — 2 Va;?/ + y. 
 
 Extracting the square root, Va — y/b— Va; — V?/. 
 
RADICALS. 207 
 
 SQUARE ROOT OF A BINOMIAL SURD. 
 
 254. The preceding principles serve to extract the square 
 root of a binomial surd whose first term is rations^. 
 
 For example, required the square root of 13 — ^160. 
 
 Assume Vl3 — \/160 = Vx — V?/. (1) 
 
 Then, by Art. 253, Vl3-f- VIOO = Vic+ V^/. (2) 
 
 Multiplying (1) by (2), V169-160 = a;-2/. 
 Or, x-y=Z, (3) 
 
 Squaring (1), 13- Vl60 = a;- 2 Vajy + y. 
 
 Whence, by Art. 252, ic + y = 13. (4) 
 
 Adding (3) and (4), 2ar= 16, oriF = 8. 
 
 Subti-actiug (3) from (4), 2?/=10, ov y = b. 
 
 Substituting in (1), Vl3-V1G0== V8 -V5 
 
 = 2V2-V5, Ans, 
 
 255. Examples like the above may often be solved by 
 inspection by expressing the given quantity in the form of a 
 perfect trinomial square (Art. 108), as follows : 
 
 Reduce the surd teinn so that its coefficient may be 2. Sepa- 
 rate the rational term into two parts whose product is the 
 quantity under the radical sign. Extract the square roots of 
 these parts, and connect them by the sign of the surd term, 
 
 1. Extract the square root of 8 -|- V48. 
 
 V8 + V48 = V8 + 2V12. 
 
 We then separate 8 into two parts whose product is 12. 
 The parts are 6 and 2 ; hence. 
 
 ■V8 4-2^12 = V 6 + 2^6x2 + 2 
 = V6 + V2, Ans. 
 
208 ALGEBRA, 
 
 2. Extract the square root of 22 — 3V32. 
 
 V22 - 3 V32 = V22 - V28S = V22 - 2 V72. 
 
 We then separate 22 into two parts whose product is 72. 
 The parts are 18 and 4 ; hence, 
 
 V22 - 3 V32 = Vl8 - 2 V72 + 4 
 
 = Vl8-V4 = 3V2-2, Ans. 
 
 EXAMPLES. 
 256. Extract the square roots of the following : 
 
 1. 12-I-2V35. 6. 8-V60. 11. 23+V360. 
 
 2. 7-2V12. 7. 15 + 4V14. 12. 24-2V63. 
 
 3. 9 + 2V8- 8- 12-V108. 13. 33 + 20V2. 
 
 4. 9-4V5. 9. 20-5V12. 14. 47-6V10. 
 
 5. 16 + 6V7. 10. 14 + 3V20. 15. 67-7V72. 
 
 16. 2m-2Vm2-n2. 17. 2 a + a? + 2 vV + a^. 
 
 SOLUTION OF EQUATIONS CONTAINING RADICALS. 
 
 257. 1 . Solve the equation Va^ — 5 — a; = — 1 . 
 
 Transposing, Va^ — 5 = a? — 1 . 
 
 Squaring both members, a^ — 5 = a:^— 2a;-f-l< 
 Transposing and uniting terms, 2x= 6. 
 
 a? = 3, ^ws. 
 
RADICALS. 209 
 
 2. Solve the equation V2x— 1 -f V2^ + 6 = 7. 
 
 Transposing V2a; — 1, 
 
 V2a;-f 6 = 7— V2a;-1. 
 
 Squaring, 2a; + 6 = 49-14V2a;-l+2a;-l. 
 
 Transposing and uniting, 
 
 14V2a;-l = 42. 
 Or, V2a;-1=3. 
 
 Squaring, 2 a; — 1 = 9. 
 
 2a;=10. 
 a; = 5, -4ns. 
 
 RULE. 
 
 Transpose the terms of the equation so that a radical teimti 
 may stand alone in one member; then raise both members to 
 a power of the same degree as the radical. 
 
 If there are still radical teimis remaining^ repeat the operor- 
 tion. 
 
 Note. The equation should be simplified as much as possible before 
 performing the involution. 
 
 EXAMPLES. 
 
 3. V5^^-2 = l. 8. ^/x''-6x'-x-\-2 = 0. 
 
 4. 6=i/2~x-h3r. 9. Va;+Va; + 5 = 5. 
 
 5. V4a; + 3 = 3. 10. Va;- 32 -f Va;= 16. 
 
 6. V4a;2-19-2a;= -1. 11. V^^^ - ■\^x-\-V2 = -3. 
 
 7. Va;2-3a;H-6 = 2-a;. 12. V2a;-7 + V2a; + 9 = 8. 
 
210 ALGEBRA. 
 
 13. V3a; -h 10 -V3a.' + 25 = -3. 
 
 14. ^ {X — ay -{- 2ab -\- b^ = X — a + b. 
 
 15. ■Vx^-3x + 5-^x''-5x-2 = l. 
 
 16. Vo? - \/^"=^3 = — . 
 
 17. Vx-l+Vx + 4=V4ir + 5. 
 
 18. Var^ + 4iK + 12 +Var^- 12.^-20 = 8, 
 
 19 ^^ — ^ _ Va? — 4 
 
 ' ' V^+'^ V^ + 1 
 
 20. V3^4-V3.'c + 13 = 
 
 91 
 
 ■VSx+ 13 
 
 21. Va;+14-Va;-2-V4a;-3=0. 
 
 22. Va; + V^r+a= ^^ . 
 
 V.T + a 
 
 23. V^9 + a;Var'-3j=:a;-3. 
 
 *4- — = = = \b — X. 
 
 Va — X ■\^b — X 
 
 25. V£c4-a4-Va; + 6 = V4a7 + (^. + 36. 
 
 26. VSl+a;Va^+16; = a^ + l. 
 
 27. Vjct2-2aa; + a;V3a-a;H«-^- 
 
QUADRATIC EQUATIONS. 211 
 
 XXI. QUADRATIC EQUATIONS. 
 
 258. A Quadratic Equation, or an equation of the second 
 degree (Art. 1G7), is one in whicti the square is the highest 
 power of the unknown quantity. 
 
 A Pure Quadratic Equation is one which contains onl}- the 
 square of the unknown quantit}' ; as, ax^ = b. 
 
 An Affected Quadratic Equation is one which contains 
 both the square and the first power of the unknown quan- 
 tity ; as, aa^ -{-bx-j-c = 0. 
 
 PURE QUADRATIC EQUATIONS. 
 
 259. A pure quadratic equation is solved by reducing it 
 
 to the form 3? = a^ and then extracting the square roots of 
 both members. 
 
 1 . Solve the equation 3 ar* + 7 = \- 35. 
 
 4 
 
 Clearing of fractions, 12ar^+ 28 = b3?-\- 140. 
 
 Transposing and uniting, 7ar^= 112. 
 
 Or, a^=lG. 
 
 Taking the square root of both members, 
 
 a; = ± 4, Ans. 
 
 Note 1. The double sign is placed before the result because the 
 square root of a number is either positive or negative (Art. 201). 
 
 2. Solve the equation 7a^-5 = 5a.-2- 13. 
 Transposing and uniting, 2a;^ = — 8. 
 
 Or, ar^ = - 4. 
 
 Whence, a; = ± V — 4 
 
 = ±2V^, Ans. 
 
 Note 2. Since the square root of a negative quantity is imaginary 
 (Art. 246), the values of x can only be indicated. 
 
212 ALGEBRA. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 3. 40^2-7 = 29. . 6. 4-V3aj2+16 = 6. 
 
 g_5 L = _5?. 8 ^ 8 5 
 
 6a^ 4a^ 16 4-x 3 4 + a; 
 
 9 2{x-\-3){x-S) = {x-]-iy-2x. 
 
 10. (3a;-2)(2a; + 5) + (5a;+l)(4a;-3)-91=0. 
 
 11. ^_3_^l^' = ^_ic2 , 335^ 
 
 2 12 24 24 
 
 12 2a;2-5 3a^4-2 a^-lQ ^Q 
 '3 7 6 
 
 13 ^ ^ _J: 14 4a;2-3 ^ 2(9a^ + 2) 
 
 15. {2x-a){x-h) + {2x^a){x-\-h) = a' + b~. 
 
 16 5ar-l Scc^ + l 89 ^^ 
 
 ^_3 i^_f_2 (a^-3)(a^+2) 
 
 17. x + V^+3 = 
 
 18. 
 
 Va^ + 3 
 1 1 V3 
 
 1-Vl-a;"^ I + VT^T^ a^ 
 
 AFFECTED QUADRATIC EQUATIONS. 
 
 260. An affected quadratic equation is solved by adding 
 to both members such a quantity as will make the first mem- 
 ber a perfect square ; au operation which is termed complet- 
 ing the square. 
 
QUADKATIC EQUATIONS. 213 
 
 FIRST METHOD OF COMPLETING THE SQUARE. 
 
 261. Every affected quadratic equation can be reduced to 
 
 the form 
 
 x^ ^px = q ; 
 
 where p and q represent any quantities whatever, positive or 
 negative, integral or fractional. 
 
 Let it be required to solve the equation a^ + 3a; = 4. 
 
 In any trinomial square (Art. 108), the middle term is 
 twice the product of the square roots of the first and tliird 
 terms ; hence the square root of the third term is equal to 
 the second term divided by twice the square root of the first. 
 
 Therefore the square root of the quantity which must be 
 
 3x 3 
 added to a^ + 3a; to make it a perfect square, is — , or — . 
 
 At X Ji 
 
 3 9 
 Adding to both members the square of - , or -, we have 
 
 ic2 + 3a; + i = 4 + - = — . 
 4 4 4 
 
 Extracting the square root of both members, 
 
 2 2 
 
 Transposing -, x = h -, or 
 
 ^ '=' 2' 2^2 2 2 
 
 Whence, x—\ or —4, Ans. 
 
 262. From the above operation we derive the following 
 rule ; 
 
 Reduce the equa^ioh to the form a^ -{-px = q. 
 Complete the square by adding to both members the square 
 of half the coefficient of x. 
 
 Extract the square root of both members^ and solve the sim- 
 ple equation thus formed. 
 
214 ALGEBRA. 
 
 1. Solve the equation Sxr — 8x= — 4:. 
 Dividing by 3, o^ _ ^' = _ i, 
 
 which is in the form aP -\-px = q. 
 
 4 16 
 Adding to both members the square of -, or — , 
 
 ^_8xl6^_416^4 
 ^ 39 39 9* 
 
 Extracting the square root, 
 
 X = ± — 
 
 3 3 
 
 Whence, x = - ±- = 2 or-, Ans. 
 
 3 3 3 
 
 Note. These values may be verified as follows : 
 Putting a: = 2 in the given equation, 12 — 16 = — 4. 
 
 Putting x = ^, -__ = _4. 
 
 If the coefficient of x^ is negative, it is necessary to change 
 the sign of each term. 
 
 2. Solve the equation —3x^—7x = — 
 
 49 
 36' 
 
 Dividing by - 
 
 -3, x^^'f = 
 
 10 
 9* 
 
 
 
 Adding to both members the 
 
 square 
 
 "'I- 
 
 or 
 
 
 ^ 3 ^36 
 
 10 
 9 ■ 
 
 -i= 
 
 _ 9 
 '36 
 
 Extracting the square root, 
 
 
 
 
 
 . + 1 = 
 
 -1- 
 
 
 
 Whence, aj= ±- = or , Ans. 
 
 6 6 3 3 
 
QUADRATIC EQUATIONS. 215 
 
 EXAMPLES. 
 Solve the following equations : 
 
 3. ar'-f4a; = 5. 8. 2a:r^ -\-ox = -2. 
 
 4. x^-5x = -4:, 9. 4a^-8x + 3 = 0. 
 6. ic2-7x = -12. 10. 4a^-3=lla;. 
 
 6. a^ + x=Q. 11. 3-a;-2ar = 0. 
 
 7. 3a.'2-4a; = 4. 12. 14 + loic- 9a;2^ 0. 
 
 263. If the coefficient of a:^ is a perfect square, it is con- 
 venient to complete the square directly by the principle of 
 Art. 261 ; that is, by adding to both members the square oj 
 the quotient obtained by dividing the second term by twice tlie 
 square root of the first. 
 
 1. Solve the equation 9 a^ — 5 a; = 4. 
 
 The quotient of the second term divided by twice the 
 
 square root of the first, is -. Adding the square of - to 
 both members, 
 
 9^'-5.x. + 25 = 4 + ?^ = l^^. 
 36 36 36 
 
 Extracting the square root, 
 
 6 6 
 
 Q 5 . 13 o 4 
 3 a; = - ± — = 3 or 
 
 6 6 3 
 
 4 
 
 Whence, a; = l or , Ans. 
 
 9 
 
 Note. If the coefficient of x"^ is not a perfect square, it may be 
 made so by multiplication. 
 
 Thus, in the equation IS x"^ -\- 6 x = 2, the coefficient of x"^ may be 
 made a perfect square by multiplying each term by 2. 
 
 If the coefficient of x- is negative, the sign of each term must be 
 changed. 
 
216 ALGEBRA. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 2. 4a^ + 3a!=10. 7. 8a^ + a;-34 = 0. 
 
 3. da^-\-2x=n. 8. llic+12-36a^ = 0. 
 
 4. 25x'-15x = -2. 9. 6a.-2-5x = -l. 
 
 5. 4.a^-7x=-3. 10. 32ic2 + 20a;- 7 = 0. 
 
 6. 2ar + 15aj = -13. 11. 48a;2_32a;= 3. 
 
 SECOND METHOD OF COMPLETING THE SQUARE. 
 
 264. Every affected quadratic can be reduced to the form 
 
 aa^ -\~bx = c. 
 
 Multiplying each term by 4 a, we have 
 
 4 aV 4- 4 abx = 4 ac. 
 
 Completing the square by adding to both members the 
 square of b (Art. 263), 
 
 4aV ^4abx + ft^ = 52 _^ 4 a<j. 
 Extracting the square root, 
 
 2ax + b = ± V6^ + 4ac. 
 
 Transposing, 2ax = — b± VF+4ac. 
 
 Dividms by 2 a, ic= — — 
 
 ^ -^ 2a 
 
 265. From the above operation we derive the following 
 rule : 
 
 Reduce the equation to the form a a? -\-bx = c. 
 
 Multiply both members by four times the coefficient of a^, 
 and add to each the square of the coefficient of x in the giveii 
 equation. 
 
 Extract the square root of both members^ and solve the sim- 
 ple equation thus formed. 
 
QUADRATIC EQUATIONS. 217 
 
 Note. The advantage of this method over the preceding is in avoid- 
 ing fractions in completing tlie square. 
 
 1. Solve the equation 2 a^ — 7ic = — 3. 
 Multiplying both members by 4 times 2, or 8, 
 
 Adding to each member the square of 7, 
 
 16 0^2 _ 5g ^ ^ 49 ^ _ 24 _^. 49 = 25. 
 
 Extracting the square root, 
 
 4a;-7 = ±5. 
 Transposing, 4a;=7±5 = 12or2. 
 
 Dividing by 4, a; = 3 or -, Aiis. 
 
 If the coefficient of a; in the given equation is an even 
 number, fractions may be avoided, and the rule modified, as 
 follows : 
 
 Mnltiply both members by the coefficient of a^, and add to 
 eojch the square of half the coefficient of x in the given equa- 
 tion. 
 
 2. Solve the equation 7ar'-|-4a;= 51. 
 Multiplying both members by 7, 
 
 49a.-2-|-28a; = 357. 
 Adding to each member the square of 2, 
 
 49ar^+28a; + 4 = 361. 
 Extracting the square root, 
 
 7a;4-2 = ±19. 
 
 7a;=-2±19 = 17or -21. 
 
 17 
 Whence, x = — or —3, Ans. 
 
218 ALGEBRA. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 3. 2x^-\-5x==3. 10. 17a; + 20=-3ar'. 
 
 4. 4:0^-x = 3. 11. 5a;2-3 = 14aj. 
 
 5. a^-3a;=18. 12. 2 + a;-Gar^ = 0. 
 
 6. 3a;2 + 4a; = 4. 13. 8a;- + 6aj + 1 = 0. 
 
 7. 8x' + 2x = 3. 14. 7a; + 3 = 60.-2. 
 
 8. 2a;2_7^^j5 15 15a^-8a; = -l. 
 
 9. 7a;2-16£c + 4 =0. 16. 41a;- 14 - 15a^ = 0. 
 
 MISCELLANEOUS EXAMPLES. 
 
 266. The following equations may be solved by either of 
 the preceding methods ; preference being given to the one 
 best adapted to the example considered. 
 
 1. 
 2. 
 
 X 
 
 5. (x-\-5){x-5)'-(nx-{-l) = 0. 
 
 6. 4x{lSx-l) = (10x-iy. 
 
 7. (3a; -5)2 -(a; + 2)2 = -5. 
 
 8. (x + 3y-{x-iy=ld. 
 
 9. (^x-iy-(3x + 8y--(2x + 5y = 0. 
 
 ^Q 2^±3_2^+9^^^ ^2 4a;-li^^=14. 
 
 8 + a;3a; + 4 a;+l 
 
 11 5 3a; + l ^l ^g 21 a; ^25 
 
 ■ a; x" A ' 6-x 1~ 7* 
 
 2^3^24 
 
 3 1 _ 7 _2 
 
 2a; 6a;2 g' 
 
 2 a; 5 
 a; 2 2 
 
 4 2 5 _ 15 
 
 5 2a; 4a.'2 
 
QUADRATIC EQUATIOXS, 219 
 
 14 3x^ ]-8x ^x ^.^ x+1 .T 4- 3 ^ 8 
 
 ' x-7 10 5' ' x-\-2 x-^4: 3' 
 
 X x — 1 3 
 
 15. -^ ±IZ±==± 18. V20 + a;-ar' = 2«-10. 
 
 x-l X 2 
 
 16. -^ ^^=1^. 19. 2V^+— =5. 
 
 5 — a; a; 4 V^ 
 
 20. ^-^^^ 30^+1 = 0. 
 
 a; 'dx— 1 2 
 
 ^^2^_ 11 
 00 2x^-{-3x — b 20^2 _ 2; _i 
 
 23. 
 
 3ar^ + 4a;-l 3x2_2a.^7 
 
 7 3_^22 
 
 0^2 — 4 x + 2 5 * 
 
 84. -JL- + -U 1 + 1 
 
 x"'-! 3 3(a;-l) a; + l 
 
 25 ^ + 3 a;-3 ^ 2a;-3 
 a;4-2 x— 2 £c— 1 
 
 ofi 124-oa; , 2 + a; 1 
 
 12 — 5a; x \ — bx 
 
 27. V4a;-3-Va;+l = l. 
 3 
 
 28. 2 Va; = V.X -f 5 + 
 29. 
 
 Vx + 5 
 a; + 2 2a;+16 a;-2 
 
 a: — 1 X -\- 5 x-{-l 
 
 30. V3a;-}-l + V2a;- 1 = V9a; + 4. 
 
220 ALGEBRA. 
 
 The same methods are applicable to the solution of literal 
 quadratic equations. 
 
 31. Solve the equation a;^ — 2 7nx = 2m -{-1. 
 Completing the square b}' adding m^ to both members, 
 
 X- — 2 mx + ?ji^ = in- -\-2m-\-l= (m + 1 )^. 
 Extracting the square root, 
 
 x — m = ± (m + 1). 
 Whence, x = m-\-{m-\-\)^ oy m—{m-\-\) 
 
 = 2m -f 1 or — 1, Ans. 
 
 32. Solve the equation a? -\-ax — hx — ab = 0. 
 The equation may be written, 
 
 ix? + {a — h)x = ah. 
 Multiplying both members by 4 times the coefficient of oj^, 
 
 4a^ + 4(a-6)x' = 4a6. 
 Adding to each member the square of a — 6, 
 4.x'-\-4.{a-h)x+{a-hy = {a-hy + 4.ah 
 
 = a?-\-2ah-\-h\ 
 Extracting the square root, 
 
 2x-\-{a-h) = ±{a-\-h). 
 Whence, 2x = -{a-h)±{a + h). ■ 
 
 Hence, 2x = — a-\-h-{-a-{-h = 2h^ 
 
 or, 2x = — a-\-h — a — h = — 2a. 
 
 Dividing by 2, a; = — a or 6, Ans, 
 
 Note. If several terms contain the same power of x, the coefficient 
 of that power should be placed in a parenthesis, as shown in Ex. 32. 
 
 Solve the following equations : 
 
 33. x^-2ax={b-{-a){h—a). 
 
 34. 9? — ax -\-hx = ah. 
 36. ar^ — (a+l)a; = — a. 
 
QUADRATIC EQUATIONS. 221 
 
 36. a^-f2(c + 8)a; = -32c. 
 
 37. 0? — 77i^( 1 — m) X = m^. 
 
 38. ac^ — hex — adx = — bd. 
 
 39. (a; + 2p)^ = (a;-hi^)' + 37i)3. 
 
 40. 6x^-\-dax-{-2bx = — 3ab, 
 
 ^^ 2x{a-x)^a^ 43 ^^l^a_^6, 
 
 3a — 2a; 4 x b a 
 
 42. — ^ = _^. 44. ^±i = fL±J. 
 
 " a; + 1 7?i H- 1 * -y/x -y/a 
 
 46. V(a-H6)a; — 4a6 = a; — 2 6. 
 
 46. Vi34^= <" + ^>("-''^ - 
 
 47. 2V^^=:^ + 3V2^=^^^±^- 
 
 Vaj — m 
 
 48. ^^ + ^ =!+?• 
 
 a + Va^ — a; a— Vo^^-^ ^ 
 
 49. Vo; + a + V« + 2 a = V2 a; + 3 a. 
 
 50. 
 
 a?-\-l a + h 
 
 x c a-\-b 
 
 a-\-b -{-X a b x 
 
 52. x^ + bx + cx = {a-^c){a-b). 
 
 53. a5ar^ + ^ = ^^' + ^^-^^'-^. 
 
 c c^ c 
 
 64. (3tt2-f- 62) (a;2 _ ^. ^ 1) ^ ^3^0 ^ ^^o^^^^ _^^_^ ^y 
 
222 ALGEBRA. 
 
 SOLUTION OF QUADRATIC EQUATIONS BY A FORMULA. 
 267. It was shown ia Art. 264 that if aa? -{-hx = c^ then 
 
 6±V// + 4ac^ .^. 
 
 2a 
 
 This result may be used as a formula for the solution of 
 quadratic equations, as follows : 
 
 1. Solve the equation 2a/'^-f-5fl;=18. 
 
 In this case, a = 2, 6 = 5, c=18; substituting these 
 values in (1), 
 
 ^ ^ ~ 5 ± V25 + 144 ^ - 5 ± Vl69 
 
 4 
 
 5 ±13 
 
 = 2 or , Ans. 
 
 4 2' 
 
 2. Solve the equation IIOj? -2lx=z-l. 
 
 In this case, a = 110, & = — 21, c = — 1 ; therefore, 
 
 ^^21±V44rr44Q_21±i^_l ^^ l_ ^^^^^ 
 220 220 10 11 
 
 Note. Particular attention must be paid to the signs of the coeffi- 
 cients in substituting. 
 
 EXAMPLES. 
 
 Solve the following equations : 
 
 3. 2x^-^bx=lS. 8. 5a52-llir = -2. 
 
 4. 3ic2-2a; = 5. 9. 4a;2-8a;-5 = 0. 
 
 5. x'-lx = -lO. 10. 6a^ + 25a; + 14 = 0. 
 
 6. 5a^ + a; = 18. 11. 30ic-16 = 9a^. 
 
 7. 6a^+7a; = -l. 12. 27 + 39a;- 10aj2= 0. 
 
QUADRATIC EQUATIONS. 223 
 
 ZXII. PROBLEMS. 
 
 INVOLVING QUADRATIC EQUATIONS. 
 
 268. 1. A man sold a watch for $21, and lost as much 
 per cent as the watch cost him. Required the cost of the 
 watch. 
 
 Let X = the cost in dollars. 
 
 Then, x = the loss per cent, 
 
 and X X -^ = -^ = the loss in dollars. 
 
 100 100 
 
 By the conditions, = x — 21. 
 
 ^ ' 100 
 
 Solving this equation, a: = 70 or 30. 
 
 That is, the cost of the watch was either $70 or $30; for each 
 of these values satisfies the given conditions. 
 
 2. A farmer bought some sheep for $72. If he had 
 bought G more for the same money, they would have cost 
 $ 1 apiece less. How many did he buy ? 
 
 Let X = the number bought. 
 
 72 
 Then, — = the price paid for one, 
 
 X 
 
 72 
 
 and = the price if there had been 6 more. 
 
 ar + 6 
 
 72 72 
 
 By the conditions, — = -^-=^ + 1. 
 
 X x + 6 
 
 Solving, X = 18 or — 24. 
 
 Only the positive value of x is admissible, as the negative value 
 does not answer to the conditions of the problem. The number of 
 sheep, therefore, was 18. 
 
 Note 1. In solving problems which involve quadratics, there will 
 always be two values of the unknown quantity ; but only those values 
 should be retained as answers which satisfy the conditions of the prob- 
 lem. 
 
224 ALGEBRA. 
 
 Note 2. If, in the given problem, the words " 6 more " had been 
 changed to " 6 fewer," and " $ 1 apiece less " to " l$ 1 apiece mwe" we 
 should have found the answer 24. 
 
 In many cases where the solution of a problem gives a negative 
 result, the wording may be changed so as to form an analogous prob- 
 lem to which the absolute value of the negative result is an answer. 
 
 PROBLEMS. 
 
 3. I bought a lot of flour for S175 ; and the number of 
 dollars per barrel was \ of the number of barrels. How 
 many barrels were purchased, and at what price? 
 
 4. Separate the number 15 into two parts the sum of 
 whose squares shall be 117. 
 
 5. Find two numbers whose product is 126, and quotient 
 
 6. I have a rectangular field of corn containing 0250 
 hills. The number of hills in the length exceeds the num- 
 ber in the breadth by 75. How many hills are there in the 
 length, and in the breadth? 
 
 7. Find two numbers whose difference is 9, and whose 
 sum multiplied by the greater isi 266. 
 
 8. The sum of the squares of two consecutive numbers i* 
 113. What are the numbers? 
 
 9. A man cut two piles of wood, whose united content? 
 were 26 cords, for $35.60. The labor on each cost as manj 
 dimes per cord as there were cords in the pile. Required 
 the number of cords in each pile. 
 
 10. Find two numbers whose sum is 8, and the sum of 
 whose cubes is 152. 
 
 11. Find three consecutive numbers su-ch that twice the 
 product of the first and third is equal to the square of the 
 second increased bv 62. 
 
PROBLEMS. 225 
 
 12. A grazier bought a certain number of oxen for $ 240. 
 Having lost 3, be sold the remainder at $8 a head more than 
 they cost him, and gained $59. How many did he bu}? 
 
 13. A merchant bought a quantity of flour for $96. If 
 he had bought 8 barrels more for the same money, he would 
 have paid $2 less per barrel. How many barrels did he 
 buy, and at what price? 
 
 14. Find two numbers, whose product is 78, such that if 
 one be divided l)y the other the quotient is 2, and the 
 remainder 1. 
 
 15. The plate of a rectangular looking-glass is 18 inches 
 by 12. It is to be framed with a frame all parts of which 
 are of the same width, and whose area is equal to that of 
 the glass. Required the width of the frame. 
 
 16. A merchant sold a quantity of flour for $39, and 
 gained as much per cent as the flour cost him. What was 
 thie cost of the flour? 
 
 17. A certain company agreed to build a vessel for 
 $ 6300 ; but, two of their number having died, the rest had 
 each to advance $ 200 more than tliey otherwise would have 
 done. Of how many persons did the company consist at 
 first? 
 
 18. Divide the number 24 into two parts, such that the 
 sum of the fractions obtained by dividing 24 by them shall 
 bef|. 
 
 19. A detachment from an army was marching in regular 
 column, with 6 men more in depth than in front. When the 
 enemy came in sight, the front was increased by 870 men, 
 and the whole was thus drawn up in 4 lines. Required the 
 number of men. 
 
 20. A merchant sold goods for $16, and lost as much per 
 cent as the goods cost him. What was the cost of the 
 goods ? 
 
226 ALGEBRA. 
 
 21. A certain farm is a rectangle, whose length is twice 
 its breadth. If it should be enlarged 20 rods in length, and 
 24 rods in breadth, its area would be doubled. Of how 
 many acres does the farm consist? 
 
 22. A square court-yard has a gravel-walk around it. 
 The side^of the coui-t lacks one 3'ard of being 6 times the 
 breadth of the walk, and the number of square yards in the 
 walk exceeds the number of yards in the perimeter of the 
 court by 340. Find the area of the court and the width of 
 the walk. 
 
 23. A merchant bought 54 bushels of wheat, and a cer- 
 tain quantity of barley. For the former he gave half as 
 many dimes per bushel as there were bushels of barley, and 
 for the latter 40 cents a bushel less. He sold the mixture 
 at $1 per bushel, and lost $57.60 b}^ the operation. Re- 
 quired the quantity of barley, and its price per bushel. 
 
 24. A certain number consists of two digits, the left- 
 hand digit being twice the right-hand. If the digits are 
 iuA^erted, the product of the number thus formed, increased 
 by 11, and the original number, is 4956. Find the number. 
 
 25. A cistern can be filled by two pipes running together 
 in 2 hours 55 minutes. The larger pipe by itself will fill it 
 sooner than the smaller by 2 hours. What time will each 
 pipe separately take to fill it? 
 
 26. A and B gained by trade $1800. A's money was in 
 the firm 12 months, and he received for his principal and 
 gain $2600. B's money, which was $3000, was in the firm 
 16 months. How much money did A put into the firm? 
 
 27. My gross income is $1000. After deducting a per- 
 centage for income tax, and then a percentage, less by one 
 than that of the income tax, from the remainder, the income 
 is reduced to $912. Find the rate per cent of the income 
 tax. 
 
PROBLEMS. 227 
 
 28. A man travelled 102 miles. If he had gone 3 miles 
 more an hour, he would have performed the journe}' in 5|^ 
 hours less time. How many miles an hour did he go? 
 
 29. The number of square inches in the surface of a 
 cubical block exceeds the number of inches in the sum of its 
 edges by 210. What is its volume? 
 
 30. A man has two square lots of unequal size, contain- 
 ing together 15,025 square feet. If the lots were contigu- 
 ous, it would require 530 feet of fence to embrace them in a 
 single enclosure of six sides. Required the area of each 
 lot. 
 
 31. A set out from C towards D at the rate of 3 miles an 
 hour. After he had gone 28 miles, B set out from D towards 
 C, and went every hour -^^ of the entire distance ; and after 
 he had travelled as many hours as he went miles in an hour, 
 he met A. Required the distance from C to D. 
 
 32. A courier proceeds from P to Q in 14 hours. A sec- 
 ond courier starts at tlie same time from a place 10 miles 
 behind P, and arrives at Q at the same time as the first 
 courier. The second courier finds that he takes half an hour 
 less than the first to accomplish 20 miles. Find the dis- 
 tance from P to Q. 
 
 33. A person bought a number of $20 mining-shares 
 when they were at a certain rate per cent discount for 
 $ 1500 ; and afterwards, when they were at the same rate per 
 cent premium, sold them all but 60 for $1000. How many 
 did he buy, and what did he give for each of them? 
 
228 ALGEBKA. 
 
 XXIII. EQUATIONS IN THE QUADRATIC 
 FORM. 
 
 269. An equation is in tiie quadratic form when it is 
 expressed in three terms, two of whicli contain the unknown 
 quantity ; and of these two, one has an exponerit twice as 
 great as the other; as, 
 
 a;«-6ar^=16; 
 
 aj3 -I- x^ = 72 ; 
 (a;2- 1)2 + 3 (0.-2-1) =18; etc. 
 
 270. The rules for the solution of quadratics are applica- 
 ble to equations having the same form. 
 
 1. Solve the equation x^— 6x^=16. 
 Completing the square, 
 
 aj«-6a^ + 9 = 16 + 9 = 25. 
 
 Extracting the square root, 
 
 a^-3 = ±5. 
 Whence, a;3^3 ^ 5^ 8 ^^j, _2. 
 
 Extracting the cube root, a;= 2 or —^2, Ans. 
 
 Note. There are also four imaginary roots, which may be obtained 
 by the method explained in Art. 282. 
 
 2. Solve the equation 2 a; + 3 -^x =27. 
 
 Since ^x is the same as x^, this is in the quadratic form. 
 Multiplying by 8, and adding 3^ or 9 to both members, 
 
 16a; + 24 Va; H- 9 = 216 + 9 = 225. 
 Extracting the square root, 
 
 4Va? + 3 = ±15. 
 Or, 4Va^ = -3±15 = 12or -18. 
 
QUADRATIC EQUATIONS. 229 
 
 9 
 
 Whence, V^= ^ ^^ 
 
 z 
 
 81 
 Squaring, . a; = 9 or — , Ans, 
 
 3. Solve the equation 16a;"' - 22a;"^ = 3. 
 Multiplying by 16, and adding IP to both members, 
 
 162a;- ' - 16 X 22a;"4 _^ 121 = 48 + 121 = 169. 
 Extracting the square root, 
 
 16a;"^-ll = ±13. 
 Or, 16a;~^ = 11 ± 13 = - 2 or 24. 
 
 _ J 1 g 
 
 Whence, a; * = or -• 
 
 8 2 
 
 Extracting the cube root, 
 a 
 Raising to the fourth power. 
 
 X * = or (- ) ■ 
 
 2 Uy 
 
 i=-i-or ''^^* 
 16 
 
 Inverting both members, a; = 16 or (-) » Ans, 
 
 V 
 
 Note. In solving equations of the form .r? = o, first extract the 
 root corresponding to the numerator, and afterwards raise to the power 
 corresponding to the denominator. Particular attention should be paid 
 to the algebraic signs ; see Arts 192 and 201. 
 
 EXAMPLES. 
 Solve the following equutious : 
 
 4. a;*-25ar^ = -144. 7. a;-*- 9a;-2 = _ 20. 
 
 5. a/'+20ar^-69 = 0. 8. %\y?^-\ = %'2. 
 
 ar 
 
 6. a;i« + 31ar''-32 = 0. 9. 8a;«- 216 = 37.rl 
 
230 ALGEBRA. 
 
 10. {3x^-2y-n{3x'-2) + 10 = 0. 
 
 11. (af^-o)- = 241-29a^. 
 
 12. af'-x^=d6. 17. 2x~^+61x~^ -96 = 0. 
 
 13. x'^-{-x^=7d6. 18. 4x- 15 = 17 V^J- 
 
 2 4 in V4x4- 2 4 ^ 
 
 14. 2a..+ 3.-n-56 = 0. 19. ^^^-^=—-1. 
 
 15. 3xt+^t=.3104. 20. 3a.t_5|!^_592. 
 
 16. 3x' + 26i»^ = -16. 21. 8x~'^-l5x~^-2 = 0. 
 
 271. An equation may be solved with reference to an 
 expression, by regarding it as a single quantity. 
 
 1. Solve the equation (x — 5)^ — 3{x~ 5) ^ = 40. 
 
 Eegarding x — 5 as a single quantity, we complete the 
 square in the usual way. Multiplying by 4, and adding 9 to 
 both members, 
 
 4(a;_5)3-12(x-o)^ + 9=160 + 9 = 169. 
 Extracting the square root, 
 
 2(05-5)2-3 = ±13. 
 Or, 2(a;-5)^ = 3±13 = 16 or -10. 
 
 Whence, {x — 5) ^ = 8 or — 5. 
 
 Extracting the cube root, 
 
 (a;-5)^ = 2 or --^5. ^ 
 
 Squaring, a; — 5 = 4 or ^25. 
 
 Transposing, a; = 9 or 5 -f^/25, ^n5. 
 
QUADRATIC EQUATIONS. 231 
 
 An equation of the fourth degree may sometimes be 
 solved by expressing it in the quadratic form. 
 
 2. Solve the equation a^ -i-l2x^ + S4:X^— V2x — Sb = 0. 
 We may write the equation as follows : 
 
 (ic^+ 12ar'^ + 36ar)-2ar-12a; = 35. 
 Or, (ar + Gx)- -2{x^-\- 6x) = 35. 
 
 Completing the square, 
 
 (xF + 6xy-2{x'-\-Gx) + l = 36. 
 Extracting the square root, (ar^ 4- 6x) — 1 = ± G. 
 Whence, ar^ + Ga; =1±6= 7 or— 5. 
 
 Completing the square, a:^ + 6a; + 9 = IG or 4. 
 
 Extracting the square root, 
 
 a; + 3= ±4or ±2. 
 Whence, a;=— 3±4or — 3±2 
 
 = 1, — 7, —1, or —5, Ans. 
 
 Note. In solving equations like the above, tlie first step is to form 
 a perfect square with the x* and x^ terms, and a portion of the x"^ 
 term. By Art. 2G1, the third term of the square is the square of the 
 quotient obtained by dividing the x^ term by twice the square root of 
 the X* term. 
 
 3. Solve the equation 2oir -\-^2x^-\- 1 = 11. 
 Adding 1 to both members. 
 
 (2a:2^1)^y2a.'2 + l = 12. 
 Completing the square, 
 
 4(2a:2-fl) + 4V2ar^4-l+l=48 4-l=49. 
 Extracting the square root, 
 
 2V2ar'+l + 1 = ± 7. 
 
 Or, 2V2ar^+l = -1 ±7 = 6 or -8. 
 
 Whence, V2.t2 + 1 = 3 or — 4. 
 
232 ALGEBRA. 
 
 Squaring, 2a^ + 1 = 9 or 16. 
 
 2a^ = 8orl5. 
 
 ar = 4 or — 
 2 
 
 Exti-acting the square root, a? = ± 2 or ± - V^^5 ^^s- 
 
 Note. In solving equations of this form, add such quantities to 
 both members tliat the expression without the radical in the first mem- 
 ber may be the same as that within, or some multiple of it. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 5. cc*-fl0a;^+17ar'-40a;-84 = 0. 
 
 6. ic2 - 10a; -2vV-10^+T8+ 15 = 0. 
 
 7. a;2 + 5+va^ + 5 = 12. 
 
 8. 2a;2+3a;-5V2#+3£c+9 = -3. 
 
 9. a;* + 2a^-25a;2-26a;-f 120 = 0. 
 
 10. a;*-6a^-29a^+114a; = 80. 
 
 11. a^ - 6a; + 5V^^^6^T20 = 46. 
 
 12. Va;+10--J/a;+10=2. 
 
 13. 4a;2^gy4^^i2a;-2 = -3-12a;. 
 
 14. (a;3+16)^-3(a;^+16)* + 2 = 0. 
 
 15. 4(a;-l)^-o(a;-l)^ + l=0. 
 
 16. a;*+14ar'^ + 47a;2_i4^_43^0. 
 
 17. 3(a.'2-f-5a')-2V^M-5^-hl=2. 
 
 18. (a;-a)^-}-2V&(«-«)^-3& = 0. 
 
SIMULTANEOUS EQUATIONS. 233 
 
 XXIV. SIMULTANEOUS EQUATIONS. 
 
 INVOLVING QUADRATICS. 
 
 272. The degi*ee of an equation containing more than one 
 unknown quantit}^ is determined by the greatest sum of the 
 exponents of the unknown quantities in any term. Thus, 
 
 2x-j-3xy = 4: is an equation of the second degree. 
 
 a^ — ayh = ab^ is an equation of the third degree. 
 
 Note. This* definition assumes that the equation has been cleared 
 of fractions, and freed from radical signs and fractional and negative 
 exponents. 
 
 273. Two equations of the second degree with two un- 
 known quantities will generally produce, by elimination, an 
 equation of the fourth degree with one unknown quantity. 
 The rules for quadratics are, therefore, not suflScient to 
 solve all simultaneous equations of the second degree. 
 
 In several cases, however, the solution may be effected by 
 the ordinary rules. 
 
 Case I. 
 
 274. When one equation is of the first degree. 
 
 Equations of this kind may always be solved by finding 
 the value of one of the unknown quantities in terms of the 
 other from the simple equation, and substituting the result 
 in the other equation. 
 
 1. Solve the equations | 2ar^-^2/ = 6y. (1) 
 
 ( x^2y=l, (2) 
 
 From (2) , 2y = 'J-x,ovy = ^^. (3) 
 
 Substituting in (1) , 2a^ - ^0-^^^ = 6 /^^~^\ 
 
234 ALGEBRA. 
 
 Clearing of fractions, 4,x^ — 7 x -\- scr = i2 — Gx. 
 Or, 5a^ — a; = 42. 
 
 Solving this equation, a? = 3 or — — . 
 
 5 
 
 . , 14 
 Substituting in (3) , y = -L--^ or 1 
 
 = 2 or — 
 10 
 
 ^ws. a; = 3, 2/ = 2; or, a; = , ?/ = — • 
 
 5 ' -^ 10 
 
 EXAMPLES. 
 Solve the following equations : 
 
 (8a; + 
 
 O 1 M/ -t- V = — 1 
 
 -32/' = -10. r^.^ = 4 
 
 ^^- = 1. 
 ^x y 
 
 I xy = — b&. 
 
 10 I «^'H- 2/^ = 152. 
 4 jaj -.y = 3. * U +1/ = 2. 
 
 1 0^4-2/2= 117. 
 
 j^ ( 3a^-2a??/=15. 
 g I 10a; + ?/ = 3ir2/. • ' X'lx +^y =12. 
 
 1 x — y = — '1 
 
 — 6xy. ■ ^ zx -\-6y 
 
 12. f8a^-2/^ = -7. 
 a;3_^ = _37. [2aj -y =-1. 
 
 6. 
 
 a; -2/ =- 1 
 
 7. f«-.v = 5. 
 laj2/ = -6. 
 
 8 f ^ +2/ = 3. 
 
 jg {a? + ^xy-y'^ = 2^. 
 
 ( a? +2?/ = 7. 
 
SIMULTANEOUS EQUATIONS. 235 
 
 Case II. 
 
 275. When the equations are symmetrical with respect to x 
 and y. 
 
 Note 1. An equation is symmetrical with respect to two quantities 
 when they can be interchanged without destroying the equality. 
 
 Thus, x^ — xi/ + y^ = S is symmetrical, for on interchanging x and _y 
 it becomes y^ ^ i/x + x"^ = 3, which is equivalent to the first equation. 
 But x — y = l is not symmetrical, for on interchanging x and y it 
 becomes y —x= 1, which is a different equation. 
 
 In solving equations by the symmetrical method, they 
 must be combined in such a wa^^ as to give the values of the 
 sum and difference of the unknown quantities. 
 
 xy = -16. (2) 
 
 Squaring (1 ) , x' + 2xy -\-y' = 4.. (3) 
 
 Multiplying (2) by 4, 4xy = - 60. (4) 
 
 Subtracting (4) from (3), x^ — 2xy + y' = 64.. 
 Extracting the square root, x — y = ±8. (5) 
 
 Adding (1) and (5), 2a; = 2±8=10 or -6. 
 
 AVhence, x= 5 or — 3. 
 
 Subtracting (5) from (1), 2y = 2 :f 8 = - 6 or 10. 
 Whence, 2/ = — 3 or 5. 
 
 Ans. ic=5, ?/ = — 3; or, x = — 3, y = b. 
 
 Note 2. The signs ± and qr before two quantities signify that when 
 the first quantity is +, the second is — ; and when the first is — , the 
 second is + . Thus, in the above solution, when2r = 2 + 8, 2y = 2— 8; 
 and when 2a: = 2 — 8, 2y = 2 + 8. That is, when x = 5, 3/ = — 3 ; and 
 when a: = — 3, y = 5. 
 
 In the operation, the sign ± is changed to =p whenever -}- would be 
 changed to — , 
 
 Note 3. The above equations may also be solved as in Case I. ; but 
 the symmetrical method is shorter, and more elegant. 
 
236 ALGEBRA. 
 
 ra;2 + / = o0. (1) 
 
 2. Solve the equations < 
 
 Ix -y = S. (2) 
 
 Squaring (2), 3^ -2xy + y^ = 64. (3) 
 
 Subtracting (3) from (1), 2xy = -U. (4) 
 
 Adding (1) and (4), a? + 2xy -{-y'^ = 36. 
 
 Whence, x-^y = ±6. (5) 
 
 Adding (2) and (5), 2x = 8±6 = 14 or 2. 
 
 Whence, x= 7 or 1 . 
 
 Subtracting (2) from (5) , 2?/ = - 8 ± 6 = - 2 or - 14. 
 
 Whence, y = —lov—7, 
 Ans. ic = 7, y = —l; or, x = l, y = — 7. 
 
 Note 4. The symmetrical method may often be used in cases like 
 the above, where the equations are symmetrical except in the signs of 
 the terms. 
 
 3. Solve the equations -< „ 
 
 (of — 
 
 aj3 + 2/3=133. (1) 
 
 xy-hf= 19. (2) 
 
 Dividing (1 ) by (2) , x-\-y=7. (3) 
 
 Squaring, x^ -\- 2 xy -^ y~ = 49 . (4) 
 
 Subtracting (4) from (2), —Sxy = — 30. 
 
 Or, -xy = -10. (5) 
 
 Adding (2) and (5) , x'-2xy-{-7f = 9. 
 
 Whence, x-y=±S. (6) 
 
 Adding (3) and (6), 2a;= 7±3 =10 or 4. 
 
 Whence, x= 5 or 2. 
 
 Subtracting (6) from (3), 2?/= 7 q: 3 =4 or 10. 
 
 Whence, y = 2 or 5. 
 
 Ans. cc = 5, 2/ = 2; or, x = 2^ y = 5. 
 
SIMULTANEOUS EQUATIONS. 237 
 
 EXAMPLES. 
 Solve the following equations : 
 
 lxy=-6. 
 
 12. ^ ^2 
 
 xy = 60. 
 
 g {x-y= 6. .3 (a^ + f=So, 
 
 ' (a^ + / = 90. ■ Xxy = A2. 
 
 6. j^-2/ = -10. 14. \^-f = 
 
 (ajv = — 21. (a;— w = 
 
 (a^ — fl;2/ + 2/^=19. ' la; +?/ = — 5. 
 
 g^ (x^ + y'=25. ^Q (a; + 2/ = 12. 
 la;2/ = 12. * \xy = — i5, 
 
 \x'-\-f' = 5S. ' \x'-\-xy-\-f- = 13. 
 
 (a;— 2/= 2. 'la;— 2/= — 5. 
 
 U -hy =3. * la; -j-i/ =- 2. 
 
 Case III. 
 
 276. WJien each equation is of the second degree, and 
 homogeneous (Art. 35). 
 
 Note. Certain examples, in which the equations are of the second 
 degree and homogeneous, may be solved by the method of Case II. 
 The method of Case III. should be used only when the equations can 
 be solved in no other way. 
 
238 ALGEBRA. 
 
 c a^ — 2xy = 5. 
 1. Solve the equations < „ 
 
 Putting y=vx in the given equations, we have 
 
 of — 2vx^= 5; or, 0;- = (1) 
 
 x2_j_^2^=3 29; or, a;^^ ^9 
 
 Equating the values of x^, 
 
 5 . 29 
 
 1-2?; l+'y2 
 5 + 5^2=29-58^. 
 
 5v2 + 58'y = 24. 
 
 2 
 
 Solving this equation, v = - or — 12. 
 
 5 
 
 Substituting these values in (1 ) , x^ = _ or 
 
 ~5 
 
 = 25 or i. 
 5 
 
 Whence, £c=±5 or ± 
 
 Substituting the values of v and x in the equation y = vx, 
 , = |(±5)or-12(±^)=±2or^i|. 
 
 Arts. x = ±5, y=±2; 
 or, a;==±-V5, y = ::f—.^o. 
 
 Note. In finding y from the equation y = vx, care must be taken to 
 multiply each pair of values of x by the corresponding value of v. 
 
SIMULTANEOUS EQUATIONS. 239 
 
 EXAMPLES. 
 Solve the following equations : 
 
 xy= 12. 
 xy-y^= 2. 
 
 \xy-\-y'=\S. ' \xy — 
 
 „ {2o? + xy = lb. y {2if-4:Xy-\-^x^ = ri. 
 
 ' X x^-f= 8. • \f 
 
 ar=lG. 
 
 
 10 1^ + ^- 2/'=l- 
 lar-a;2/H-2?/- = 8. 
 
 "■{ 
 
 4:Xy — x^= 5. 
 
 13ic2 - 31 a;// + 16?/- = 2^^. 
 
 MISCELLANEOUS EXAMPLES. 
 
 277. No general rules can be given for the solution of 
 examples which do not come under the cases just consid- 
 ered. Various artifices are employed, familiarity with which 
 can only be obtained by experience. 
 
 ( x'-f = ld. (1) 
 
 1. Solve the equations i „ „ 
 
 ^ \x^y-xf= 6. (2) 
 
 Multiplying (2) by 3, Sx^y -3xy' = 18. (3) 
 
 Subtracting (3) from (1), 
 
 x^-nx'y + Sxy^-f^ 1. 
 
 Extracting the cube root, x — y= 1. (4) 
 
 Dividing (2) by (4), xy = 6. (5) 
 
 Equations (4) and (5) may now be solved by the method 
 of Case II. We shall find x=d or — 2, and 2/ = 2 or — 3. 
 Ans. X = 3, y = 2 ; or, a; = — 2, y = — 3. 
 
240 ALGEBRA. 
 
 X 
 
 of -^y^'z= 9 xy. 
 -\-y =G. 
 
 Putting x = u-\-v and y = u — v, we have 
 
 (u -\- vy-h (u.- vy = 9 (u-j-v) (u-v) . (1) 
 (u-\-v) -j-{u-v) =6. (2) 
 
 Reducing (2) , 2u = 6, or u = S. 
 
 Reducing ( 1 ) , 2u^ + 6 uv^ = d{u-- v^) . 
 
 Substituting the value of ?i, 
 
 Whence, -y^ = 1 , or 'y = ± 1 . 
 
 Therefore, x = u-\-v = S±l = 4:or2, 
 
 y=zu — v = S^l = 2 or 4:. 
 
 Ans. £c = 4, 2/ = 2 ; or, ic = 2, ?/ = 4. 
 
 Note. The artifice of substituting u -\- v and u — v for x and i/ is 
 advantageous in any case where the given equations are symmetrical. 
 
 3. Solve the equations 
 
 ra-^ + / + 2a^ + 22/ = 23. (1) 
 
 I xy= Q. (2) 
 
 Multiplying (2) by 2, 2xy=12. (3) 
 
 Adding (1) and (3), 
 
 x^-[-2xy-\-y^-\-2x-}-2y = 3D. 
 Or, (x-^yy-{-2{x-^y)=3o. 
 
 Completing the square, 
 
 {x-\-yy-\-2(x-^y)-hl=36. 
 Whence, (x-^y) -{-1 = ± 6, 
 
 x-{-y = — l±G = 5or—7. (4) 
 
 Squaring (4), x^ -{-2xy -{-y^ = 2ry or 49. (5) 
 
 Multiplying (2) by 4, Axy = 2i. (6) 
 
 Subtracting (6) from (5), 
 
 x^ — 2xy-{-y^=l or 25. 
 
SIMULTANEOUS EQUATIONS. 241 
 
 Wheuce, x — y=±loY±D. (7) 
 
 Adding (4) and (7), 2a;= 5 ± 1 or -7 ± 5, 
 
 a; =3, 2, —1, or — 6. 
 Subtracting (7) from (4), 2?/ = 5 q: 1, or — 7t5, 
 
 2/ = 2, 3, -6, or -1. 
 Ans. a;=3, 2/ = 2; x=2^ y = S ; 
 
 a: = -l, 2/ = -6; ov, x = -6, y = -l. 
 
 cai*-\-y*= 97. 
 4. Solve the equations < 
 
 Putting x = u -^v and y = ii — v, we have, 
 
 {u-\-vy-^{u-vy= 97. (1) 
 
 (u-^v) -h{u-v) =-1. (2) 
 
 Reducing (2) , 2 w = — 1 , or ?* = — -• 
 
 Reducing (1), 2w* + 12i^V + 2'y*= 97. 
 Substituting the value of w, 
 
 i-h3'y2 + 2t'^= 97. 
 8 
 
 25 31 
 
 Solving this equation, 1/^ = — , or • 
 
 4 4 
 
 Whence, vrzii-, or ± 
 
 2 2 
 
 ,^, » , 1 _i_ ^ 1 j_ V— 31 
 Therefore, x — u-\-v — ±- or ± 
 
 2 2 2 2 
 
 2 or — 3 or — ; 
 
 , 15 1 V-31 
 
 and y=iu — v = if- or if 
 
 2 2 2 2 
 
 -3 or 20.-1^^-^1. 
 
242 
 
 ALGEBRA. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 ^ (xy-2x = 
 lxy-\-3y = - 
 
 = 5. 
 2. 
 
 6. 1^ + 2/ = 
 
 y ( 4x2_3^: 
 
 = 9. 
 
 2 = -ll. 
 2- 301. 
 
 \a^y + xif = 30. 
 
 9 U a; 10 
 [3x-2y = 4:, 
 
 10. j «'•' + /= 5m2. 
 la; —2/ = m. 
 
 * la;i/=6. 
 
 * l2a;t/ + 2/' = -3. 
 
 13 f a^ + 2/^=18a;2/. 
 la; 4-2/ =12. 
 
 la;2/ + 42/-= 30. 
 
 15. 
 
 ^1+1=11. 
 a; 2/ 
 
 1 = 18. 
 
 xy 
 
 16. f«^-2/=«-&- 
 
 ^y (x' + y' = 9-x, 
 la;2-2/2=6. 
 
 
 18. ^ 
 
 2/^ 
 
 1_1 
 
 ^ y 
 
 11. 
 
 18. 
 19. 
 
 19. j ^H-2/^-a;-2/ = 
 1 a;^ + a; + 2/ = 
 
 20*1^ + ^^ +2/2= 7. 
 U^ + a;y + 2/'=133. 
 
 22. 1-^ + 2/^=17- 
 ( X —y = 3. 
 
 23. |^-^ = 7a'. 
 (a; —2/ = a. 
 
 Divide the second equation by the first. 
 
24 
 
 SIMULTANEOUS EQUATIONS. 243 
 
 (x +x^y^-\-y = 19. g^ { x-\-y = S(a-b). 
 Xx'-^xy +^=133. * lxy = '2a'-oab-\'2b\ 
 
 \xy-\-y^=2a{a + b). ' {x +y = 3 
 
 x^y -\-xy^ = 30. 23 | .t/ +y = 1. 
 
 a^y^-\-afy* = iQ8. ' Xa^y^-\-y''=5, 
 
 27. 
 28. 
 
 (a^-f = 6a'b + 2b^ rx-\-z=7. 
 
 \xy(x-y)=2o?b-2b\ 34. ^ 2?/- 32 = - 5. 
 
 ^ ^ (.0^ + 2/^-2^=11. 
 
 (a^+3a;+2/= 73-2a:^. 
 
 ^ ^ l3x?/-a^ +62/2=44. 
 
 x-_^4V^^33^ r^-2/_^+2/^3^ 
 
 2^ V2/ 4 3g J a; ^ 2/ ic - 2/ 2* 
 
 30 |a^-if2/ + 2/'=19. 3- j ar' + 2/2= 7 +a^. 
 ' l2a^-2/' = -17. * la^+2/«=6a;2/-l. 
 
 PROBLEMS. 
 
 278. Note. In the following problems, as in those of Chap. XXII., 
 only those answers are to be retained whicli satisfy the conditions of 
 the problem. 
 
 1. The sum of the squares of two numbers is 106, and 
 the difference of their squares is f the square of their differ- 
 ence. Find the numbers. 
 
 2. What two numbers are those whose difference multi- 
 plied by the less produces 42, and by their sum, 133? 
 
 3. The sum of the areas of two square fields is 1300 
 square rods, and it requires 200 rods of fence to enclose 
 both What are the areas of the fields? 
 
244 ALGEBRA. 
 
 4. The difference of the squares of two numbers is 7, 
 and the product of their squares is 144. Find the numbers. 
 
 5. If the length of a rectangular field were increased by 
 2 rods, and its breadth by 3 rods, its area would be 108 
 square rods ; and if its length were diminished by 2 rods, 
 and its breadth by 3 rods, its area would be 24 square rods. 
 Find the length and breadth of the field. 
 
 6. The sum of the cubes of two numbers is 407, and the 
 sum of their squares exceeds their product by 37. Find 
 the numbers. 
 
 7. A man bought 6 ducks and 2 turkeys for $15. He 
 bought four more ducks for $14 than turkeys for $9. 
 What was the price of each? 
 
 8. Find a number of two figures, such that if its digits 
 are inverted, the sum of the number thus formed, and the 
 original number, is 33, and their product is 252. 
 
 9. The sum of the terms of a fraction is 8. If 1 is 
 added to each term, the product of the resulting fraction 
 and the original fraction is |. Required the fraction. 
 
 10. A rectangular garden is surrounded by a walk 7 feet 
 wide; the area of the garden is 15,000 square feet, and of 
 the walk 3696 square feet. Find the length and breadth 
 of the garden. 
 
 11. A rectangular field contains 160 square rods. If its 
 length be increased by 4 rods, and its breadth by 3 rods, its 
 area is increased by 100 square rods. Find the length and 
 breadth of the field. 
 
 12. A man rows down stream 12 miles in 4 hours less 
 time than it takes him to return. Should he row at twice 
 his ordinary rate, his rate down stream would be 10 miles an 
 hour. Find his rate in still water, and the rate of tlie 
 stream. 
 
SIMULTANEOUS EQUATIONS. 245 
 
 13. A and B bought a farm of 104 acres, for which they 
 paid $3^0 each. On dividing the land, A said to B, "If 
 you will let me have my portion in the situation which I 
 shall choose, you shall have so much more land than I, that 
 mine shall cost $3 an acre more than yours." B accepted 
 the proposal. How much did each have, and at what price 
 per acre? 
 
 14. If the product of two numbers be added to their sum, 
 the result is 47 ; and the sum of their squares exceeds their 
 sum by 62. Find the numbers. 
 
 Note. Let the numbers be represented hy x -\- y and x — y. 
 
 ^» 
 
 15. The sum of two numbers is 7, and the sum of their 
 fourth powers is 641. Required the numbers. 
 
 16. The fore- wheel of a carriage makes 15 more revolu- 
 tions than the hind-wheel in going 180 yards ; but if the 
 circumference of each wheel were increased by 3 feet, the 
 fore-wheel would make only 9 more revolutions than the 
 hind- wheel in the same distance. Find the circumference 
 of each wheel. 
 
 17. A man has $1300, which he divides into two por- 
 tions, and loans at different rates of interest, so that the two 
 portions produce equal returns. If the first portion had 
 been loaned at the second rate, it would have produced $36 ; 
 and if the second portion had been loaned at the first rate, 
 it would have produced $49. Ijiequired the rates of interest. 
 
 18. Cloth, when wetted, shrinks \ in its length and ^ in 
 its width. If the surface of a piece of cloth is diminished 
 by 5 J square yards, and the length of the four sides by 4 J 
 yards, what were the length and width of the cloth origi- 
 nally ? 
 
246 ALGEBRA. 
 
 XXV. THEORY OF QUADRATIC EQtJA- 
 TIONS. 
 
 279. Denoting the roots of the equation a^ +px = q by Vy 
 and r2, we have (Art. 267), 
 
 2 ' ' 2 
 
 Adding these values, 
 
 -2p 
 
 Multiplying them together, 
 
 ^,^, = ^ZlO^±i£) (Art. 95) = ^iS = _ 5. 
 4 4 
 
 That is, z/ a quadratic equation be reduced to the form 
 o?-\-px = q^ the algebraic sum of the roots is equal to the 
 coefficient of x with its sign changed, and the product of the 
 roots is equal to the second member, with its sign changed. 
 
 Example. Required the sum and product of the roots of 
 the equation 2a^ — 7a; — 15 = 0. 
 
 The equation may be written in the form 
 
 ^_7^^15 
 2 2 
 
 7 15 
 Whence, the sum of the roots is -, and their product is 
 
 280. The principles of Art. 279 may be used to form a 
 quadratic equation which shall have an}- required roots. 
 
 For, denoting the roots of the equation a;^ + j)a; — ^ = by 
 Ti and rg, we have, by the preceding article, 
 
 i> = - (n + ^2) , and - g = r^r^. 
 
THEORY OF QUADRATIC EQUATIONS. 247 
 
 We may therefore write the equation in the form 
 a^ — (ri + 7-2)0; 4- ?v'2 = 0, 
 or, a? — riX — r^-\- r{i\ = 0- 
 
 That is (Art. 105) , {x- r{) {x - r^) = 0. 
 
 Hence, to form an equation which shall have any required 
 roots. 
 
 Subtract each of the roots from x, and place the product of 
 the resulting expressions equal to zero, 
 
 7 
 
 Example. Form the equation whose roots are 4 and . 
 
 4 
 
 By the rule, (x — 4) /^a; -f - ] = 0. 
 
 Multiplying by 4, {x — 4) (4a; + 7) = 0. 
 
 Or, 40?*- 9 a; -28 = 0, Ans. 
 
 EXAMPLES. 
 
 281. Find by inspection the sum and product of the roots 
 of : 
 
 1. a;2_|_5a; + 2 = 0. 6. Ha? -x-\-4: = {). 
 
 2. x'-lx + n = 0. 6. 6a;-4a;2_^3^Q 
 
 3. o?-\-Q>x-l = 0. 7. 7-12a;-14a;2^0. 
 
 4. 27?-Zx-2 = 0. 8. A.x^-'iax + a^-b'^z=0. 
 Form the equations whose roots are : 
 
 9. 4,5. 11. 3, -?. 
 5 
 
 "I-!- 
 
 15. -^,-1. 
 3' 2 
 
 10. 1, -3. 12. 7, -i^. 
 
 
 »■ -If 
 
 16. -n,o. 
 
 17. a — 6, a + 26. 
 
 19. 2 + V3 
 
 , 2-V3. 
 
 18. m(l+m),m(l-m). 20. ?2d:V» mWn. 
 
248 ALGEBRA. 
 
 282. By Art. 280, the equation a:r-+-px — q = may be 
 written in the form {x — Vj) {x — ^-g) = 0, where rj and /g are 
 its roots. 
 
 It will be observed that the roots may be obtained by 
 placing the factors of the first member separately equal to 
 zero, and solving the simple equations thus formed. 
 
 This principle is often used in solving equations : 
 
 1 . Solve the equation (2 aj - 3) (3 a; + 5) = 0. 
 Placing the factors separately equal to zero, 
 
 2a;-3 = 0, ora; = 5; 
 
 2 
 
 and 3 ic + 5 = 0, or a; = 
 
 o 
 
 Ans, x — -,ov — . 
 2 3 
 
 2. Solve the equation aj^— 5a^— 24ic=0. 
 Factoring the first member, x{x — 8) (a? + 3) = 0. 
 Therefore, a; = ; 
 
 aj — 8 = 0, ora; = 8; 
 and a; 4- 3 = 0, or a; = — 3. 
 
 Ans. a; = 0, 8, or —3. 
 
 ' 3. Solve the equation a^ + 4a^ — a; — 4 = 0. 
 
 Factoring the first member (Art. 105), 
 
 {x + ^){a?-\)z=^. 
 Therefore, a;-f4 = 0, ora; = — 4; 
 
 and ar^ — 1 = 0, ora7=±l. 
 
 Ans. a? = — 4 or ± 1 . 
 
THEORY OF QUADRATIC EQUATIONS. 249 
 
 4. Solve the equation or^ — 1 = 0. 
 
 Factoring the first member, 
 
 Therefore, a; — 1 = 0, or a* = 1 ; 
 
 and a:r-\-x-\-l=0. (1) 
 
 2 
 
 Solving (1) by the rules for quadratics, x = 
 
 A71S, a; = 1 or 
 
 EXAMPLES. 
 Solve the following equations : 
 
 5. A._|Yaj_|_I'\ = o. 10. 2a^-18x = 0. 
 
 6. 2xr-x = 0. 11. (3x-hl){4a?-25) = 0. 
 
 7. {ax-\-b){bx-a) = 0. 12. Sic^-h 12ar = 0. 
 
 8. (x2-4)(a^-9) = 0. 13. {x^-a-){x'-ax-b) = 0, 
 
 9. (a;-2)(a^-f-9a;+20) = 0. 14. 24:a^^2x'--12x = 0. 
 
 15. x{2x-^5){^x-7){-ix-{-l) = 0. 
 
 16. (a.-2 - 5a; + 6) (a^ 4- 7x -f- 12) (a;^^ 3^ _ 4) ^ q^ 
 
 17. a^ + l=0.. 19. x^-x'-dx-{-d = 0, 
 
 18. a«-l=0. 20. 2a;3 + 3a;2_2a;_3==o. 
 
 Note. Tlie above examples are illustrations of the important prin- 
 ciple that, the degree of an equation indicates the number of its roots ; 
 thus, an equation of the third degree has three roots ; of the fourth 
 degree, four roots ; etc. 
 
 It should be observed that the roots are not necessarily unequal; 
 thus, the equation z^ — 2x + l = may be written (x — l)(a: — 1) = 0, 
 and therefore the two roots are 1 and 1. 
 
250 ALGEBRA. 
 
 FACTORING. 
 
 283. A Quadratic Expression is a trinomial expression of 
 the form ax^-\- bx -\- c. The principles of the preceding articles 
 serve to resolve any such expression into two simple factors. 
 
 The expression ax^ + bx + c may be written 
 
 \ a a) 
 
 bx c 
 But, by Art. 280, aj^ + — + - = (a; - rj) (a; - rg) , 
 
 where Tj and rg are the roots of the equation, 
 
 bx c 
 
 a^H (-- = 0, or ax"^ + bx-\-c = 0; 
 
 a a 
 
 which, it will be observed, may be formed l\y placing the 
 given expression equal to zero. Hence, 
 
 axP -\- bx -j- c = a(x — Vi) (x — r^ . 
 
 1. Factor 6a^ + 11a; -f- 3. 
 Placing the expression equal to 0, 
 
 6a^+lla; + 3 = 0. 
 Solving, as in Art. 267, 
 
 -11 ±Vl21-72 -11±7 1 3 
 
 X = = — = or • 
 
 12 12 3 2 
 
 1 3 
 
 Then, a = 6, ri = --, r2 = - -• 
 
 Hence, 6a;2-f lla;-}-3 = 6(^a; + iYa; + -) 
 
 3 
 
 = (3a;+l)(2a; + 3), Avs, 
 
 ^ + 2 
 
-13±Vl69-hl92 
 
 .-13±19_ 14. 
 
 -24 
 
 ■ -24 - 4^'3 
 
 4:-\-13x-l2x^ = 
 
 -'i-*'i)i'-t) 
 
 = 
 
 K-i)(-)ei) 
 
 THEORY OF QUADRATIC EQUATIONS. 251 
 2. Factor 4 + 130; -12a.'2. 
 Solving the equation 4 + 13a; — 12x^ = 0, we have 
 
 Hence, 
 
 = (1 +4x)(4-3a;), Ans. 
 
 Note. It must be remembered, in using the formula a{x — r^){x — r^), 
 that a represents the coeflScient of x^ in the given expression ; thus, in 
 Ex. 2, we have a = — 12. 
 
 EXAMPLES. 
 Factor the following : 
 
 3. ar^ + 13a; + 40. 16. 70^ + 50a; + 7. 
 
 4. .T^- 11 a; 4- 18. 17. 6ar^- 13aa;- 15a«. 
 6. a^-4tx-60. 18. 5 + 4a;-12a2 
 
 6. 2ar^-7a;-15. 19. dx'-Ux + l, 
 
 7. 4::x^-15x-\-9. 20. 12:t? -7xy -10y\ 
 
 8. ^x' + SGx + l. 21. Sa^-hlSxy-df, 
 
 9. 4:x'-{-15x-A. 22. 10ic2_ 23^;+ q. 
 
 10. 39-10a;-ar'. 23. 20a.-2 4- 41 ma;-}- 20 m^ 
 
 11. 2-\~x-Gx^. 24. Ux'-SAx-^-U. 
 
 12. x'-4.x-{-l. 25. l-8a;-a;2^ 
 
 13. 9a;2_g^_4^ 26. 1562-|- 266a;- 24a:2 
 
 14. 8a;2_ig^_,_9^ 27. 21 a;2 + 58 mwa;-|- 21 mV 
 
 15. 6-a;-2a;2. 28. 25a;2_20flj_ 2. 
 
252 ALGEBRA. 
 
 284. Many expressions may be factored by the artifice of 
 completing the square, in connection with Art. 111. 
 
 1. FsictoY a'^ + a^b^-{-b\ 
 
 By Art. 108, the expression will become a perfect square 
 if the middle term is 2a^b^. Hence, 
 
 a^ 4- a'b' + 6* = (a^ + 2 d'b' + b') - a'b' 
 = {d' + by-a'b' 
 
 = {a^ + b' + ab){a' -i-b' -ab), (Art. Ill) 
 = (a2 + a6 + &')(a'-«& + <^'), Ans. 
 
 2. Factor 9 a;* -39 0)2 + 25. 
 
 9ic^ - 39 a^ H- 25 = (9a^ - 30a^ + 25) - 9a^ 
 • =(3ic2-5)2_9aj2 
 
 = (3a^ + 3x-5)(3a^-3a;-5), Ans, 
 
 3. Factor aj^ — ar^ + l. 
 
 a^ _ 0^ + 1 = (£c4 4- 2a?2 + 1) - 30^2 
 = (a^+l)2_(^y3)2 
 
 = (a^-{-x^S-^l){x'-x^S-\-l), Ans. 
 
 EXAMPLES. 
 Factor the following : 
 
 4. a^ + a^ + i. 12. a*-5aV-fa^. 
 
 5. x' -7x^+1. 13. x^+1. 
 
 6. 4a*-8a252 + 5^ 14. 4a' + 15 a^b"" -\- 16 b\ 
 
 7. m'-Um-n^-^n\ 15. 16 x' - 49 m^a^ -\- ^ m\ 
 
 8. 1-13 62 + 454^ 16. 9a;^-6a^+4. 
 
 9. a;4-12.T2/ + 4/. . 17. da*-\-Ua'm^-^2om\ 
 
 10. 4a* + 8a2 4_9. 18. 4-32^2 + 49^1^ 
 
 11. 4m*-24m2H-25. 19. 16a;^- 49a^?/2 + 25?/^ 
 
THEORY OF QUADRATIC EQUATIONS. 253 
 
 DISCUSSION OF THE GENERAL EQUATION. 
 285. The roots of the equation o^ -\-2^^ = 9 ^I'e 
 
 We will now discuss these values for different values of p 
 and q. 
 
 I. Suppose q positive. 
 
 Since p^ is essentially positive (Art. 192), the quantity 
 under the radical sign is positive and greater than ;>^. 
 Therefore the value of the radical is greater than p. Hence, 
 Ti is positive and rj is negative. 
 
 If p is positive, 7\ is numerically greater than Vi ; that is, 
 the negative root is numerically the greater. 
 
 If p is zero, the roots are numerically equal. 
 
 If p is negative, ?*i is numerically greater than r2 ; that is, 
 the positive root is numerically the greater. 
 
 II. Suppose q = 0. 
 
 The quantity under the radical sign is now equal to p^ so 
 that the value of the radical is p. 
 
 If p is positive, ^i = 0, and ?*2 is negative. 
 If p is negative, Vi is positive, and r^ = 0. 
 
 m. Suppose q negative and Aq numerically <p^. 
 
 The quantity under the radical sign is now positive and 
 less than p^. Therefore the value of the radical is less 
 than p. 
 
 If p is positive, both roots are negative. 
 If p is negative, both roots are positive. 
 
254 ALGEBRA. 
 
 IV. Suppose q negative and 4 q numerically = |7^. 
 
 The quantity under the radical sign is now equal to zero. 
 Therefore the roots are equal ; being negative if p is posi- 
 tive, and positive if p is negative. 
 
 V. Suppose q negative and iq numerically >p^. 
 
 The quantit}' under the radical sign is now negative ; 
 hence, by Art. 201, both roots are imaginary. 
 
 The roots are both rational or both irrational according as 
 p^ + 4g is or is not a perfect square. 
 
 EXAMPLES. 
 
 286. 1. Determine by inspection the nature of the roots 
 of the equation 2a;^ — 5a;— 18 = 0. 
 
 The equation may be written x^ = 9. 
 
 Since q is positive and p negative, the roots are one posi- 
 tive and the other negative ; and the positive root is numeri- 
 cally the greater. 
 
 In this case, j)^ .+ 4g = — 4- 36 = — ; a perfect square. 
 4 4 
 
 Hence the roots are both rational. 
 
 Determine by inspection the nature of the roots of the 
 following : 
 
 2. y?-\-2x-\b = 0. 6. 6a^-7a;-5 = 0. • 
 
 3. .^•2 + 5a;-f 6 = 0. 7. 9a^ + 30a; = - 25. 
 
 4. x'-10x = -2b. 8. 9x2 + 8 = 18a;. 
 
 5. 3a.'2-5a;-f4 = 0. 9. 10- 3a;- 18a^ = 0. 
 
RATIO AND PROPORTION. 255 
 
 XXVI. RATIO AND PROPORTION. 
 
 287. The Ratio of one quantity to another of the same 
 kind is the quotient obtained by dividing the first quantity 
 by the second. 
 
 Thus, the ratio of a to 6 is - ; which is also expressed a:h. 
 
 h 
 
 288. The first term of a ratio is called the antecedent, and 
 the second term the consequent. 
 
 Thus, in the ratio a : 6, « is the antecedent and h the con- 
 sequent. 
 
 ). A Proportion is an equality of ratios. 
 
 Thus, if the ratio of a to 6 is equal to the ratio of c to fZ, 
 they form a proportion, which may be written in either of 
 
 the f oims : 
 
 Ci C 
 
 a:b=:c: d, - = -, or a : b : : c : d. 
 b d 
 
 290. The first and fourth terms of a proportion are called 
 the extremes; and the second and third terms the means. 
 
 Thus, in the proportion a : 6 = c : d, a and d are the 
 extremes, and b and c the means. 
 
 291. In a proportion in which the means are equal, either 
 mean is called a Mean Proportional between the first and 
 last terras, and the last term is called a Third Proportional 
 to the first and second terms. 
 
 A Fourth Proportional to three quantities is the fourth 
 term of a proportion whose first three terms are the three 
 quantities taken in their order. 
 
 Thus, in the proportion a'.b=^b: c, b is a mean propor- 
 tional between a and c, and c is a third proportional to a and 
 b. In the proportion a:6 = c:c?, (Zisa fourth proportional 
 to a, 6, and c. 
 
256 ALGEBRA. 
 
 292. A Continued Proportion is one in which each conse- 
 quent is the same as the next antecedent ; as, 
 
 a '. b = b : c = c : d = d : e. 
 
 PROPERTIES OF PROPORTIONS. 
 
 293. In any proportion the product of the extremes is equal 
 to the product of the means. 
 
 Let the proportion he a : b = c : d. 
 
 Then, by Art. 289, -=-. 
 
 b d 
 
 Clearing of fractions, ad = be. 
 
 294. A mean proportional between two quantities is equal 
 to the square root of their product. 
 
 Let the proportion be a:b=^b:c. 
 Then, by Art. 293, 6^ = ac. 
 
 Whence, b=-\/ac. 
 
 295. From the equation ad = be, we obtain 
 
 be J r. ad 
 a — — and b = — 
 d c 
 
 That is, in any proportion either extreme is equal to the 
 product of the means divided by the other extreme ; and 
 either mean is equal to the product of the extremes divided 
 by the other mean. 
 
 296. (Converse of Art. 293.) If the product of two 
 quantities be equal to the prod^ict of two others, one pair may 
 be made the extremes, and the other pair the means, of a pro- 
 portion. 
 
 Let ad — be. 
 
 T\' 'J* V T.J ad be a c 
 
 Dividmg by bd, — = — , or - = - 
 
 bd bd b d 
 
 Whence, a : b = c : d. 
 
RATIO AND PROPORTION. 257 
 
 In a similar manner we may prove that : 
 a: c = 6 : cZ, 
 
 c: d = a:h, etc. 
 
 297. In any propoHion the terms are in j)t'oportion by 
 Alternation; that is, the first term is to the third, as the sec- 
 ond term is to the fourth. 
 
 Let a : h — c : d. 
 
 Then, by Art. 293, ad=bc. 
 Whence, by Art. 296, ci : c = 6 : d. 
 
 298. In any proportion the terms are in proportion by 
 Inversion; that is, the second term is to the first, as the fourth 
 term is to the third. 
 
 Let a : b = c : d. 
 
 Then, ad = be. 
 
 Whence, b: a = d: c. 
 
 299. In any proportion the terms are in proportion by 
 Composition ; that is, the su7n of the first two terms is to the 
 first term, as the sum of the last two terms is to the third term. 
 
 Let a : b = c : d. 
 
 Then, ad = be. 
 
 Adding both members to ac, 
 
 ac-\- ad = ac -f- be, 
 or, a(c-f-d) = c(a4-6)• 
 
 Whence (Art.. 296), 
 
 a-\-b: a = c-{-d: c. 
 Similarly we may prove that 
 
 a-{-b:b = c-\-d:d. 
 
258 ALGEBRA. 
 
 300. In any proportion the terms are in proportion by Divi- 
 sion; that is, the difference of the first two terms is to the first 
 term, as the difference of the last two terms is to the third term. 
 
 Let a:h==c:d. 
 
 Then, ad = bc. 
 
 Subtracting both members from etc, 
 
 ac — ad = ac — bc^ 
 or, a{c — d) = c{a — b). 
 
 Whence, a — b : a = c — d : c. 
 
 Similarly, a — b:b = c — d:d. 
 
 301. In any proportion the terms are in proportion by 
 Composition and Division ; that is, the sum of the first tivo 
 terms is to their difference, as the sum of the last two terms is 
 to their difference. 
 
 Let a: b = c: d. 
 
 Then, by Art. 299, ^±^ = ^±^. (1) 
 
 ^ a c ^ ^ 
 
 And, by Art. 300, ^'—^ = ^^- (2) 
 
 Dividing (1) by (2), 
 
 a-\-b c-{-d 
 
 Oj — b c — d 
 Whence, a-\-b: a — b = c-{-d: c — d. 
 
 302. In a series of equal ratios, any antecedent is to its 
 consequent, as the sum of all the antecedents is to the sum of 
 all the consequents. 
 
 Let a:b = c:d = e:f. 
 
 Then, by Art. 293, ad = 6c, 
 
 and af=be. 
 
 Also, -' ab = ba. 
 
 Adding, a{b-\-d-{-f) = b{a-\- c + e). 
 
 Whence (Art. 296) , a:b = a-^c-\- e : b + d +/. 
 
RATIO AND PROPORTION. 259 
 
 303. In any number of proportions^ the products of ike 
 corresponding terms are in proportio7i. 
 
 Let a : b = c : d, 
 
 and e : f=g: h. 
 
 rr.1- a c ^ e a 
 
 Then, - = -, and - = |. 
 
 b d f h 
 
 Multiplying these equals, 
 
 b^f~d^h' ^ bf~dh 
 Whence, ae : bf= eg: dh. 
 
 304. In any proportion, like powers or like roots of tin 
 terms are in proportion. 
 
 Let a : b = c: d. 
 
 Then, 
 
 Therefore, 
 
 Whence, a** : 6" = c" : d". 
 
 In a similar manner we may prove that 
 
 V a : yb = Vc : -y/d. 
 
 305. In any proportion, if the first two terms be multiplied 
 by any quantity, as also the last two, the resulting quantities 
 will be in proportion. 
 
 Let a : b = c : d. 
 
 Then, 
 
 Therefore, 
 Whence, 
 
 a 
 
 c 
 
 — = 
 
 = — 
 
 b 
 
 d 
 
 «**_ 
 
 - ^. 
 
 
 
 6«" 
 
 "d"' 
 
 a _ 
 b^ 
 
 _ c 
 
 
 ma 
 
 — = 
 
 mb 
 
 ~nd 
 
 
 : mb - 
 
 = nc: 
 
 nd. 
 
260 ALGEBRA. 
 
 In a similar manner we may prove that 
 ah c d 
 mm n' n 
 
 Note. Either m or n may be unity ; that is, either couplet may be 
 multiplied or divided without multiplying or dividing the other. 
 
 306. In any proportion^ if the first and third terms be 
 multiplied by any quantity, as also the second and fourth 
 terms, the resulting quantities imll be in pi-oportion. 
 
 Let a: b = c : d. 
 
 Then, 
 
 a 
 b~ 
 
 _ c 
 ~ d 
 
 ma 
 nb 
 
 _mc 
 nd 
 
 Therefore, 
 
 Whence, ma : nb = mc: nd. 
 
 In a similar manner we may prove that 
 
 a b _ c d 
 m ' n m ' n 
 
 Note. Either m or n may be unity. 
 
 307. If three quantities are in continued proportion, the 
 first is to the third as the square of the first is to the square of 
 the second. 
 
 Let 
 
 Then, 
 
 Therefore, 
 
 . Or, 
 
 Whence, 
 
 a: 
 
 b = 
 
 :b: 
 
 c. 
 
 
 a 
 b~ 
 
 b^ 
 c 
 
 
 > 
 
 b_ 
 
 c 
 
 b^'b 
 
 
 a _ 
 
 c 
 
 a' 
 ' b'' 
 
 
 a: 
 
 : c = 
 
 ■.a': 
 
 b\ 
 

 
 a: 
 
 :6 = 
 
 a 
 b~ 
 
 c 
 
 c = c 
 c 
 
 : cZ. 
 
 a 
 b 
 
 X 
 
 c 
 
 , c _ 
 
 ■Mx 
 
 a 
 
 
 
 
 a __ 
 
 6«' 
 
 
 
 
 
 a : 
 
 d = 
 
 a^ 
 
 ;6». 
 
 
 RATIO AND PROPORTION. 261 
 
 308. Jf four quantities are in continued proportion, the 
 first is to the fourth as the cube of the first is to the cube of 
 the second. 
 
 Let 
 Then, 
 
 Therefore, 
 
 Or, 
 
 Whence, 
 
 Note. The ratio a^ : b^ is called the duplicate ratio, and the ratio 
 a* : b^ the triplicate ratio, of a : b. 
 
 PROBLEMS. 
 
 309. 1. Solve the equation, 
 
 x-^1 : X — 1 = a -{- b : a — b. 
 
 By Art. 301, 2a:: 2 = 2a: 26. 
 
 Whence, by Art. 305, x: 1 =a:b. 
 
 Therefore, x = -, Ans. 
 
 b 
 
 2. If x:y = (x-\-zy: {y-}-z)-, prove that 2 is a mean 
 proportional between x and y. 
 
 From the given proportion, by Art. 293, 
 y{x + zy=x{y-\-zy. 
 Or, x^y -{-2xyz-{-yz^ = xif -\-2xyz-\-icz^. 
 
 Or, ' ay^y — xy^ = xz^ — yz^. 
 
 Dividing b}^ x — y, xy = 2'. 
 
 Therefore 2 is a mean proportional between x and y. 
 
262 ALGEBRA. 
 
 3. Find the first term of the proportion whose last three 
 terms are 18, 6, and 27. 
 
 4. Find the second term of the proportion whose first, 
 third, and fourth terms are 4, 20, and 55. 
 
 5,. Find a fourth proportional to |, f , and f. 
 
 6. Find a third proportional to | and |. 
 
 7. Find a mean proportional between 8 and 18. 
 
 8. Find a mean proportional between 14 and 42. 
 
 9. Find a mean proportional between 2f and y\. 
 Solve the following equations : 
 
 10. 2x — 5:Sx-\-2 = x—l:7x + l, 
 
 11. x^-4::a^-9 = x'-5x + 6:a^-{-4x-i-S. 
 
 •12. a; + Vl -x^ : x-~-Vl-x- = a-^-^b'^-a^ : a-^^b^—a^, 
 
 13 1^-2/= ^''^' 14 {x-\-y:x-y = a+h:a-h, 
 '. U':4 = 15:2/. * \ x"" + f^ = o?b\a? + h^) . 
 
 16. Find two numbers in the ratio of 2|- to 2, such that 
 when each is diminished by 5, they shall be in the ratio of 
 litol. 
 
 16. Divide 50 into two parts such that the greater in- 
 creased by 3 shall be to the less diminished by 3, as 3 to 2. 
 
 17. Divide 12 into two parts such that their product shall 
 be to the sum of their squares as 3 to 10. 
 
 18. Find two numbers in the ratio of 4 to 9, such that 12 
 is a mean proportional between tnem. 
 
 19. The sum of two numbers is to their difference as 10 
 to 3, and their product is 364. What are the numbers? 
 
 20. If a — b:b — c=b:c, prove that 6 is a mean propor- 
 tional between a and c. 
 
RATIO AND PliOPORTION. 263 
 
 21. K5a + 45:9a + 26 = 564-4c:96 + 2c, prove that 
 b is a mean proportional between a and c. 
 
 22. If (a + 6 + c -f d) (a - 6 - c + d) = (a - 6 H- c - cZ) 
 (a-\-b — c — d), prove that a : b = c : d. 
 
 23. If ax — by: ex — dy = ay — bzicy — dz, prove that y 
 is a mean proportional between x and z. 
 
 24. Find two numbers such that if 3 be added- to each, 
 they will be in the ratio of 4 to 3 ; and if 8 be subtracted 
 from each, they will be in the ratio of 9 to 4. 
 
 25. There are two numbers whose product is 96, and the 
 difference of their cubes is to the cube of their difference as 
 19 to 1. What are the numbers? 
 
 26. Divide $564 between A, B, and C, so that A's share 
 may be to B's in the ratio of 5 to 9, and B's share to C's in 
 the ratio of 7 to 10. 
 
 27. A railway passenger observes that a train passes him, 
 moving in the opposite direction, in 2 seconds ; whereas, if 
 it had been moving in the same direction with him, it would 
 have passed him in 30 seconds. Compare the rates of the 
 two trains. 
 
 28. Each of two vessels contains a mixture of wine and 
 water. A mixture, consisting of equal measures from the 
 two vessels, contains as much wine as water ; and another 
 mixture, consisting of four measures from the first vessel 
 and one from the second, is composed of wine and water in 
 the ratio of 2 to 3. Find the ratio of wine to water in each 
 vessel. 
 
 29. Divide a into two parts such that the first increased 
 by b shall be to the second diminished by 6, as a -{-3 b is to 
 a-3b. 
 
264 ALGEBRA. 
 
 XXVII. ARITHMETICAL PROGRESSION. 
 
 310. An Arithmetical Progression is a series of terms, 
 each of which is derived from the preceding by adding a 
 constant quantity called the common difference. 
 
 Thus 1, 3, 5, 7, 9, 11, ... is an increasing arithmetical 
 progression, in which the common difference is 2. 
 
 12, 9, 6, 3, 0, —3, ...is a decreasing arithmetical progres- 
 sion, in which the common difference is — 3. 
 
 311. Given the first term., a, the common difference^ d, and 
 the number of terms., ti, to find the last term, I. 
 
 The progression is 
 
 a, a + d, a-\-2d, a-}- 3d, ••• 
 
 It will be observed that the coefficient of d in any term is 
 one less than the number of the term. Hence, In the nth, or 
 last term, the coefficient of d will be ?i — 1. That is, 
 
 Z = a+(n-l)d. (I.) 
 
 312. Given the first term., a, the last term., Z, and the num- 
 ber of terms., n, to find the sum of the series^ S. 
 
 JS = a + {a-{-d)-\-(a-\-2d) -\ \-{l-d) +Z. 
 
 Writing the series in reverse order, 
 
 S = l -\-(l-d) +(l-2d) +...+(a + cZ) + a. 
 Adding these equations, term by term, 
 
 2>S = (a+0 + (a+0 + (a+0 + ---+(«+0 + (a+0 
 = n(a-{-l). 
 
 Therefore, S=~{a + l). (II. ) 
 
 313. Substituting in (II.) the value of I from (T.) , we have 
 
 ^ = |[2a + (n-l)d]. 
 
ARITHMETICAL PROGRESSION. 265 
 
 EXAMPLES. 
 
 314. 1. In the series 8, 5, 2, —1, —4, ... to 27 terms, 
 find the last term and the sum. 
 
 In this ease, a = 8, cZ = — 3, ?i = 27. 
 Substituting in (I.) and (II.)) 
 
 Z = 8+(27-l)(-3) =8-78 = - 70. 
 
 .S = — (8-70) =27 X (-31) =-837. 
 id 
 
 Note. The common difference may always be found by subtracting 
 the first term from the second. Thus, in the series 
 
 5 _1, -2, ...,wehaverf = -l-^=-ll. 
 3' 6' ' ' 6 3 6 
 
 In each of the following, find the last term and the sum of 
 the series : 
 
 2. 1, 6, 11, ... to 15 terms. 
 
 3. 7, 3, -1, ... to 20 terms. 
 
 4. -9, -6, -3, ... to 23 terms. 
 
 5. -5, -10, -15, ... to 29 terms. 
 
 6. -, -, -, ... to 35 terms. 
 
 4' 2' 4' 
 
 3 _8^ 
 
 5' 15^ 
 
 2 3 
 
 3' 4' 6 
 
 1 _5^ 
 
 2' 11 
 
 7. -, — , ... to 19 terms. 
 
 8* -' 7? :;? ••• to 16 terms. 
 
 1 5 
 9. -, — , ... to 22 terms. 
 
 10. —3, — -, ... to 17 terms. 
 
 2 1 
 
 11. , -, ... to 14 terms. 
 
 5' 3 
 
266 ALGEBRA. 
 
 315. If any three of the five elements of an arithmetical 
 progression are given, the other two may be found by sub- 
 stituting the known values in the fundamental formulae (I.) 
 and (II.)? and solving the resulting equations. 
 
 5 5 
 
 1 . Given a = , ti = 20, JS = ; find d and L 
 
 3 3 
 
 Substituting the given values in (I.) and (II.), we have 
 ; = _| + 19d. (1) 
 
 _|=10(_| + z);o..-i = -| + . (2) 
 
 From (2), ^ = |-i = |- 
 
 Substituting in (1), 
 
 - = --+ldd; or d = -' 
 2 3 6 
 
 Ans. d= , Z = — 
 6 2 
 
 2. Givenc? = -3, l=-39, S = -2G4; find a and n. 
 Substituting in (I.) and (II.), 
 
 — 39 = a + (w-l)(-3), or a = 3n-42. (1) 
 
 -264 = -(a-39), or an - 39n = - 528. * (2) 
 
 Substituting the value of a from (1) in (2), 
 3n2_42ri-39n = -528, 
 or, n^-27n = -l76. 
 
 wu 27±V729-704 27 ± 5 ,. ,. 
 
 Whence, n= = — - — = 16 or 11. 
 
 2 2 
 
 Substituting in (1), 
 
 a = 48 - 42, or 33 - 42 = 6 or - 9. 
 
 Ans. a = (j, 71=16; or, a = — 9, n=ll 
 
ARITHMETICAL PROGRESSION. 267 
 
 Note. The interpretation of these answers is as follows : 
 
 If a = 6, and n = 16, the series is 
 
 0, 3, 0, - 3, — 6, - 9, - 12, - 15, - 18, - 21, - 24, - 27, - 30, 
 
 _33, -36, -39. 
 If a = — 9, and n = 11, the series is 
 
 _9, -12, -15, -18, -21, -24, -27, -30, -33, -36, -39. 
 In each of these results the last term is — 39, and the sum — 264. 
 
 113 
 
 3. Givena = -, d = , 8 = ; find Z and n. 
 
 3 12' 2 
 
 Substituting in (I.) and (II. )» 
 
 Z = i + (n-l)/^-— Y or Z = ^^^. (1) 
 
 3 ^ \ 12/ 12 ^ ^ 
 
 -| = ^Q + ?\ or n + 3Z7i = -9. (2) 
 
 Substituting the value of ^ from (1) in (2), 
 
 ^ 5n-n^ ^_c| Q^ ,i2_9?i = 36. 
 4 
 
 Solving this equation, n= 12 or — 3. 
 
 The second value is inapplicable, as no significance can be 
 attached to a negative number of terms. 
 Substituting the value of ?i in (1), 
 
 ^^ 5-12 ^ 7 . 
 12 12' 
 
 .7 
 
 Ajis. 1 = , n = 12. 
 
 12 
 
 . Note. A negative or fractional value of n is always inapplicable, 
 aiid should be rejected together with all other values dependent on it. 
 
 EXAMPLES. 
 
 4. Givenc? = 4, Z=75, 71= 19; find a and >S^. 
 
 6. Given cZ = — l, 71= Id, S = ; find of and /. 
 
 2 
 
268 ALGEBRA. 
 
 2 
 
 6. Given a = , w = 18, Z=5; find c? and /S. 
 
 o 
 
 7. Given a = , w = 7, >S' = — 7; find d and ?. 
 
 4 
 
 8. Given a = -, Z = , S = — ; find d and n. 
 
 9. Given Zz=-31, 71=13, >S' = -169; find a and d. 
 
 10. Given (^ = - 3, /S' = -328, a=2; find Z and n. 
 
 11. Given a = 3, ? = 42|, d = 2i; find w and /S". 
 
 12. Given (l = - 4, ?i = 17, >S' = -493; find a and Z. 
 
 7 1 
 
 13. Given Z = -, d = -, >S' = 20 ; find a and n. 
 
 2 3 
 
 14. Given / = — , 71 = 21, S = — ; find a and c?. 
 
 2 2 
 
 15. Given a = , I — , /S = ; find d and n. 
 
 3 3 3 
 
 3 
 
 16. Given a = , ti = 15, >iS = 120; find d and Z. 
 
 4 
 
 17. Given Z = -47, d = -l, >S = -1118; find a and w. 
 
 18. Given a = 6, d = , S = ; find ?i and Z. 
 
 o o 
 
 From (I.) and (II.) general formulae for the solution of 
 cases like the above may be readily derived. 
 
 19. Given a, c?, and 8 ; derive the formula for n. 
 Substituting the value of/ from (I.) in (II.), 
 
 2S = n[2a+{n-\)d'], ov dn^ -^{2a-d)n^2S. 
 
 This is a quadratic in n, and may be solved by the method 
 of Art. 265. 
 
ARITHMETICAL PROGRESSION. 269 
 
 Multiplying by Ad, and adding (2a — ciy to both members 
 
 4dV + 4d (2 a - d)n + (2« - d)- = 8d>S + (2a - dy. 
 Extracting the square root, 
 
 2dn-{-2a-d = ±-VSdS-^{2a-dy. 
 
 2 
 
 Whence, ,^^d-2a±y8dS + {2a-d)\ 
 
 2d 
 
 20. Given a, /, and ?i ; derive the formula for d. 
 
 21. Given a, n, and S ; derive the formula* for d and Z. 
 
 22. Given cZ, ?/, and S ; derive the formulae for a and /. 
 
 23. Given a, cZ, and Z ; df live the formulie for n and S. 
 
 24. Given d, Z, and n ; derive the formulss for a and S. 
 26. Given Z, n, and >S' ; derive the formulae for a and d. 
 
 26. Given a, d, and S ; derive the formula for Z. 
 
 27. Given a, Z, and >S' ; derive the formulae for d and n. 
 
 28. Given cZ, Z, and /iS ; derive the formulae for a and n. 
 
 316. jTo iwse?'Z any number of arithmetical means between 
 two given terms. 
 
 1. Insert 5 arithmetical means between 3 and —5. 
 
 This signifies that we are to find 7 terms in arithmetical 
 progression, such that the first term is 3, and the last term 
 -5. 
 
 Substituting a = 3, Z = — 5, and w = 7 in (1) , we have 
 
 -5 = 3 + 6d, orcZ = --. 
 o 
 
 Hence, the required series is 
 
 3 5 1 7 11 , 
 
270 ALGEBRA. 
 
 EXAMPLES. 
 
 2. Insert 5 arithmetical means between 2 and 4. 
 
 3. Insert 7 arithmetical means between 3 and — 1. 
 
 4. Insert 4 arithmetical means between — 1 and — 7. 
 
 5. Insert 6 arithmetical means between — 8 and — 4. 
 
 1 13 
 
 6. Insert 8 arithmetical means between - and 
 
 2 10 
 
 Note. The arithmetical mean between two quantities, a and b, may 
 be found as follows : 
 
 Let X denote the required mean ; then, by the nature of the progres- 
 sion, X — a = h — X, or 2x = a -\-b. 
 
 Whence, x — ^— t — 
 
 2 
 
 That is, the arithmetical mean between two quantities is equal to one-half 
 
 their sum. 
 
 7 9 
 
 7. Find the arithmetical mean between - and 
 
 3 5 
 
 8. Find the arithmetical mean between(a+6)^and— (a — 6)^. 
 
 9. Find the arithmetical mean between — ^^^— and 
 
 a — b a-{-b 
 
 PROBLEMS. 
 
 317. 1. The sixth term of an arithmetical progression is 
 
 5 16 
 
 -, and the fifteenth term is — Find the first term. 
 
 6 3 
 
 By Art. 311, the sixth term is a-\-5d, and the fifteenth term is 
 a + 14 c? ; hence, 
 
 «+ 5c/ = 1 . (1) 
 
 a+Ud = '^-t . (2) 
 
 Subtracting (1) from (2), 9d=-, or d = -- 
 
 Substituting in (2), a -f 7 — — ; whence, a = — -, Ans. 
 
 3 3 
 
ARITHMETICAL PROGRESSION. 271 
 
 2. Find four quantities in arithmetical progression such 
 that the product of the extremes shall be 45, and the product 
 of the means 77. 
 
 Let the quantities hex — Sy,x — y,x-\- y, and a: + 3^. Then, by the 
 conditions, 
 
 3:2-9^2,^45. 
 
 a:2- y^=ri. 
 Solving these equations, a: = db 9 and y = ± 2. 
 Therefore the quantities are 3, 7, 11, and 15; or, —3,-7,-11, and 
 -15. 
 
 Note. In problems like the above it is convenient to represent the 
 unknown quantities by symmetrical expressions. Thus, if five quanti- 
 ties had been required, we should have represented them by x — 2 ^, 
 
 x — y,x,x-\-y, and x + 2y. 
 
 3. Find tjie sum of the odd numbers from 1 to 100. 
 
 4. Tne seventh term of an arithmetical progression is 27, 
 and the thirteenth term is — 3. Find the twenty-first term. 
 
 6. Find four numbers in arithmetical progression such 
 that the sum of the fiist and third shall be 22, and the sum 
 of the second and fourth 36. 
 
 6. A person saves $270 the first year, $245 the second, 
 and so on. In how many years will a person who saves 
 every ^ear $145 have saved as much as he? 
 
 7. In the progression m, 2m — 3 n, 3m — Grj, ... to 10 
 terms, find the last term and the sum of the series. 
 
 8. The seventh term of an arithmetical progression is 
 5a + 46, and the nineteenth term is 9a — 26. Find the 
 fifteenth term. 
 
 9. Find the sum of the even numbers beginning with 2 
 and ending with 500. 
 
 10. The sum of the squares of the extremes of four num- 
 bers in arithmetical progression is 200, and the sum of the 
 squares of the means is 136. What are the numbers? 
 
272 ALGEBRA. 
 
 11. The seventh term of an arithmetical progression is 
 
 1 3 y 
 
 , the thirteenth term is -, and the last term is -. Find 
 
 2 2 2 
 
 the number of terms. 
 
 12. Find five quantities in arithmetical progression such 
 that the sum of the first, third, and fourth is 3, and the 
 product of the second and fifth is — 8. 
 
 13. Two persons start together. One travels 10 leagues 
 a day ; the other 8 leagues the first day, which he augments 
 daily b}^ half a league. After how many days, and at what 
 distance from the point of departure, will they come together? 
 
 14. A body falls 16^2 ^^^^ ^^^ ^^'^^ second, and in each 
 succeeding second 321 feet more than in the next preceding 
 one. How far will it fall in 16 seconds? 
 
 15. Find three quantities in arithmetical progression such 
 that the sum of the squares of the first and third exceeds the 
 second by 123, and the second exceeds one-third the first 
 by 6. 
 
 16. After A had travelled 2| hours at the rate of 4 miles 
 an hour, B set out to overtake him, and went 4i miles the 
 first hour, 4| the second, 5 the third, and so on, increasing 
 his speed a quarter of a mile every hour. In how many 
 hours would he overtake A ? 
 
 17. If a person should save $100 a year, and put this sum 
 at simple interest at 5 per cent at the end of each year, to 
 how much would his property amount at the end of 20 years? 
 
 18. The digits of a number of three figures are in arith- 
 metical progression ; the first digit exceeds the sum of the 
 second and third by 1 ; and if 594 be subtracted from the 
 number, the digits will be inverted. Find the number. 
 
GEOMETRICAL PROGRESSION. 273 
 
 XXVIII. GEOMETRICAL PROGRESSION. 
 
 318. A Geometrical Progression is a series of terms, each 
 of which is derived from the preceding by multiplying by a 
 constant quantity called the ratio. 
 
 Thus, 2, 6, 18, 54, 162, ••• is an increasing geometrical 
 progression in which the ratio is 3. 
 
 9, 3, 1, -,-,... is a decreasing geometrical progression in 
 
 which the ratio is -• 
 3 
 
 Negative values of the ratio are also admissible ; thus, 
 
 — 3, 6, — 12, 24, — 48, ... is a progression in which the ratio 
 
 is -2. 
 
 319. Given the first term^ a, the ratio, r, and the number 
 
 of terms, ?i, to find the last term, I. 
 
 The progression is a, ar, an^, a?-^, ... 
 
 It will be obsei-ved that the exponent of r in any term is 
 one less than the number of the term. Hence, in the nih or 
 last term, the exponent of r will be n — 1. That is, 
 
 l=zaT^''^. (I.) 
 
 320. Given the first term, a, the last term, Z, and the ratio, 
 r, to find the sum of the series, S. 
 
 S= a-\-ar +ar^-] \- ar""-^ + ar^^^ 4- ar**" ^ 
 
 Multiplying each term by r, 
 
 rS = ar + ai^ -\- ar^ -\- \- ar'^~^ -\- af^~'^ -j- ar"*. 
 
 Subtracting the first equation from the second, 
 
 rS-S = ar^-a; or, S = ^^" ~ ^ . 
 
 r— 1 
 
 But from (I.) , Art. 319, rl — ar"". Hence, 
 
 5=^- (II.) 
 
 ?* — 1 
 
274 ALGEBRA. 
 
 EXAMPLES. 
 
 321. 1. In the series 3, 1, - , ... to 7 terms, find the last 
 term and the sum. 
 
 In this case, a = 3, r = -, n = 7. Substituting in (I.) and 
 
 (11-), ^ 
 
 3J 3' 243 
 
 lx_i__3 J 3 2186 
 
 ^^ 3 243 _^ 729 ^ 729 ^1093 
 
 3 3 3 
 
 Note. The ratio may always be found hy dividing the second term 
 by the first. 
 
 2. In the series —2, 6, — 18, 54, ... to 8 terms, find the 
 last term and the sum. 
 
 In this case, a = — 2, r = = — 3, n = 8. Hence, 
 
 Z = -2(-3)" = -2x(- 2187) = 4374. 
 
 ^ ^ - 3 X 4374 - (- 2) ^ - 13122 + 2 ^ g^g^ 
 -3~1 -4 
 
 In each of the following find the last term and the sum of 
 the series : 
 
 3. 1, 2, 4, ... to 9 terms. 
 
 4. 3, 2, ^, ... to 7Xerms. 
 
 o 
 
 5. -2, 8, -32, ... to G terms. 
 
 6. 2, -1, i, ... to 10 terms. 
 
 7. -,-,-,... to 11 terms. ' 
 2 4 8 
 
GEOMETRICAL PROGRESSION. 275 
 
 9 Q 
 
 8. -, — 1, , ... to 8 terms. 
 3 2 
 
 9. 8, 4, 2, ... to 9 terms. 
 
 10. -, , — , ... to 6 terms. 
 
 4' 4' 12' 
 
 11. 3, —6, 12, ... to 7 terms. 
 
 12. , , , ... to 10 terms. 
 
 3' 3' 6' 
 
 322. If any three of the five elements of a geometrical 
 progression are given, the other two may be found b}' sub- 
 stituting the known values in the fundamental formula? (I.) 
 and (II.)? ^^^ solving the resulting equations. 
 
 But in certain cases the operation involves the solution of 
 an equation of a degree higher than the second ; and in 
 others the unknown quantity appears as an exponent, the 
 solution of which form of equation can usually only be 
 effected by the use of logarithms (Art. 363). In all such 
 examples in the present chapter, the equations may be solved 
 by inspection. 
 
 1. Given a = — 2, ?i = 5, Z = — 32 ; find r and S. 
 Substituting the given values in (I.), we have 
 
 — 32 = — 2 r^ ; whence, 9*^ = 16, or r = ± 2. 
 Substituting in (II.), 
 
 If r= 2, ^- =^ ^( - ^^^) - ( - ^) = - 64 + 2 =- 62. 
 It, = _2, ^^(-2)(-32)-(-2)^6i+2^_^^ 
 
 — ^ — 1 — O 
 
 Ans. r = 2, ^ = - 62 ; or, r = -2, S = -2'2. 
 
 Note. The interpretation of these answers is as follows : 
 If r = 2, the series is — 2, — 4, — 8, — 16, — 32, in which the sum is - 62. 
 If r = — 2, the series is - 2, 4, — 8, 16, — 32, in which the sura is — 22. 
 
276 ALGEBRA. 
 
 2. Given a = 3, r = - -, ^ = i^ ; find n and L 
 3 729 
 
 U-S 
 
 Substituting in (U . ) ? 
 
 1640^ 3 ^^ + 9 
 
 729 _l_i 4 
 
 3 
 
 Whence, Z + 9 = -^; or, / = - — . 
 
 Substituting in (I.), 
 
 729 \ 3y V 3/ 2187 
 
 Whence, by inspection, 
 
 71— 1 = 7, or n = 8. • 
 
 EXAMPLES. 
 
 3. Given r = 2, 7i = 10, / = 256 ; find a and /S'. 
 
 4. Given r = - 2, n = 6, /S = — ; find a and Z. 
 
 5. Given a = 2, ti = 7, 1= 1458 ; find r and S. 
 
 6. Given a=l, r = 3, 1=81; find ?i and >^. 
 
 7. Given r = i, n = 8, >S = 5^; find a and Z. 
 
 o 6o61 
 
 Q 
 
 8. Given a = 3, n = G. 1 = ; find r and /S'. 
 
 1024 
 
 9. Given a = 2, Z = — , S = — ; find n and ?•. 
 
 32 32 
 
 10. Given a = i, r = -3, /S = - 91 ; find n and ?. 
 
 11. Given ? = -128, r = 2, .S' = -255; find a and >i. 
 
GEOMETRICAL PROGRESSION. 277 
 
 From (I.) and (II.) general formulae may be derived for 
 the solution of cases like the above. 
 
 12. Given a, r, and S ; derive the formula for L 
 
 13. Given a, I, and S ; derive the formula for r. 
 
 14. Given r, Z, and S ; derive the formula for a. 
 
 16. Given r, w, and I ; derive the formulae tor a and S. 
 
 16. Given r, ?i, and S ; derive the formulae for a and Z. 
 
 17. Given a, ?i, and Z ; derive the formulae for r and S. 
 
 Note. If the given elements are n, /, and S, equations for a and r 
 may be found, but there are no definite formulae for their values. The 
 same is the case when the given elements are a, n, and 5. 
 
 The general formulae for n involve logarithms ; these cases are dis- 
 cussed in Art. 363. 
 
 323. The limit to which the sum of the tenns of a decreas- 
 ing geometrical progression approaches, as the number of 
 terms increases indefinitely, is called the sum of the senes to 
 infinity. 
 
 The value of S in formula (II.) may be written 
 
 ^ a — r l 
 
 In a decreasing geometrical progression, the greater the 
 number of terms taken, the smaller will be the value of 
 the last term. Hence, as the number of terms increases 
 indefinite^, the term rl decreases indefinitely and approaches 
 the limit 0. 
 
 CL —^ tIi CI 
 
 Therefore the fraction approaches the limit 
 
 1 — r \ —r 
 
 Hence, the sum of a decreasing geometrical progression 
 to infinity is given by the formula 
 
 ^ = -^. (III.) 
 
 1 — r 
 
278 ALGEBRA. 
 
 EXAMPLES. 
 
 1. Find the sum of the series 4, — , — , ... to infinity 
 
 9 
 
 In this case, a = 4, r = — -• 
 
 3 
 
 4 12 
 
 Substituting in (III.), S = = — , Ans. 
 
 l-h? ^ 
 3 
 
 Find the sum of the following to infinity : 
 
 2. 
 
 2, 1, |, ... 
 
 
 6. 
 
 3 1 1 
 
 4' 2' 3' - 
 
 3. 
 
 4, -2,1, ... 
 
 
 7. 
 
 o 3 3 
 
 ' 10' 100' •" 
 
 4. 
 
 -1 1 -1 
 ''3' 9' 
 
 ... 
 
 8. 
 
 _8, -?, -.1, 
 5 50 
 
 5. 
 
 o 3 
 -3, --,- 
 
 3 
 25'- 
 
 9. 
 
 '' ~^^'^'- 
 
 324. TojiiKl the value of a repeating decimal. 
 
 This is a case of finding the sum of a geometrical pro- 
 gression to infinity, and may be solved by the formula of 
 Art. 323. 
 
 1. Find the value of .85151 ... 
 
 .85151 ... = .8 4- .051 4- -00051 + ... 
 
 The terms after the first constitute a geometrical progres- 
 sion in which a= .051, and r= .01. 
 Substituting in (III.), 
 
 ^ ^ _^05]_ ^ ^051 ^ ^ ^ 21 
 l-.Ol .99 990 330* 
 
 Hence the value of the given decimal is 
 
 8 , 17 281 . 
 
 10 330 330 
 
GEOMETRICAL PROGRESSION. 279 
 
 EXAMPLES. 
 Find the values of the following : 
 
 2. .7272... 4. .7333... 6. .110303... 
 
 3. .407407... 6. .52121... 7. .215454-.. 
 
 325. To insert any number of geometrical means between 
 two given terms. 
 
 C A 
 
 1. Insert 4 geometrical means between 2 and 
 
 ^ 243 
 
 This signifies that we are to find G terms in geometrical pro- 
 gression such that the first term is 2, and the last term 
 
 ^ 243 
 
 64 
 
 Substituting a = 2, ?i = 6, 1= in (I.) , we have 
 
 243 
 
 64 o. . .32 2 
 
 = 2r*: whence ?'^ = , or r= — 
 
 243 243 3 
 
 Hence the required series is 
 
 g 4 8 16 32 64 
 ' 3' 9' 27' 81' 243* 
 
 EXAMPLES. 
 2. Insert 6 geometrical means between 3 and 
 
 128 
 729* 
 
 3. Insert 5 geometrical means between - and 3641. 
 
 4. Insert 6 geometrical means between ^ 2 and — 4374. 
 
 q q 
 
 6. Insert 7 geometrical means between - and -— . 
 
 2 512 
 
 6. Insert 5 geometrical means between — 2 and — 128. 
 
 7. Insert 4 geometrical means between 3 and — ^ 
 
 1024 
 
280 ALGEBRA. 
 
 Note. The geometrical mean between two quantities, a and 6, may 
 be found as follows : 
 
 Let X denote the required mean ; then, by the nature of the pro- 
 
 a X 
 Whence, x = Vol), 
 
 That is, the geometrical mean hetiveen two quantities is equal to the square _ 
 root of their product. 
 
 8. Find the geometrical mean between 11| and 2^. 
 
 9. Find the geometrical mean between 4 a^-f 12^7?/ + 92/^ 
 and 4a^— l-lxy-\- dy-. 
 
 10. Find the geometrical mean between and -• 
 
 PROBLEMS. 
 
 326. 1. Find three numbers in geometrical progression, 
 such that their sum shall be 14 and the sum of their squares 84. 
 
 Let the quantities be a, ar, and ar"^ ; then, by the conditions, 
 
 a + rtr + ar"^ = 14. (1) 
 
 a2 ^ ah'2 + ah-^ = 84. (2) 
 
 (3) 
 
 (4) 
 
 Dividing (2) by (1), 
 
 a — ar -{- ar"^ = 6. 
 
 Subtracting (3) from (1), 
 
 4 
 
 2ar = 8, or r = -. 
 a 
 
 Substituting in (1), 
 
 a + 4 + L6=14. 
 
 Or, 
 
 a^ -10a = -16. 
 
 Solving this equation, 
 
 a = 8 or 2. 
 
 Substituting in (4), 
 
 8 2 2 
 
 Therefore, the numbers are 
 
 2, 4, and 8. 
 
 2. The fifth term of a geometrical progression is 48, and 
 the eighth term is — 384. Find the first term. 
 
 3. The sum of the first and second of four quantities in 
 geometrical progression is 15, and the sum of the third and 
 fourth is 60. What are the quantities? 
 
GEOMETRICAL PROGRESSION. 281 
 
 4. Find three quantities in geometrical progression, such 
 that the sum of the first and second is 20, and the third 
 exceeds the second by 30. 
 
 5. The fourth term of a geometrical progression is — 108, 
 and the eighth term is — 8748. Find the first term. 
 
 6. A person who saved every year half as much again as 
 he saved the previous year, had in seven years saved $2059. 
 How much did he save the first year ? 
 
 7. The elastic power of a ball, which falls from a height 
 of a hundred feet, causes it to rise 0.9375 of the height from 
 which it fell, and to continue in this way diminishing the 
 height to which it will rise, in geometrical progression, until 
 it comes to rest. How far will it have moved? 
 
 8. The sum of four quantities in geometrical progression 
 is 30, and the quotient of the fourth quantity divided by the 
 
 sum of the second and third is -. Find the quantities. 
 
 o 
 
 9. The third term of a geometrical progression is — , and 
 
 9 ^^ 
 
 the sixth term is — -. Find the eighth term. 
 Oi2 
 
 10. Divide the number 39 into three parts in geometrical 
 progression, such that the third part shaU exceed the first 
 by 24. 
 
 11. The product of three numbers in geometrical progres- 
 sion is 64, and the sum of the squares of the first and third 
 is 68. What are the numbers? 
 
 12. The product of three quantities in geometrical pro- 
 gression is 8, and the sum of their cubes is 73. What are 
 the quantities? 
 
282 ALGEBRA. 
 
 XXIX. BINOMIAL THEOREM. 
 
 327. The Binomial Theorem is a formula by means of 
 which a binomial may be raised to any required power with- 
 out going through the process of actual multiplication. 
 
 Examples of its application have been given in Art. 196. 
 
 PROOF OF THE THEOREM FOR A POSITIVE INTEGRAL 
 EXPONENT. 
 
 328. Assuming the laws of Art. 196 to hold for the expan- 
 sion of (a + ^Yi where n is any positive integer : 
 
 The exponent of a in the first term is w, and decreases by 
 1 in each succeeding term. 
 
 The exponent of x in the second term is 1, and increases 
 by 1 in each succeeding term. 
 
 The coefficient of the first term is 1 ; of the second term, 
 n ; multiplying the coefficient of the second term by n — 1 , 
 the exponent of a in that term, and dividing the result by 
 the exponent of x increased by 1, or 2, we have n{n— 1) 
 as the coefficient of the third term ; and so on. 2 
 
 Thus, {a + x)" = a'* + na^'-'^x -f ViiHull a^'-^x^ 
 
 ^(^_1)(^_2) .3 
 1.2.3 
 This result is called the Binomial Theorem. 
 
 329. To prove that it holds for any positive integral 
 exponent, multiply both members by a+ a;. Then, 
 
 n(n-l)(n-2)^._,^ 
 1.2.3 
 
BINOMIAL THEOREM. 283 
 
 Collecting the terms which contain like powers of a and x, 
 the result 
 
 = a'^+i + (n -f- l)a-a; + r^^^^^-^ + n~| a'-^a^ 
 
 r n{n - 1) {n - 2) n{n - 1) 1 ^„- 2^ , ... 
 |_ 1.2.3 1.2 J 
 
 = a"+i + {u 4- l)a^x + ?/^^-^ + iVt'^^^x-^ 
 
 n(n-21/n-2 \ ._,^ ^^ 
 
 1.2 V 3 y 
 
 = a"+^ + (?i + l)a"a; + ?i /'^^ii'ja^-ia;^ 
 n(n-l)/n+j\ ,.,^ 
 
 1.2 V 3 y 
 
 = a'*+^ + (71 + 1 )a"a; + i^i±J^a«- V 
 
 (n + l)M(n-l) 2 o 
 1.2.3 
 
 It will be observed that the result is in the same form as 
 tlue expansion of {a-\-x)'\ having ?i+ 1 in place of 7i. 
 
 Hence, if the theorem holds for any positive integral 
 exponent, w, it also holds when that exponent is increased 
 by 1. 
 
 But, in Art. 196, the theorem was shown to hold for 
 (a -|- xY ; hence it also holds for (a + xY ; and since it holds 
 for (« + a;)^, it also holds for (a -f xY ; and so on. . There- 
 fore the theorem holds when the exponent is any positive 
 
 Note 1. The above method of proof is known as the Method of 
 Induction. 
 
 Note 2. In place of the denominators 1-2, 1 • 2 • 3, etc., it is cus- 
 tomary to write [2, [8, etc. The symbol [n, read " factorial n," signifies 
 the product of the natural numbers from 1 to n inclusive. 
 
284 ALGEBRA. 
 
 330. If X is negative, the terms in the expansion will be 
 alternately positive and negative. Thus, 
 
 (a - xy = a" - na^-^x-^ ^^^^^ ~ ^^ a^-^x" 
 
 II. 
 
 331. If a = 1 , we have 
 
 EXAMPLES. 
 332. 1. Expand (m"^ - V^)^- 
 Proceeding as in Art. 196, we have 
 {m~^-^ny=l{m~^)-{7i^)Y 
 
 = (m~3)5_ b{m~^y{'n?) +\0{m~^y{n^~y 
 
 -10(m~^)2(n^)3+ 5(m~*) {n^y-{n^y 
 
 = 
 
 = m 
 
 t-5m-^n^ 
 
 Expand the following 
 
 2. 
 
 (cS 
 
 + (^-')^ 
 
 3. 
 
 (m' 
 
 -^-Tl^)^ 
 
 4. 
 
 /a; 
 
 4'J- 
 
 5. 
 
 (af"+2.r)^ 
 
 6. 
 
 (a^ 
 
 + 3V«^)'. 
 
 7. 
 
 
 ■ Y 
 
 — Vmn J • 
 
 
 Ans. 
 
 8. 
 
 
 9. 
 
 e-iT 
 
 10. 
 
 {ah-i-arhiy. 
 
 11. 
 
 {y/a?-3^ay. 
 
 12. 
 
 /2x y V 
 
 13. 
 
 (»--H' 
 
BINOMIAL THEOREM. 285 
 
 14. {x^-^3y-fy. 16. {3a-^■^yh-b-^^ay. 
 
 15. f«-V6_2i^V ^^ (J^^2j% 
 
 A trinomial may be raised to any power by the Binomial 
 Theorem if two of its terms be enclosed in a parenthesis and 
 regarded as a single term. (Compare Art. 195.) 
 
 Expand the following : 
 
 18. (1-x-a^y. 20. (l-\-2x-x'y. 
 
 19. (a^-^x-iy. 21. (l-x + ar')^ 
 
 333. To find the rth or general term in the expansion of 
 {a-\-xy\ 
 
 The following laws will be observed to hold for any term 
 in the expansion of (a -ha)" : 
 
 1. The exponent of x is less by 1 than the number of the 
 term. 
 
 2. The exponent of a is n minus the exponent of x. 
 
 3. The last factor of the numerator is greater by 1 than 
 the exponent of a. 
 
 4. The last factor of the denominator is the same as the 
 exponent of a;. 
 
 Therefore, in the rth term, 
 
 The exponent of x will be r — 1 . 
 
 The exponent of a will be ?i — (r — 1) , or w — r + 1 . 
 
 The last factor of the numerator will be n — r-\-2. 
 
 The last factor of the denominator will be r — 1. 
 
 Hence, the rth term 
 
 ^ 7i (n-l)(yi-2)...(n-r + 2) ^nr+i^-i 
 1.2.3...(r-l) 
 
286 ALGEBRA. 
 
 EXAMPLES. 
 
 1. Find the eighth term of (3a^-b-y\ 
 
 In this case, r = 8, n = 11 ; hence, the eighth term 
 
 ^11.10.9.8.7.6.5 ^^ • 
 
 1.2.3.4.5.6.7 ^ ^ ^ ^ 
 
 = 330(81 a^) (- b~') = - 267^0 a'b~\ Ans. 
 
 Note. If the second term of tlie binomial is negative, it is con- 
 venient to enclose the term, sign and all, in a parenthesis, as shown in 
 Ex.1. 
 
 Or, the absolute value of the term may be found by the formula, 
 and the sign determined in accordance with the principle that the odd 
 terms of the expansion are positive, and the even terms negative. 
 
 Find the 
 
 2. Seventh term of (a -|- a;) ^^ 
 
 3. Sixth term of (1 -j-m)io. 
 
 4. Eighth term of (c - dyK 
 
 5. Fifth term of (l-a^K 
 
 6. Seventh term of f^ + -^^ 
 
 8. Sixth term of I a"^ ab 
 
 7. Fifth term of {x — ^xy\ 
 
 1 
 2 
 
 9. Eighth term of {x''^ + 22/-)^^ 
 
 10. Fourth term of (a^ -3x^y\ 
 
 11. Ninth term of ( ^m -f — 
 
 V^, 
 
LOGARITHMS. 287 
 
 XXX. LOGARITHMS. 
 
 334. Every positive number may be expressed, exactly or 
 approximately, as a power of 10 ; thus, 
 
 100 = 102; 13 = 10i-i^" ; etc. 
 
 When thus expressed, the corresponding exponent is called 
 its Logarithm to the base 10 ; thus, 2 is the logarithm of 100 
 to the base 10, a relation which is written 
 
 logio 100 = 2, or simply log 100 = 2. 
 And, in general, if 10* = m, then a; = logm. 
 
 335. Any positive number except unity may be taken as 
 the base of a system of logarithms ; thus, if a*= m, x is the 
 logarithm of m to the ba3e a. 
 
 Logarithms to the base 10 are called Common Logarithms^ 
 and are the only ones used for numerical computations. If 
 no base is expressed, the base 10 is understood. 
 
 336. By Arts. 220 and 221, we have 
 
 10« = 1, 
 
 '''=ro =-1' 
 
 10^=10, 
 
 '"'-^lo =-''^ 
 
 102 = 100, 
 
 ''"^iko--'''^'*'- 
 
 Whence, by Art. 334, 
 
 
 log 1=0, 
 
 log.l =-1 = 9-10, 
 
 loglO = l, 
 
 log.01=-2 = 8-10. 
 
 log 100 = 2, 
 
 log.001=-3 = 7-10, etc, 
 
 Note. The second form of the results for log .1, log .01, etc., is prefer- 
 able in practice. 
 
288 ALGEBRA. 
 
 337. It is evident from thi» preceding article that the 
 logarithm of a number greater than 1 is positive, and the 
 logarithm of a number less than 1, and greater than 0, is 
 negative. ^ 
 
 338. If a number is not an exact power of 10, its common 
 logarithm can only be expressed approximately ; the integral 
 part of the logarithm is called the characteristic^ and the deci- 
 mal part the mantissa. 
 
 For example, logl3 = 1.1139. 
 
 In this case the characteristic is 1, and the mantissa is .1139. 
 
 339. It is evident from the first column of Art. 336 that 
 the logarithm of any number between 
 
 1 and 10 is equal to plus a decimal ; 
 10 and 100 is equal to 1 plus a decimal ; 
 100 and 1000 is equal to 2 plus a decimal ; etc. 
 
 Hence, the characteristic of the logarithm of a number, 
 with one figure to the left of its decimal point, is ; with two 
 figures to the left of the decimal point, is 1 ; with three figures 
 to the left of the decimal point, is 2 ; etc. 
 
 340. Similarly, from the second column of Art. 336, the 
 logarithm of a decimal between 
 
 1 and .1 is equal to 9 plus a decimal — 10 ; 
 .1 and .01 is equal to 8 plus a decimal — 10 ; 
 .01 and .001 is equal to 7 plus a decimal — 10 ; etc. 
 
 Hence, the characteristic of the logarithm of a decimal, 
 with no ciphers between its decimal point and first significant 
 figure, is 9, with — 10 after the mantissa ; of a decimal with 
 one cipher between its point and first figure, is 8, with — 10 
 after the mantissa ; of a decimal with two ciphers between 
 its point and first figure, is 7, with — 10 after the mantissa ; 
 etc. 
 
LOGARITHMS. 289 
 
 341. For reasons which will be given hereafter, only the 
 mantissa of the logarithm is given in the table ; the charac 
 teristic must be supplied b}' the reader. The rules for 
 characteristic are based on Arts. 339 and 340. 
 
 I. If the number is greater than 1 , the characteristic is one 
 less than the number ofplaxies to the left of the decimal point. 
 
 II. Jf the number is less than 1, subtroxt the number of 
 ciphers between the decimal point and first significant figure 
 from 9, writing — 10 after the mantissa. 
 
 Thus, characteristic of log 906328.5 = 5 ; 
 
 characteristic of log .00702 =7, with —10 after 
 the mantissa. 
 
 Note. Some writers, in dealing with the characteristics of the loga- 
 rithms of numbers less than 1, combine the two portions of the 
 characteristic, writing the result as a negative characteristic before 
 the mantissa. Thus, instead of 7.6036 —10, the student will frequently 
 find 3.6036; a minus sign being written over the characteristic to 
 denote that it alone is negative, the mantissa being always positive. 
 
 PROPERTIES OF LOGARITHMS. 
 
 342. The logarithm of a product is equal to Cite sum of the 
 logarithms of its factors. 
 
 Assume the equations 
 
 10' 
 10* 
 
 ^' = ^ I ; whence, by Art. 334, \ ^ = ^^^^' 
 
 Multiplying, 10' x 10^ = wm, or 10*+^ = mw. 
 Whence, log mn = x-\-y. 
 
 Substituting the values of x and 2/, 
 
 logmw = logm + log 71. 
 
 In a similar manner the theorem may be proved for the 
 product of three or more factors. 
 
290 ALGEBRA. 
 
 343. By aid of the theorem of Art. 342, the logarithm of 
 any composite number may be found when the logarithms 
 of its factors are known. 
 
 1. Given log 2 = .3010, log 3 = .4771 ; find log 72. 
 log 72 = log(2 X2x2x3x3) 
 
 = log 2 + log 2 + log 2 -h log 3 + log 3 
 = 3 xlog2 + 2 xlog3 
 = .9030 + .9542 
 = 1.8572, Ans. 
 
 EXAMPLES. 
 
 Given log 2 = .3010, log 3 = .4771, log 5 = .6990, log 7 
 = .8451 ; find the values of the following : 
 
 2. log 6. 7. log 21. 12. log 98. 17. log 135. 
 
 3. log 14. 8. log 63. 13. log 105. 18. log 168. 
 
 4. log 8. 9. log 56. 14. log 112. 19. log 147. 
 
 5. log 12. 10. log 84. 15. log 144. .20. log 375. 
 
 6. logl5. 11. log45. 16. log216. 21. log343. 
 
 344. The logarithm of a fraction is equal to the logarithm 
 of the numerator minus the logarithm of the denominator. 
 
 Assume the equations 
 10^ = 
 
 10^ 
 
 = ^ I ; whence, j^'^^ogm. 
 = n ) yy = logw. 
 
 Dividmg, r^- = - , or 10^"^ = — 
 10^ n n 
 
 Whence, log — = x — y. 
 
 Substituting the values of x and y^ 
 
 log — = log m — logn. 
 
LOGARITHMS. 291 
 
 345. 1. Given log 2 = .3010; find log 5. 
 log 5 = logy = log 10 - log 2 
 = 1-. 3010 =.6990, Ans. 
 
 EXAMPLES. 
 
 Given log 2 = .3010, log 3 =.4771, log 7 = .8451; find the 
 values of the following : 
 
 2. log-. 5. log 35. 8. log^. 11. log7f 
 
 3 25 
 
 3. logi5. 6. log—. 9. log 175. 12. log— . 
 
 4. log3J. 7. log 125. 10. logllf 13. log 5f 
 
 346. The logarithm of any power of a quantity is equal to 
 the logarithm of the quantity multiplied by the exponent of the 
 power. 
 
 Assume the equation 
 
 10"= = m ; whence, x = logm. 
 Raising both members to the pth power, 
 
 10^ = m^ ; whence, logm^ = j9a; =/)logm. 
 
 347. The logarithm of any root of a quantity is equal to the 
 loganthm of the quantity divided by the index of the root. 
 
 For, log -{/m = log {rrf) = -\ogm. (Art. 346.) 
 
 r 
 
 348. 1. Given log2 = .3010 ; find the logarithm of 2i 
 
 log2^=-Xlog2 = 5 X .3010 =.501 7, Ans. 
 3 3 
 
 Note. To multiply a logarithm by a fraction, multiply first by the 
 numerator, and divide the result by the denominator. 
 
292 ALGEBRA. 
 
 2. Given logS = .4771 ; find the logarithm of ^3. 
 
 log ^3 = i^ = ^i^l = .0596, Ans. 
 8 8 
 
 EXAMPLES. 
 
 Given log2 = .3010, log3 = .4771, log7 = .8451 ; find the 
 values of the following : 
 
 3. log3i 7. logl2^ 11. logl5'. 15. log ^5. 
 
 4. log2^ 8. log212. 12. logV7. 16. log ^35. 
 
 5. log7^ 9. logl4^ 13. log -^3. 17. log ^98. 
 
 6. logSi 10. log25i 14. log ^2. 18. log ^126. ' 
 
 19. Find the logarithm of (2' x 3^). 
 
 By Art. 342, log(2* x 3^) = log2^ + log3^ 
 
 = ^log2 -f- ^log3 = .6967,^715. 
 
 Find the values of the following : 
 20. log(^yJ- 22.1og(3'x2t). 24.1og^|. 26. log ^ 
 
 21. log^'. 23. log 3^7. 25. log^. 27. log^- 
 
 5^ V^ 10^ 
 
 349. The mantissce of the logarithms of numbers having 
 the same sequence of figures are the same. 
 
 To illustrate, suppose that log 3.053 = .4847. 
 Then, log30.53 = log(10 x 3.053) = log 10 + log 3.053 
 = 1-1- .4847 =1.4847. 
 
 log 305.3 =log(100 X 3.053) = log 100 -h log 3.053 
 
 = 2 + .4847 = 2.4847. 
 
 log .03053 = log(.01 X 3.053) = log .01 -f- log3.053 
 
 , =8-10-h.4847 =8.4847-10; etc. 
 
LOGARITHMS. 293 
 
 It is evident from the above that, if a number be multi- 
 plied or divided by any integral power of 10, thus producing 
 another number with the same sequence of figures, the man- 
 tissse of their logarithms will be the same. 
 
 Thus, if log 3. 053 = .4847, 
 then, log 30.53= 1.4847, log .3053 =9.4847-10, 
 log 305.3 = 2.4847, log .03053 =8.4847-10, 
 log 3053. = 3.4847, log .003053 = 7.4847-10, etc. 
 
 Note. The reason will now be seen for the statement made in 
 Art. 341, that only the mantissas are given in the table. For, to find 
 the logarithm of any number, we have only to take from the table the 
 mantissa corresponding to its sequence of figures, and the character- 
 istic may then be prefixed in accordance with the rules of Art. 341. 
 
 This proj)erty of logarithms is only enjoyed by tlie common system, 
 and constitutes its superiority over all others for numerical compu- 
 tations. 
 
 350. 1. Given log 2 =.3010, log 3 =.4771 ; find log .00432. 
 log432 = log(2^x33) 
 
 = 41og2-f-31og3 
 
 = 1.2040 4-1.4313=2.6353. 
 
 Whence, by Art. 341, log .00432 = 7.6353 - 10, Ans. 
 
 EXAMPLES. 
 
 Given log2 = .3010, log3 = .4771, log7=.8451 ; find the 
 values of the following : 
 
 2. log 1.8. 7. log. 0054. 12. Iog302.4. 
 
 3. log2.25. 8. log. 000315. 13. log.06174. 
 
 4. log. 196. 9. Iog7350. 14. log(8.1)^ 
 
 5. log. 048. 10. Iog4.05. 15. log -^9^. 
 
 6. log 38.4. 11. log. 448. 16. log\/r62. 
 
 17. log (22.4) ». 
 
294 
 
 ALGEBRA. 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 oooo 
 
 0043 
 
 0086 
 
 0128 
 
 0170 
 
 0212 
 
 0253 
 
 0294 
 
 0334 
 
 0374 
 
 11 
 
 414 
 
 453 
 
 492 
 
 531 
 
 569 
 
 607 
 
 645 
 
 682 
 
 719 
 
 , 755 
 
 12 
 
 792 
 
 828 
 
 864 
 
 899 
 
 934 
 
 969 
 
 1004 
 
 1038 
 
 1072 
 
 1 106 
 
 13 
 
 1 139 
 
 1173 
 
 1206 
 
 1239 
 
 1271 
 
 1303 
 
 335 
 
 367 
 
 399 
 
 430 
 
 14 
 
 461 
 
 492 
 
 523 
 
 553 
 
 584 
 
 614 
 
 644 
 
 673 
 
 703 
 
 732 
 
 15 
 
 I76I 
 
 1790 
 
 1S18 
 
 1847 
 
 1875 
 
 1903 
 
 1931 
 
 1959 
 
 1987 
 
 2014 
 
 16 
 
 2041 
 
 2068 
 
 2095 
 
 2122 
 
 2148 
 
 2175 
 
 2201 
 
 2227 
 
 2253 
 
 279 
 
 17 
 
 304 
 
 330 
 
 |55 
 
 380 
 
 405 
 
 430 
 
 455 
 
 480 
 
 504 
 
 529 
 
 18 
 
 553 
 
 577 
 
 601 
 
 625 
 
 648 
 
 672 
 
 695 
 
 718 
 
 742 
 
 765 
 
 19 
 
 7SS 
 
 
 833 
 
 ^56 
 
 878 
 
 900 
 
 923 
 
 945 
 
 967' 
 
 989 
 
 £0 
 
 3010 
 
 3032 
 
 3054 
 
 3075 
 
 3096 
 
 3118 
 
 3139 
 
 .3160 
 
 31S1 
 
 3201 
 
 21 
 
 222 
 
 243 
 
 263 
 
 284 
 
 304 
 
 324 
 
 345 
 
 365 
 
 385 
 
 404 
 
 22 
 
 424 
 
 444 
 
 464 
 
 483 
 
 502 
 
 522 
 
 541 
 
 560 
 
 579 
 
 598 
 
 23 
 
 617 
 
 636 
 
 655 
 
 674 
 
 692 
 
 711 
 
 729 
 
 747 
 
 766 
 
 784 
 
 24 
 
 802 
 
 820 
 
 838 
 
 856 
 
 874 
 
 892 
 
 909 
 
 927 
 
 945 
 
 962 
 
 25 
 
 3979 
 
 3997 
 
 4014 
 
 4031 
 
 4048 
 
 4065 
 
 4082 
 
 4099 
 
 4116 
 
 4133 
 
 26 
 
 4150 
 
 4166 
 
 183 
 
 200 
 
 216 
 
 232 
 
 249 
 
 265 
 
 281 
 
 
 27 
 
 314 
 
 330 
 
 346 
 
 362 
 
 378 
 
 393 
 
 409 
 
 425 
 
 440 
 
 456 
 
 28 
 
 472 
 
 487 
 
 502 
 
 518 
 
 533 
 
 548 
 
 564 
 
 579 
 
 594 
 
 609 
 
 29 
 
 624 
 
 639 
 
 654 
 
 669 
 
 683 
 
 698 
 
 7^3 
 
 728 
 
 742 
 
 757 
 
 30 
 
 4771 
 
 4786 
 
 4800 
 
 4814 
 
 4829 
 
 4843 
 
 4857 
 
 4871 
 
 4886 
 
 4900 
 
 31 
 
 914 
 
 928 
 
 942 
 
 955 
 
 969 
 
 983 
 
 997 
 
 501 1 
 
 5024 
 
 5038 
 
 32 
 
 5051 
 
 5065 
 
 5079 
 
 5092 
 
 5105 
 
 5"9 
 
 5132 
 
 145 
 
 159 
 
 172 
 
 33 
 
 185 
 
 198 
 
 211 
 
 224 
 
 237 
 
 250 
 
 263 
 
 276 
 
 289 
 
 302 
 
 34 
 
 315 
 
 328 
 
 340 
 
 353 
 
 366 
 
 378 
 
 391 
 
 403 
 
 416 
 
 428 
 
 35 
 
 5441 
 
 5453 
 
 5465 
 
 5478 
 
 5490 
 
 5502 
 
 55H 
 
 5527 
 
 5539 
 
 5551 
 
 36 
 
 5^^ 
 
 575 
 
 587 
 
 599 
 
 611 
 
 623 
 
 635 
 
 647 
 
 658 
 
 670 
 
 37 
 
 682 
 
 694 
 
 705 
 
 717 
 
 729 
 
 740 
 
 752 
 
 763 
 
 775 
 
 786 
 
 38 
 
 798 
 
 809 
 
 821 
 
 832 
 
 843 
 
 855 
 
 866 
 
 877 
 
 888 
 
 899 
 
 39 
 
 911 
 
 922 
 
 933 
 
 944 
 
 955 
 
 966 
 
 977 
 
 988 
 
 999 
 
 6010 
 
 40 
 
 6021 
 
 6031 
 
 6042 
 
 6053 
 
 6064 
 
 6075 
 
 6085 
 
 6096 
 
 6107 
 
 6117 
 
 41 
 
 128 
 
 138 
 
 149 
 
 160 
 
 170 
 
 180 
 
 191 
 
 201 
 
 212 
 
 222 
 
 42 
 
 232 
 
 243 
 
 253 
 
 263 
 
 274 
 
 284 
 
 294 
 
 304 
 
 3H 
 
 325 
 
 43 
 
 335 
 
 345 
 
 355 
 
 365 
 
 375 
 
 3?5 
 
 395 
 
 405 
 
 415 
 
 425 
 
 44 
 
 435 
 
 444 
 
 454 
 
 464 
 
 474 
 
 484 
 
 493 
 
 503 
 
 513 
 
 522 
 
 45 
 
 6532 
 
 6542 
 
 6551 
 
 6561 
 
 6571 
 
 6580 
 
 6590 
 
 6599 
 
 6609 
 
 6618 
 
 46 
 
 628 
 
 637 
 
 646 
 
 656 
 
 665 
 
 675 
 
 684 
 
 693 
 
 702 
 
 712 
 
 47 
 
 721 
 
 730 
 
 739 
 
 749 
 
 758 
 
 767 
 
 776 
 
 785 
 
 794 
 
 803 
 
 48 
 
 812 
 
 821 
 
 830 
 
 839 
 
 848 
 
 857 
 
 866 
 
 875 
 
 884 
 
 893 
 
 49 
 
 902 
 
 911 
 
 920 
 
 928 
 
 937 
 
 946 
 
 955 
 
 964 
 
 972 
 
 981 
 
 50 
 
 6990 
 
 6998 
 
 7007 
 
 7016 
 
 7024 
 
 7033 
 
 7042 
 
 7050 
 
 7059 
 
 7067 
 
 61 
 
 7076 
 
 7084 
 
 093 
 
 lOI 
 
 no 
 
 
 126 
 
 135 
 
 143 
 
 152 
 
 62 
 
 160 
 
 168 
 
 177 
 
 185 
 
 ^93 
 
 202 
 
 210 
 
 
 226 
 
 235 
 
 53 
 
 243 
 
 251 
 
 259 
 
 267 
 
 27s 
 
 284 
 
 292 
 
 300 
 
 308 
 
 316 
 
 54 
 
 324 
 
 332 
 
 340 
 
 348 
 
 356 
 
 364 
 
 372 
 
 380 
 
 388 
 
 396 
 
 No. 
 
 
 
 1 
 
 2 3 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
LOrxARTTIIMS. 
 
 295 
 
 No. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 1 9 
 
 66 
 
 7404 
 
 7412 
 
 7419 
 
 7427 
 
 7435 
 
 7443 
 
 7451 
 
 7459 
 
 7466 
 
 7474 
 
 56 
 
 482 
 
 490 
 
 497 
 
 505 
 
 513 
 
 520 
 
 528 
 
 536 
 
 543 
 
 551 
 
 57 
 
 559 
 
 566 
 
 574 
 
 582 
 
 589 
 
 597 
 
 604 
 
 612 
 
 619 
 
 627 
 
 68 
 
 634 
 
 642 
 
 649 
 
 657 
 
 664 
 
 672 
 
 679 
 
 686 
 
 694 
 
 701 
 
 59 
 
 709 
 
 716 
 
 723 
 
 731 
 
 738 
 
 745 
 
 752 
 
 760 
 
 767 
 
 774 
 
 60 
 
 7782 
 
 7789 
 
 7796 
 
 7803 
 
 7810 
 
 7818 
 
 7825 
 
 7832 
 
 7839 
 
 7846 
 
 61 
 
 853 
 
 860 
 
 868 
 
 875 
 
 882 
 
 889 
 
 896 
 
 903 
 
 910 
 
 917 
 
 02 
 
 924 
 
 931 
 
 938 
 
 945 
 
 952 
 
 959 
 
 966 
 
 973 
 
 980 
 
 987 
 
 G3 
 
 993 
 
 8000 
 
 8007 
 
 8014 
 
 8021 
 
 8028 
 
 8035 
 
 8041 
 
 8048 
 
 8055 
 
 64 
 
 8062 
 
 069 
 
 075 
 
 082 
 
 089 
 
 096 
 
 102 
 
 109 
 
 116 
 
 122 
 
 65 
 
 8129 
 
 8136 
 
 8142 
 
 8149 
 
 8156 
 
 8162 
 
 8169 
 
 8176 
 
 8182 
 
 8189 
 
 66 
 
 195 
 
 202 
 
 209 
 
 215 
 
 222 
 
 228 
 
 235 
 
 241 
 
 248 
 
 254 
 
 67 
 
 261 
 
 267 
 
 274 
 
 280 
 
 287 
 
 293 
 
 299 
 
 306 
 
 312 
 
 319 
 
 68 
 
 \'5. 
 
 331 
 
 338 
 
 344 
 
 351 
 
 357 
 
 363 
 
 370 
 
 376 
 
 382 
 
 69 
 
 3^S 
 
 395 
 
 401 
 
 407 
 
 414 
 
 420 
 
 426 
 
 432 
 
 439 
 
 445 
 
 70 
 
 8451 
 
 8457 
 
 8463 
 
 8470 
 
 8476 
 
 8482 
 
 8488 
 
 8494 
 
 8500 
 
 8506 
 
 71 
 
 513 
 
 5^9 
 
 525 
 
 531 
 
 537 
 
 543 
 
 549 
 
 555 
 
 561 
 
 567 
 
 72 
 
 573 
 
 579 
 
 585 
 
 591 
 
 597 
 
 ■ 603 
 
 609 
 
 615 
 
 621 
 
 627 
 
 73 
 
 (^33 
 
 639 
 
 645 
 
 651 
 
 657 
 
 663 
 
 669 
 
 675 
 
 681 
 
 686 
 
 74 
 
 692 
 
 698 
 
 704 
 
 710 
 
 716 
 
 722 
 
 727 
 
 733 
 
 739 
 
 745 
 
 76 
 
 8751 
 
 8756 
 
 8762 
 
 8768 
 
 8774 
 
 8779 
 
 8785 
 
 8791 
 
 8797 
 
 8S02 
 
 76 
 
 808 
 
 814 
 
 820 
 
 825 
 
 831 
 
 837 
 
 842 
 
 848 
 
 854 
 
 859 
 
 77 
 
 865 
 
 871 
 
 876 
 
 882 
 
 887 
 
 893 
 
 899 
 
 904 
 
 910 
 
 915 
 
 78 
 
 921 
 
 927 
 
 932 
 
 938 
 
 943 
 
 949 
 
 954 
 
 960 
 
 965 
 
 971 
 
 79 
 
 976 
 
 982 
 
 987 
 
 993 
 
 998 
 
 9004 
 
 9009 
 
 9015 
 
 9020 
 
 9025 
 
 80 
 
 9031 
 
 9036 
 
 9042 
 
 9047 
 
 9053 
 
 9058 
 
 9063 
 
 9069 
 
 9074 
 
 9079 
 
 81 
 
 085 
 
 090 
 
 096 
 
 lOI 
 
 106 
 
 112 
 
 117 
 
 122 
 
 128 
 
 133 
 
 82 
 
 I3« 
 
 143 
 
 149 
 
 154 
 
 159 
 
 165 
 
 170 
 
 175 
 
 180 
 
 186 
 
 83 
 
 191 
 
 196 
 
 201 
 
 206 
 
 212 
 
 217 
 
 222 
 
 227 
 
 232 
 
 238 
 
 84 
 
 243 
 
 248 
 
 253 
 
 258 
 
 263 
 
 269 
 
 274 
 
 279 
 
 284 
 
 289 
 
 85 
 
 9294 
 
 9299 
 
 9304 
 
 9309 
 
 9315 
 
 9320 
 
 9325 
 
 9330 
 
 9335 
 
 9340 
 
 86 
 
 345 
 
 350 
 
 355 
 
 360 
 
 365 
 
 370 
 
 375 
 
 380 
 
 385 
 
 390 
 
 87 
 
 395 
 
 400 
 
 405 
 
 410 
 
 415 
 
 420 
 
 425 
 
 430 
 
 435 
 
 440 
 
 88 
 
 445 
 
 450 
 
 455 
 
 460 
 
 465 
 
 469 
 
 474 
 
 479 
 
 484 
 
 489 
 
 89 
 
 494 
 
 499 
 
 504 
 
 509 
 
 513 
 
 518 
 
 523 
 
 528 
 
 533 
 
 538 
 
 90 
 
 9542 
 
 9547 
 
 9552 
 
 9557 
 
 9562 
 
 9566 
 
 9571 
 
 9576 
 
 9581 
 
 95S6 
 
 91 
 
 590 
 
 595 
 
 600 
 
 605 
 
 609 
 
 614 
 
 619 
 
 624 
 
 628 
 
 633 
 
 92 
 
 638 
 
 643 
 
 647 
 
 632 
 
 657 
 
 661 
 
 666 
 
 671 
 
 675 
 
 680 
 
 93 
 
 685 
 
 689 
 
 694 
 
 697 
 
 703 
 
 708 
 
 713 
 
 717 
 
 722 
 
 727 
 
 94 
 
 731 
 
 736 
 
 741 
 
 745 
 
 750 
 
 754 
 
 759 
 
 763 
 
 768 
 
 773 
 
 95 
 
 9777 
 
 9782 
 
 9786 
 
 9791 
 
 9795 
 
 9800 
 
 9805 
 
 9809 
 
 9814 
 
 9818 
 
 96 
 
 «23 
 
 827 
 
 832 
 
 836 
 
 841 
 
 845 
 
 850 
 
 854 
 
 859 
 
 S63 
 
 97 
 
 868 
 
 872 
 
 877 
 
 881 
 
 886 
 
 890 
 
 894 
 
 899 
 
 903 
 
 908 
 
 98 
 
 912 
 
 917 
 
 921 
 
 926 
 
 930 
 
 934 
 
 939 
 
 943 
 
 948 
 
 952 
 
 99 
 
 956 
 
 961 
 
 965 
 
 969 
 
 974 
 
 978 
 
 983 
 
 987 
 
 991 
 
 996 
 
 No. 
 
 
 
 1 2| 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 8 
 
 9 
 
296 ALGEBKA. 
 
 USE OF THE TABLE. 
 
 351. The table (pages 294 and 295) gives the mantissae 
 of the logarithms of all numbers from 100 to 1000, calcu- 
 lated to four decimal places. 
 
 352. To find the logarithm of any number of three figures. 
 Find in the column headed " No." the first two fiorures of 
 
 the given number. Then the mantissa required will be found 
 in the corresponding horizontal line, in the vertical column 
 headed by the third figure of the number. 
 
 Finally, prefix the characteristic by the rules of Art. 339. 
 
 For example, log 168 = 2.2253. 
 
 If only the last three figures of the mantissa are found, 
 the first figure may be obtained from the nearest mantissa 
 above, in the same column, consisting of four figures. 
 
 Thus, log.344= 9.5366-10. 
 
 353. For numbers of one or two figures, the column 
 headed may be used ; for, by Art. 349, log 83 has the same 
 mantissa as log 830, and log 9 the same mantissa as log 900. 
 
 Thus, log83 = 1.9191, and log9 = 0.9542. 
 
 354. To find the logarithm of a number of more than three 
 figures. 
 
 For example, required the logarithm of 327.6. 
 
 We find from the table, log 327 = 2.5145. 
 
 log 328 = 2.5159. 
 
 That is, an increase of one unit in the number produces an 
 increase of .0014 in the logarithm. Therefore an increase 
 of .6 of a unit in the number will produce an increase of 
 .6 X .0014 in the logarithm, or .0008 to the nearest fourth 
 decimal place. Hence, 
 
 log327.6 = 2.5145 + .0008 = 2.5153. 
 
LOGARITHMS. 297 
 
 Note. The difference in the table between any mantissa and the 
 mantissa of the next higher number of three figures, is called the tabu- 
 lar difference. It may always be obtained mentally. 
 
 The following rule is derived from the above : 
 
 Find from the table the mantissa ofthejirst three significant 
 figures^ and the tabular difference. 
 
 Multiply the latter by the remaining figures of the number, 
 with a decimal j^oint before them. 
 
 Add the result to the mantissa of the first three figures, and 
 prefix the proper chara^iteristic. 
 
 355. 1. Find the logarithm of .021508. 
 
 Mantissa of 215 = 8324 
 
 Tabular difference = 21 2 
 
 .08 3326 
 
 Correction = 1 .68 = 2 nearly. 
 
 Ans. 8.3326 - 10. 
 
 EXAMPLES. 
 Find the logarithms of the following : 
 
 2. 
 
 80. 
 
 6. 
 
 .7723. 
 
 10. 
 
 20.08. 
 
 14. 
 
 5.1809. 
 
 3. 
 
 6.3. 
 
 7. 
 
 1056. 
 
 11. 
 
 92461. 
 
 15. 
 
 1036.5. 
 
 4. 
 
 298. 
 
 8. 
 
 3.294. 
 
 12. 
 
 .40322. 
 
 16. 
 
 .086676. 
 
 5. 
 
 .902. 
 
 9. 
 
 .05205. 
 
 13. 
 
 .007178. 
 
 17. 
 
 .11507. 
 
 356. To find the mimber corresponding to a logarithm. 
 1. Required the number whose logarithm is 1.6571. 
 
 Find in the table the mantissa 6571. In the correspond- 
 ing line, in the column headed " No.,'* we find 45, the first 
 two figures of the required number, and at the head of the 
 column we find 4, the third figure. 
 
 Since th^ characteristic is 1 , there must be two figures to 
 the left of the decimal point (Art. 339). Hence, 
 Number corresponding to 1.6571 = 45.4. 
 
298 ALGEBRA. 
 
 2. Required the number whose logarithm is 2.3934. 
 
 We find in the table the mantissa 3927, of which the 
 corresponding number is 247, and the mantissa 3945, of 
 which the corresponding number is 248. 
 
 That is, an increase of 18 in the mantissa produces an 
 increase of one unit in the number corresponding. Tlierefore, 
 an increase of 7 in the mantissa will produce an increase of 
 ^8 of a unit in the number, or .39 nearly. Hence, 
 
 Number corresponding = 247 + .39 = 247.39. 
 
 We derive the following rule from the above operation : 
 
 Find from the table the next less mantissa^ the three figures 
 corresponding^ and the tabular difference. 
 
 Subtract the next less from the given mantissa, and divide 
 the remainder by the tabular difference. 
 
 Annex the quotient to the first three figures of the number, 
 and point off the result. 
 
 357. 1. Find the number whose logarithm is 7.5264 — 10. 
 
 5264 
 Next less mantissa = 5263 ; three figures corresponding = 336. 
 Tabular difference = 13) 1.000(. 077 = .08 nearly. 
 
 91 
 
 90 
 
 Since the characteristic is 7 — 10, there must be two 
 ciphers between the decimal point and first significant figure 
 (Art. 339). Hence, 
 
 Number corresponding = .0033608, Ans. 
 
 Note. In computations with four-place logarithms, the results can- 
 not usually be depended upon to more than ybur significant figures. 
 
LOGABITHMS. 299 
 
 EXAMPLES. 
 
 Find the numbers correspondiDg to the following loga- 
 rithms : 
 
 2. 1.8055. 7. 8.1648-10. 12. 1.6482. 
 
 3. 9.4487-10. 8. 7.5209-10. 13. 7.0450-10. 
 
 4. 0.2165. 9. 4.0095. 14. 4.8016. 
 
 5. 3.9487. 10. 0.9774. 15. 8.1144-10. 
 
 6. 2.7364. 11. 9.3178-10. 16. 2.7015. 
 
 APPLICATIONS. 
 
 358. The value of an arithmetical quantity, in which the 
 operations indicated involve only multiplication, division, 
 involution, or evolution, may be most conveniently found by 
 logarithms. 
 
 The utility of the process consists in the fact that addition 
 takes the place of multiplication, subtraction of division, 
 multiplication of involution, and division of evolution. 
 
 In operations with negative characteristics the rules of 
 Algebra must be followed. 
 
 1. Find the value of .0631 X 7.208 x .51272. 
 
 By Art. 342, log(.0631 x 7.208 x .51272) 
 
 = log .0631 + log 7.208 + log .51272. 
 log .0631= 8.8000-10 
 log 7.208= 0.8578 
 log. 51272= 9.7099-10 
 
 Adding, .'. log of result = 19.3677 — 20 
 
 = 9.3677-10 (see Note 1). 
 
 Number corresponding to 9.3677 — 10 = .2332, Ans. 
 
 Note 1. If the sum is a negative logarithm, it should be reduced so 
 that the negative part of tlie characteristic may be — 10. 
 Thus, 19.3677-20 is reduced to 9.3077-10. 
 
300 ALGEBRA. 
 
 336.8 
 
 2. Find the value of 
 
 7984 
 
 By Art. 344, log ^|^ = log 336.8 - log 7984. 
 
 log 336.8 = 12.5273 -10 (see Note 2). 
 los 7984= 3.9022 
 
 Subtracting, .*. log of result = 8.6251 — 10 
 
 Number corresponding = .04218, Ans. 
 
 Note 2. To subtract a greater logarithm from a less, or to subtract 
 a negative logarithm from a positive, increase the characteristic of tlie 
 minuend by 10, writing — 10 after the mantissa to compensate. 
 
 Thus, to subtract 3.9022 from 2.5273, write the minuend in the form 
 12.5273-10; subtracting 3.9022 from this, the result is 8.6251 - 10. 
 
 3.^ Find the value of (.07396)^ 
 By Art. 346, log (.07396)^ = 5 x log .07396. 
 log.07396 = 8.8690 -10 
 
 44.3450 - 50 
 = 4.3450-10 
 = log .000002213, ^Tzs. 
 
 4. Find the value of A/m5063. 
 
 By Art. 347, log ^.035063 = ilog .035063. 
 
 o 
 
 log .035063 = 8.5449 - 10 
 
 20. -20 (see Note 3). 
 
 3)28.5449-30 
 
 9.5150-10 
 
 = log .3274, Ans, 
 
 Note 3. To divide a negative logarithm, add to both parts such a 
 multiple of 10 as will make the quantity standing after the mantissa 
 exactly divisible by the divisor, with — 10 as the quotient. 
 
 Thus, to divide 8.5449 — 10 by 3, add 20 to both parts of the loga- 
 rithm, giving the result 28.5449 — 30. Dividing this by 3, the quotient 
 is 9.5150 - 10. 
 
LOGARITHMS. 301 
 
 ARITHMETICAL COMPLEMENT. 
 
 359. The Arithmetical Complement of the logarithm of a 
 number, or, briefly, the cologarithm of the number, is the 
 logarithm of the reciprocal of that number. 
 
 Thus, colog 409 = log-i = log 1 - log 409. 
 
 log 1 = 10. - 10 (Note 2, Art. 358.) 
 
 Io2 409= 2.6117 
 
 .-. colog 409 = 7.3883 - 10. 
 colog .0G7 = log— i^ = log 1 — log .067. 
 
 log 1 = 10. -10 
 
 log .067= 8.8261-10 
 .-.colog .067= 1.1739. 
 
 Note. The cologarithm may be calculated mentally from the loga- 
 rithm by subtracting the last signijicant figure from 10, and each of the 
 others from 9. 
 
 360. Example. Find the value of ^-;r^^r— — 777^' 
 
 , .51384 \ ( nao^ ^ 1 ., 1 
 
 8.709 X .0946 V 8.709 .0946 
 
 log .51384 + log -4t;-+ log ^ 
 
 8.709 ^.0946 
 = log .51384 + colog 8.709 + colog .0946. 
 log .51384 = 9.7109 -10 
 colog8.709 = 9.0601 -10 
 coloff.0946 = 1.0241 
 
 9.7951 — 10 = log .6239, Ans. 
 
 It is evident from the above that the logarithm of a frac- 
 tion is equal to the logarithm of the numerator plus the 
 cologarithm of the denominator. 
 
302 ALGEBRA. 
 
 Or, in general, to find the logarithm of a fraction whose 
 terms are composed of factors. 
 
 Add together the logarithms of the factors of the numerator^ 
 and the cologarithms of the factors of the denominator. 
 
 Note. The value of the above fraction may be found without using 
 cologarithms, as follows : 
 
 '°« 8.709x'o946 = '"^ ■'''^ ' "« <»'™« ^ "^"^ 
 
 = log .51384 - (log 8.709 + log .0946). 
 
 The advantage of the use of cologarithms is that it exhibits the writ- 
 ten work of computation in a more compact form. 
 
 EXAMPLES. 
 
 361. Note. A negative quantity can have no common logarithm, 
 as is evident from the definition of Art. 334. If negative quantities 
 occur in computation, they may be treated as if they were positive, and 
 the sign of the result determined irrespective of the logarithmic work. 
 See Ex. 34. 
 
 Find. by logarithms the values of the following : 
 
 1. 9.238 X. 9152. 4. (-4.3264) x (-.050377). 
 
 2. 130.36 X .08237. 5. .27031 x .042809. 
 
 3. 721.3 x(- 3.0528). 6. (- .063165) x 11.134. 
 
 7 401.8 g - .3384 ^^ 225^8^ 
 
 52.37' * .08659 * ' 64327* 
 
 g 7.2321 ^Q 9.163 ^g .007514 
 
 10.813* ' .0051422* * -.015822* 
 
 13 3.3681 j^ 15.008 X (-.0843) 
 
 12.853 X .6349* * .06376x4.248 
 
 ^g (-2563)X .03442 
 714.8 X(- .511) * 
 
 16 121.6 X(- 9.025) 
 
 (-48.3) X 3662 X (-.0856)* 
 
LOGARITHMS. 303 
 
 17. (23.86)3. 22. (.8)1 27. V'^. 
 
 18. (.532)». 23. (-3.16)i 28. V^im. 
 
 19. (-1.0246)^ 24. (.021)i 29. v.02305. 
 
 20. (.09323)^ 25. V2. 30. a/IoOO. 
 
 21. 5^. 26. \/5. 31. -y/-.00\)Di. 
 
 32. -{/.OOOlOll. 
 2^5 
 
 33. Find the value of 
 
 3^ 
 
 log?:^ = log 2 + log ^5 + colog 3'^ (Art. 360.) 
 3^ 
 
 = log2 + ilog5-l-fcolog3. 
 o o 
 
 log 2= .3010 
 
 log 5 = .6990 ; divide by 3 = .2330 
 
 colog 3 = 9.5229 - 10 ; multiply by ^ = 9.6024 - 10 
 
 .1364 
 = log 1.369, Ans. 
 
 34. Find the value of a/ J ,,,.,^ — 
 
 .902 
 
 of#^ 
 \ 7.962 
 
 W «M)3296 ^ 1 J ^^03296 ^ 1 ,j ^g^gg _ ^ 7 ggg) . 
 ^ \ 7.962 3 ^ 7.962 3 ^ "^ & ^ 
 
 log .03296= 8.5180-10 
 log 7.962 = 0.9010 
 
 . 3 )27.6170^30 
 
 9.2057 -10 = log. 1606. 
 
 Alls. -.1606. 
 
304 ALGEBRA. 
 
 Find the values of the following : 
 
 
 4f \13 
 
 ol 3 >.« 100- 
 
 37 __^I_. 42. xV-- ^7. 
 
 (_10)t ^ 7 (.7325)^ 
 
 6M 43. 5l2_^ si 3. 4g^ V. 0001289 
 
 ^l^ \^ * -</.0008276 
 
 39. f-^y. 44. ^2x^3x^^/:oI. 49. (-7^^^)l 
 V^^^/ -(.2345)^ 
 
 ^Q V:Q073 gg^ (18.9503)" x(-.l)". 
 
 (.68291)^ 
 
 51. V5.9 55 X V gTTg^ 54^ ^3734.9 x. 00001108. 
 
 ■</298.54 
 
 52. (538.2 X .0005969)*. 55. (2.6317)^ x (.71272)1 
 
 56 v -'008193 X (.06285) 
 -.98342 
 
 57. Vj035 X V.62667 X V.0072103. 
 
 EXPONENTIAL EQUATIONS. 
 
 362. An Exponential Equation is one in which the un- 
 known quantit}^ occurs as an exponent. 
 
 To solve an equation of this form, take the logarithms of 
 both members ; the result will be an equation which can be 
 solved by ordinary algebraic methods. 
 
LOGARITHMS. 305 
 
 363. 1. Given SP = 23 ; find the value of a;. 
 Taking the logarithms of both members, 
 
 log(3P) = log23. 
 Or (Art. 346), 
 
 a; log 31= log 23. 
 
 Whence, x = 1^^ = ^^^^ = .91303, Ans. 
 
 log 31 1.4914 
 
 2. Given .2=^ = 3 ; find the value of x. 
 
 Taking the logarithms of both members, 
 
 .r log.2 = log 3. 
 
 ,^^ log 3 .4771 .4771 
 
 Whence, x=—^ 
 
 log.2 9.3010-10 -.6990 
 
 = -.6825, Ans. 
 
 EXAMPLES. 
 Solve the following equations : 
 
 3. ir = 3. 6. 13* = .281. 7. a'^=6"'c". 
 
 4. .3^=. 8. 6. .703^=1.096. 8. ma-=n. 
 
 9. Given a, r, and I ; derive the formula for n. ( Art.322.) 
 
 10. Given a, ?', and S ; derive the formula for n. 
 
 11. Given a, I, and S ; derive the formula for 7i. 
 
 12. Given r, I, and S ; derive the formula for n. 
 
ANSWERS. 
 
 Note. In the following collection of answers, all those are omitted 
 which, if given, would destroy the utility of the example. 
 
 
 
 Art. 40 ; pages 
 
 8 and 9. 
 
 
 
 1. 
 
 19. 
 
 6. 
 
 9 
 
 16* 
 
 
 11. 
 
 48. 
 
 16. 
 
 4. 
 
 2. 
 
 1. 
 
 7. 
 
 6*. 
 
 
 12. 
 
 114. 
 
 17. 
 
 9 
 5* 
 
 3. 
 
 24. 
 
 8. 
 
 9. 
 
 
 13. 
 
 24. 
 
 18. 
 
 50 
 39 
 
 4. 
 
 15f. 
 
 9. 
 
 628 
 135 
 
 
 14. 
 
 204. 
 
 19. 
 
 36 
 
 6. 
 
 47 
 12* 
 
 10. 
 21. 
 
 1369 
 144 
 
 3. 
 
 
 16. 
 22. 
 
 310. 
 
 35 
 62* 
 
 20. 
 
 48, 
 
 
 
 
 Art. 43 ; 
 
 page 12 
 
 
 
 
 4. 35 and 7. 8. A, S20; B, $60; C, $180. 
 
 6. A, 31 ; B, 37. 9. 52 and 73. 
 
 6. A, $536; B, $664. 10. 13, 39, and 46. 
 
 7. $0.93. 11. A, $28; B, $43; C, $56. 
 
 12. Horse, $275; carriage, $100; harness, $25. 
 
 13. 15, 45, and 48. 14. 35, 70, and 105. 
 15. Cow, $45; sheep, $18; hog, $12. 
 
 Art. 57 ; page 19. 
 4. 2a-2b-h2d. 6. 4.aK 7. 2a^-ah. 
 
308 ALGEBRA. 
 
 8. 5a^-8a^-6. 12. -a^ -\-3x-\- 2. 
 
 9. 6mn-ab-4:C—x+S7)i\ 13. 3a + 36 4-3c + 3d. 
 
 10. x-6y. 14. 6a'-3ab'-Ab\ 
 
 11. -x + 3m-7n. 15. a3-4a^. 
 
 Art. 64 ; pages 22 and 23. ' 
 
 6. 4a6. 10. 66 + 1. 
 
 7. — 4a6c— 14a;— 22/— 148. 11. 4m — Sw — r + 3s. 
 
 8. 2m-42/- + 12a + l. 12. 6d-26-3a-3c. 
 
 9. 14af-82/2 + 5a6-7. 13. 6m' -{-9 n"- 71 x, 
 
 14. -3a-26 + 10c-13d + 2a;. 
 
 15. 2a- 6- 3c. 20. 2x-3z. 
 
 16. a;*~a^-3a^-8aj+ll. 21. Ga^ + Oa^ + a^- a- 6. 
 
 17. 4:aP-6w^b-2ab' -9b\ 22. -Ss(^-5x'y+8xy'+2f. 
 
 18. 4a^-3aH6a2-6a+3. 23. 5i»3 + 7^ + 18a;. 
 
 19. —x^ — xy. 24. -a2-4a6 + 52. 
 
 25. 5a^ + 6a-2/-7/-6a;-72/ + 6. 
 
 26. 3aj'-5a;*-a^-lla:2_|_4^_2. 
 
 27. 4a^+9a^2/-4a?2/'-32/'. 28. Aa^ + la^ -2a-4. 
 
 Art. 69 ; pages 25 and 26. 
 
 4. 5x — y. 10. a+&— c+(^— e. 16. — 3a — 1. 
 
 5. a-b + c. 11. 2^. 17. 4a; -2. 
 
 6. om'-6n-4:a. 12. 14a;+2. 18. 6m + 2. 
 
 7. _a2-36^ 13. 6m -371. 19. 0. 
 
 8. 2a + 2. 14. 2x + 4:y. 20. a + 26. 
 
 9. a— 6+c— d+e. 15. a — c. 
 
 Art. 82 ; pages 32 to 34. 
 
 4. 15a2_ii^_14^ 6^ _6a2+16a&-862. 
 
 5. -12a;2_^28a;-15. 7. 50a;2^2_ig^ 
 
ANSWERS. 309 
 
 8. h^-o?. 11. Qx^-lQ>o?y-^^xy--\-^f. 
 
 9. I0a''h''-^a?h^-l^a^h\ 12. 2m^-8m»ri + 18mw^ 
 10. a:(^ — a. 13. a;* + 4a; + 3. 
 
 14. 30a^-43a26 + 39a62-2063. 
 15. 8a^-14«2_ig^_^21. 16. a^ -62 + 26c-c2. 
 
 17. 2x^-a^H-8a;-5. 
 
 18. 6a;*+13a.'3-70a^ + 71a;-20. 
 
 19. 6a;*-19ar^ + 22a; + 5. 20. -m^-37m2-f-70m-50. 
 
 21. -6ar'-25a;4-f7ar''4-81a^ + 3a;-28. 
 
 22. 2 a^fe^ - 3 a^W - 7 a^d^ + 4 a^h'. 
 
 23. 4iB2'«+V-lCa;'"+«2/""'' + 12x«/'»-^ 
 
 24. a^ + a:^2/"4-2/^. 26. 12af'-|-7ic*+5a:3+10a;-4. 
 
 25. 16a^ + 4a262 4-6^ 27. m'' -2im'n + mn'-Zn\ 
 
 28. 243ar^-81a;*2/-3a^ + 2/^. 
 
 29. a^ - ba'h + lOa^fe^ - lOtrft^ + 5a6* - 6*. 
 
 30. x^ + f + 7^-Zxyz. 
 
 31. 6a:«-llar'4-14a;*-12a7^ + 6ar^-23a;4-5. 
 
 32. 0^52 + c2(Z2-a2c2- 62(^2. 36. 36m3-49??i + 20. 
 
 33. 8a«-98a^+152a2-32. 37. 9a;*- 13ar^ + 4. 
 
 34. 3?-Qx^-\^x+M. 38. x« + a;* + l. 
 
 35. a^-h\ 39. a«-6«. 
 
 40. m*-5m2 + 4. 
 
 41. 120a;* + 20a:3_ 111 a^_ 143.^24. 
 
 42. a«-6a*6- + 9a26*-46«. 
 
 Art. 83 ; pages 34 and 35. 
 
 2. a^-\-lr + (?-\-d^'-\-'iab + 2ac-^2ad-\-2hc-^2hd + 2cd. 
 
 3. ah-^ac — 2ad — 2hc-\-hd-\-cd. 4. a;^. 
 
 6. 4a6H-46c. 7. a''-2a'ly' + h\ 9. 4a + 2a2 + 2a3. 
 6. h\ 8. l-a:«. 10. a;2_4^.^_^4y2_9^2_ 
 
31C 
 
 ) 
 
 ALGEBRA. 
 
 
 
 11. 
 
 a^-f. 
 
 13. 4a2+462+4c2. 
 
 15. 
 
 -iyz. 
 
 12. 
 
 y'-dK 
 
 14. ab-\-bc+ca-a^-b^-c^. 
 
 16. 
 
 0. 
 
 17. 
 
 Sac. 
 
 18. -9m* + 82mV-9?i^ 
 
 19. 
 
 0. 
 
 Art. 93 ; pages 41 to 43. 
 
 4. 2a;-5. 7. 8-5a;. 10. Sx-1. 
 
 5. 2 + ax. 8. Sb^-4aK 11. 4m2-10m + 7. 
 
 6. a -2b. 9. -2a^-2ax. 12. dx'-Sxy-i-y^ 
 
 13. 8m^ + 4m2 + 2m + l. 15. 4^2 + 12a6 + 962. 
 
 14. a — 6 — c. 16. ay^ — xy-\-y^. 
 
 17. a;-3. 19. 2x^-Sx-6. 21. 2a + 3. 
 
 18. 4m2-3n2. 20. 2rc2-7fl;-8. 22. a^-Sa;-?/. 
 23. a^-Sa*b + 9a^b^-27b\ 24. a;-2/ + ^- 
 
 25. 3a^-{-6x'-2x-A. 
 
 26. 2/ (a^^ — a^2/ + ^/ — •'^2/''^ + 2/^) • 
 27. 5m2-4m + 3. 28. 3a^-2a; + l. 
 
 29. 3(a^-{-a^b + a'b^ + ab^ + b*). 
 
 30. 8a^ + 4a; + l. 33. a^-2ic-l. 36. 2ar^- a:+ 1. 
 
 31. 9a^-9a-4. 34. 3aj2-5iC2/-22/^ 37. 3^.-2 + 60^ + 9. 
 
 32. x'-2xy + y\ 35. a'+Sab-^db^ 38. a;2_2aj_3. 
 
 40. x-\-y. 
 
 41. x^y — fl??/^. 
 
 42. a^-2x^-^x-2. 
 
 43. a^-2a26-5a62 + 753^ 
 
 44. a^-2a^-a; + l. 
 
 45. x + 2y-Sz. 
 
 46. a" — 6'" + c^ 
 
 47. a^-^2x^-x-\-l. 
 
 48. 2a2-a6 + 262. 
 
 50. 
 
 x-\-a. 
 
 51. 
 
 {b-\-c)a^-}-bc. 
 
 52. 
 
 (a; + 2/)-3. 
 
 53. 
 
 (a + 6)2-(a + &) + l. 
 
 54. 
 
 x + a. 
 
 55. 
 
 (m — ny—2(m — n)+l 
 
 56. 
 
 a^-]-(a-b)x-ab. 
 
 57. 
 
 x^ — bx-\-c. 
 
 58. 
 
 a (6 + c) — be. 
 
ANSWERS. 311 
 
 Art. 96 ; page 46. 
 
 30. a^^f-\-2xz-^z\ 34. a'-b^-2ac + c'. 
 
 31. x'-f-^yz-z'. 35. a*-2a2 + l. 
 
 32. l-o?-\-2ab-h\ 36. a;*- 4.^^+ 12 a;- 9. 
 
 33. a^_a^-2a;-l. 37. m^-{-mhi^ -\-n^. 
 
 Art. Ill ; pages 56 and 57. 
 
 26. {x-\-y-\-2){x-{-y-2), 29. {a + h-\-c){a-h -c). 
 
 27. {a-h-\-c){a-h-c). 30. (c + d-h 1) (c + d- 1). 
 
 28. (a + 6-c)(a-&4-c). 31. {2,-\-x-y){^ -x + y), 
 
 32. (2m2 4.26-l)(2m2-26 + l). 
 
 33. (2a-6-f-3d)(2a-6-3c^). 
 
 34. (a + & — wi + w)(a — 6 — m — 7i). 
 
 35. (a? + y — c — d) (a; — »/ — c + (J) . 
 
 36. (a H- 6 + m — ?i) (a — ^ + m H- ?i) . 
 
 37. (a-\-b-\-c-^d){a — b-\-c — d). 
 
 Art. 120 ; page 64. 
 
 27. 2xy(x-{-y + z){x-\-y-z). 
 
 40. {x-\-y-\-z){x-{-y-z){x-y + z) {x -y — z). 
 
 46. {a + b){a-b){x^y){x-y). 
 
 50. (a;-2)(a; + 3)(a^-a;+6). 
 
 51. (a-l)(a-2)(a + 4)(a + 5). . - 
 53. (a 4- ?> + c) (a -f 6 - c) (a - 6 4- o) (a - & - c) . 
 
 55. (.x-l)2(a; + l). 58. (a; + 2)'(a;-2)2. 
 
 56. 2(a-6)(a + 26). 59. {x-yf. 
 
 60. (a-l)(a + 2)(a-2)(a + 3). 
 
 Art. 131 ; pages 73 and 74. 
 
 l..aj — 2. 4. 8a:— 7a. 7. 5m + 3. 10. 2a;-l. 
 
 2. 2a; + 3. 5. 2a-5. 8. 2a-3a;. 11. 2m-5n. 
 
 3. a— 1. 6. a:2_^ja;. 9. a;— 2. 12. a; +2. 
 
312 ALGEBRA. 
 
 13. ax^-ax. 15. a^-a-1. 17. 3a;4-2. 19. x-\-l. 
 
 14. x^-\-x-\-l. 16. x^-2x. 18. a-x. 20. a;-2y. 
 
 21. 2aj-3. 22. Sa-{-b. 
 
 Art. 132 ; page 74. 
 
 1. 2a;+7. 2. 2flj-5. 3. 3m + 2%. 4. 3a-l. 
 5. x + 4:. 6. ic-l. 7. 2a + 3. 
 
 Art. 136 ; page 76. 
 
 4. 270mV. 7. 336r^?/«;3. 9. 480m^n^ci^y\ 
 
 5. 210 a5c. 8. 360aVc?\ 10. 252 a^f:^, 
 
 11. 1080a26V(i^ 
 
 Art. 137 ; pages 76 and 77. 
 
 2. y{x^-y^). 9. (a;+3a) (a;-5a)(a!+7a). 
 
 3. (a^-l)(a;-8). 10. ^(m-^-Ti^). 
 
 4. 2Aab{a^-b^). 11. (a+ &) («- 36)(ic- 2). 
 
 5 . (m + ?i) (m^ — ?i^) . 12. aa; (a; + a) {x^ — a^) . 
 
 6. a2-4a6 + 362. 13. 24:{a + by{a-by. 
 
 7. a;2/(aJ + 2/)(i»-«/)^. 14. (/a?(a;-3) (aj-7) (ic+8). 
 
 8. 12abc{a'-b'). 15. a;^-2a^-f-l. 
 
 16. 24(1 -a;^). 
 
 17. {x + l){x-2)(x-\-S)(x + 4). 
 
 19. (2m + l)(2m-l)2(4m2 + 2mH-l). 
 
 20. a\a-l){a'-^l). 
 
 21. (a-l)(a-3)(a + 4)(a-5). 
 ^ 22. (14-a;)2(l- 0^)2(1+0^)2. 
 
 23. (a + 64-c)(a + 6 -c)(a-&-c). 
 
 24. So.b{a-b){x-yy. 
 
 25. 2aa;2(3a; + 2)2(9a^-6ic+4). 
 
 26. (x + 2/ + 2;)(a;+7/-2;)(x-2/4-^). 
 
ANSWERS. 313 
 
 Art. 139 ; page 79. 
 
 3. 24:x''-^22x'-177x+U0. 5. 2ax{ea^+ar-4:2x-4o). 
 
 6. a*- 16(i36 + 86a262_ 1760^3 _|. 1056*. 
 
 7. 2n(2m^-5m* + 3m«-5m2 + 4m-|-4). 
 
 8. a(30 x" - 11 ax^-bda'x-\- 12 a^). 
 
 9. 2a* + a-^-17a2-4a+6. 
 
 10. 2ar'+3a;*-4a^+5a;-6. 11. a' -a^b- -\-d'b^-b\ 
 
 12. aa;(a^ + iB*-a53-3a^-3iB-l). 
 
 13. a;(6ar'-31a^-4a;3 + 44a^ + 7a;-10). 
 
 14. iB«+2ic5-4a;*-7a:3_iga^_l_32a._8. 
 
 Art. 140 ; page 79. 
 
 1. 4a;4_4a^-39ar^H-4ic + 35. 
 
 2. l8a*-33a3+14a2 + 3a-2. 
 
 3. 20a;*-24a:3-51a^ + 41a;-6. 
 
 4. 2ic2(12aj*-32iB»-29ar^ + 57a;-18). 
 
 5. a*4-3a*-23a3-27a2+ 166a- 120. 
 
 12. 
 13. 
 
 14. 
 15. 
 
 Art. 149 ; pages 83 and 84. 
 
 cd -g m — 2 24 9v^+loy+25 
 
 3a;.y' * w-|-9' * 3^ — 5 " 
 
 2/ • 6(37i-2) ' 2a; + l* 
 
 1^. 20. ^ + ^^ 26. — i— . 
 
 ic — 5 a + 36 aa;(a;4-4) 
 
 «-5 21. ^^ + y . 27. ^ + ^ + ^ . 
 
 a + 7 a; , ' a — b-^c 
 
 16. ?.^^!±^. 22. ^-^^ 28 1. 
 
 ab + 2b' a'-^ab + b' 
 
 17. ''^--^ -. 23. 
 
 a(a; + 2) oq ct — ft + c — d 
 
 c(2c + 5) a;(a; — 7) a+6 — c — d 
 
314 ALGEBKA. 
 
 Art. 150 ; page 85. 
 
 2 x — 5 M m — 1 g 6m — n « Sx — 2 
 
 3aj + 7 * 6m — 5 * 5m — In ' .r-\-s' 
 
 5a + 7 ' cc^+ic+l' * x — S ' 2y — 5 
 
 Art. 155 ; pages 87 and 88. 
 
 5. af — xy-\-y^-i — ^- — 10. 2m^— 5mn4-7%^ 
 
 x-\-y 2m— M 
 
 6. 2x+6--^^. 11. a2_3_ 3a-5 
 
 iK-3 2a--a-3 
 
 7. a-2+-^£^. 12. . + l + -^^±^. 
 
 a^ -\-a — l ar + oj + l 
 
 8. 30.-5^^^. 14. 20.-3 2^±3_. 
 
 4a; -1 3a;2-2aj+l 
 
 Art. 156 ; page 89. 
 
 3 (x+iy g 2a6 j^^ x^-Sx^ 
 
 X ' a-\-b 
 
 M fl.-^ + 4a; — 1 Q 6x^ — x ^e 
 
 x-\-S ' ' 2a.+ l* 
 
 f. 5m^—2mn—4:n^ jq a^-\-b^ -^ 
 
 3mH-7i a — 6 
 
 6. i^zii. 11. ^JL. 17. 
 
 8 x-y 
 
 7. ^2?_. 12. -1^. 18. 
 m + 7i m+w xF-\-3x — 2 
 
 x-2 
 
 2n^ 
 
 m^ + ??i?i + n^ 
 
 9ar^ 
 
 2aj-l 
 
 6o;.V 
 2y — X 
 
 x^-5o^-l 
 
 13. 
 
 a 
 
 6« 
 
 a + 6 
 
ANSWERS. 315 
 
 Art. 157 ; page 91. 
 
 g 2a2 + 4a AaP-^12a 
 
 7. 
 
 (a -1-3) (a- -4)' (a-h3)(a2-4) 
 x^ + x-^l x-\-l 
 
 g mn (m^ — n^) m^ (m + n) mn^ 
 
 -- 2a^6 m^ — n^ 
 
 12. 
 
 2a(a — 6)(m H-?i) 2a(a — 6)(m -f-w) 
 
 (a;-l)(a;-2)(a;-3)' (»- l)(a;- 2)(»-3)' 
 
 a^-4 
 
 {x-l){x-2){x-3)' 
 
 Art. 158 ; pages 92 to 97. 
 
 3 12a; H-7 g 3mV-4 ^ 4CT6-6-4a^ 
 
 36 * * 6??iV * • 12 a3^, 
 
 ^ 6 g — 5 6 » 5 6^ -f- 4 g'^ in -L li 2??i 
 
 lOa'^^ • • i20ab ' 15* * 42* 
 
 5 _£Ldl?. 8 5a -i-6 -o 3a; — 2 
 
 24 * ' 24 * ' 18ar^ * 
 
 jg 1_^ jj^ 4 6cc? + 6 acd — 3 abd — 2 abc 
 
 60* * 48 a6cd 
 
 17. ^2 . 20. ^±A!. 23. ^^ 
 
 1-x^ a'-b^ 1-ar* 
 
 18. ^__. 21. -i^. 24. Sm^-hn^ 
 
 Q^x-a^ a^-b"^ {m+n)(m-ny 
 
 19. -^ ^ 22. -J^. 25. ^ 
 
 ar^H-15a;H-56 a;-|-y (aH-3)(a-2)2 
 
316 
 
 
 26. 
 
 Ax 
 x-\-y 
 
 27. 
 
 a-^b 
 a-b 
 
 28. 
 
 0. 
 
 nck 
 
 4a^ 
 
 ALGEBRA. 
 
 30. ? 34. ^^ 
 
 x{4:X^-l) x^-^3x-10 
 
 31. 4^.' 35. 2. 
 (t^ — b* 
 
 32 L+^_±-^. 36 a^— 2 
 
 (1 -a;)3 ' * 05(0)2-1)* 
 
 29. -;^^^- 33. 0. 37. 0. 
 
 38 (^^ + ^)' 40 2(^-^) 
 
 (a, + l)(a^_l) • (,, + 2/)(^ + ^) 
 
 39 13 -18a; ^g 4&-3a 
 
 (o; + l)(a; + 2)(«-3) ab{(i-b) 
 
 44. l^^-l 46. _A!!L+^L_. 43. ^^ 
 
 6(a-l) m(m2-n2) a + & 
 
 45. -^. 47. ^ ^'^ "^ 
 
 dx-a^ (a;+2)(o;+a) a^ - 1 
 
 60. -—1^ 51. 0. 52. - '^ 
 
 a^ - 5a; + 6 (a;— 2) (a;-3) (a?-4) 
 
 Art. 159 ; pages 98 and 99. 
 
 4. L 8.-^. 12. ^^-^' . 16. i±l. 
 
 a 10 aa; + 6a? a; 
 
 K 1 q 3a;— 1 -„ xy 17. — 
 
 « 2; irt a^— ic— 20 ,* aa;— 2a 18. 
 
 b. 10. 14. — -• x—y-j-z 
 
 4 xy x^ a 4- 1 
 
 19. a^ + xy. 
 
 7. a^ 11. ^-^^ 15. ^^±1^. 
 
 a^ ar'+2a; 20. 1. 
 
 Art. 160; pages 100 and 101. 
 a- M 3 nif g a-\-6 
 
 4. ^^^. 5. 
 
 1 Wm?n^ 5 ma; a + 4 
 

 
 ANSWERS. 
 
 
 
 31 
 
 6. 
 
 3a;-12 
 
 8. 
 
 (a-b)\ 
 ab + W 
 
 
 10. 
 
 a+2 
 a^-2a 
 
 7. 
 
 x^ 4- ic _ 20 
 
 9. 
 
 m- — mn 
 
 -\-n- 
 
 11. 
 
 3x-2y 
 
 
 a;-2 
 
 
 w? — mn 
 
 
 x-\-y 
 
 
 12. 
 
 a-2b 
 rt + 6 
 
 13. 
 
 2x 
 2x 
 
 + 32/ 
 
 
 
 
 Art. 161; 
 
 pages 102 and 103. 
 
 
 4. 
 
 iC-l. 
 
 10. 
 
 m -f- n 
 n 
 
 
 16. 
 
 a — b — c 
 a-\-b — c 
 
 5. 
 
 1 
 
 a + 6 
 
 11. 
 
 a; 4-5 
 x + 1 
 
 
 17. 
 
 1 
 
 6. 
 
 a^-x+l. 
 
 12. 
 
 5a:-16 
 3a;-10 
 
 
 18. 
 
 ac — bd 
 ad -f be 
 
 7. 
 
 a-1. 
 
 13. 
 
 a-b 
 a 
 
 
 19. 
 
 4 
 3x + 3 
 
 8. 
 
 rc2_2a^. 
 
 14. 
 
 xy 
 
 
 20. 
 
 1. 
 
 9. 
 
 X-4: 
 
 x + 6 
 
 15. 
 
 X. 
 
 
 21. 
 
 x — a 
 
 x->r2a 
 
 
 22. 
 
 ab 
 
 23. 
 
 2{m — n) 
 {m-\-ny' 
 
 
 Art. 162 ; pages 104 and 105. 
 - bx — a 2 m — 1 « a^ 
 
 ar* 3m -15 (1+a;)* 
 
 am-\-an 2 (1 + a^) (1 +a^) 
 
 6. 2ia-h6); 3^ a_b^ j2 
 
 a^ 5 a 3-3a^ 
 
 7. a^+i+1. 10. -i— . 13. ^P^' 
 
 ^ \-\-7? 371 
 
 8. -i-. 11. -^. 14. a^ + a^. 
 1 + ar aa; 4- 6 
 
318 
 
 
 16. 
 
 4b 
 
 16. 
 
 
 17. 
 
 4a^-2 
 
 ALGEBRA. 
 
 19. (^^ + yy . 23. 3. 
 
 20. ^:^2. 24. ^ + ^ 
 
 (a; -2)3 
 
 4a;^ + l' ' a-3b ' ' a 
 
 18. -1^. 22. ^' . 26. ?^±ll. 
 
 scf^ — a* a;2 — 2/^ x — ly 
 
 oy a6 + ^c + co^ QQ 2a^ — 2 
 
 (a + 6)(6-f-c)(c + a) a;* + a^ + i 
 
 14-9»' ' ah{a-b)' 
 
 Art. 174 ; pages 109 and 110. 
 
 3. 
 
 14. 
 
 8. 2. 
 
 13. 
 
 2. 
 
 19. 0. 
 
 24. 2. 
 
 4. 
 
 -5. 
 
 «•! 
 
 14. 
 
 _2 
 
 20. 1. 
 
 25. 1. 
 
 5. 
 
 3. 
 
 10. -If 
 
 15. 
 
 -2. 
 
 21. 2. 
 
 26. 2. 
 
 6. 
 
 -3. 
 
 11. 1. 
 
 16. 
 
 16. 
 
 22. 2. 
 
 27. -8. 
 
 7. 
 
 -8. 
 
 29. 4. 
 
 18. 
 
 3 
 2* 
 
 23. -4. 
 30. -3. 
 
 28. -1 
 
 Art. 175 ; pages 111 to 115. 
 
 2. 
 
 -6. 
 
 7. 
 
 -If. 13. 
 
 -2. 
 
 18. 7. 
 
 23. 
 
 -2. 
 
 3. 
 
 2 
 
 3* 
 
 8. 
 
 \ "■ 
 
 1 
 
 3* 
 
 19. -1,^. 
 
 24. 
 
 2 
 
 3* 
 
 4. 
 
 2 
 3* 
 
 9. 
 
 -2f 15. 
 
 5. 
 
 20. -5. 
 
 25. 
 
 __1^ 
 2* 
 
 5. 
 
 3. 
 
 10. 
 
 56. 16. 
 
 -3. 
 
 21. 4. 
 
 26. 
 
 2 
 8* 
 
 6. 
 
 5. 
 
 11. 
 
 2. 17. 
 
 1^ 
 
 22. -5. 
 
 27. 
 
 1. 
 

 
 
 AJiSWERS. 
 
 
 31 
 
 30. 
 
 7. 
 
 34. - 
 
 7. 38. 
 
 5. 
 
 42. -4. 
 
 
 «.i. 
 
 31. 
 
 - ItV- 
 
 35. - 
 
 2. 39. 
 
 3. 
 
 43. - 7. 
 
 
 "•1 
 
 32. 
 
 2. 
 
 36. - 
 
 1 «. 
 
 0. 
 
 44. -2f 
 
 
 48. -1 
 
 33. 
 
 1 
 2* 
 
 37. - 
 
 ? «■ 
 
 1. 
 
 45. - 1|. 
 
 
 «i- 
 
 
 
 Art. 176 ; pages 116 and 117. 
 
 
 
 3. 
 
 dc—d 
 2a + b 
 
 8. 
 
 2m2 
 3n 
 
 13. 
 
 3a-3. 
 
 18. 
 
 mw. 
 
 4. 
 
 5a 
 2 b 
 
 9. 
 
 a-b. 
 
 14. 
 
 71 
 
 2* 
 
 19. 
 
 7 a. 
 
 5. 
 
 a-1 
 
 10. 
 
 a + &. 
 
 15. 
 
 a 
 b 
 
 20. 
 
 a 
 3* 
 
 6. 
 
 Sb-{-4a, 11. 
 
 a-b 
 2 
 
 16. 
 
 \2a\ 
 
 21. 
 
 a 
 36* 
 
 7. 
 
 4a2 
 5 
 
 12. 
 
 b-2c. 
 
 17. 
 
 1 
 a+2 
 
 22. 
 
 36^ 
 
 a ' 
 
 
 
 23. 
 
 1 
 
 _. 
 
 24. n. 
 
 
 
 2{a-\-b) 
 
 Art 177 ; page 118. 
 
 4. 2. 6. .7. 8. 8. 10. 0. 
 
 5. 50. 7. 5. 9. -.04. 
 
 Art 179 ; pages 120 to 131. 
 
 7. 35, 24. 10. A,S30;B,$60. 
 
 8. A, 60; B, 15. 11. 116, 91. 
 
 9. 23. 12. 7 and 10. 
 13. 12 oxen, 24 cows. 14. 47,33. 
 
 15. Wife, $864 ; a daughter, $288 ; a son, $144. 
 
 16. A, $18; B, $48; C, $4. 
 
 2. 
 
 .8. 
 
 3. 
 
 -3. 
 
 4. 
 
 30. 
 
 5. 
 
 20, 14. 
 
 6. 
 
 120. 
 
320 ALGEBRA. 
 
 17. Infantry, 2450; cavalry, 196; artillery, 98. 
 
 18. A, 62 ; B, 28. 
 
 19. On foot, 880; by water, 1540; on horseback, 616. 
 
 21. — - — , ^^^ • 22. 22, 23, 24, 25. 23. 29, 14 
 
 1 + mn 1 + *^^ 
 
 24. A, $14; B, $13; C, $11; D, $9. 25. 115. 
 26. 120, 60, 20, 5. 27". A, 59 ; B, 23. 28. 22, 23. 29. 36. 
 
 30. A,^^^^^-^>; B, ^(^-^). 33. 8^. 
 
 m—n m—n 
 
 rt-i an^ an a 04 11. 
 
 n^ + n-i-l' n^ + n + l' n^-\-n-^l ' 12* 
 
 ab 
 
 35. 144 sq. yds. 36. A, $35; B, $38. 37. 
 
 a-{-b 
 abc 
 
 40. 2 dollars, 20 dimes, 4 cents. 
 
 ab + bc-^ ca 
 41. 17 two-penny pieces, 36 farthings. 42. $2.75. 
 
 43. Worked, 20; absent, 16. 44. $58. 45. ^^^. 
 
 c+1 
 
 46. 48 minutes. 48. 84. 49. 93. 
 
 50. 30 bushels at 9 shillings ; 10 at 13 shillings. 
 51. Gold, 3377 oz. ; silver, 783 oz. 62. 36. 54. 8| miles. 
 
 ec an en K ^^ -1 T^ ^ . -1 _ 
 
 
 uu. • yu. ii, x^ iiiiius ; 15, ±4: nines. 
 
 b — a 
 
 
 57. $1?00 in 5 per cents ; $2000 in 6 per cents. 
 
 58. 
 
 100a . ^^ lOO(a-p). ^^ 
 r« + 100 pr 
 
 62. 
 
 82, 31. 63. 12,121 men; 110 on a side at first. 
 
 64. 
 
 a-c ab + c^ 65. ^. 66.51. 67. lQQ(^-i>). 
 6 + 16 + 1 13 pt 
 
 
 68. Picture, $5.28; frame, $3.96. 69. ?. 
 
 4 
 
 71. b^ minutes after 7. 73. 27y\ minutes after 5. 
 
 72. 43 j^ minutes after 2. 74. 5^\- minutes after 1. 
 
ANSWERS. 321 
 
 75. 7. ' 77. -^ miles. 
 
 6 + c 
 
 76. 7 dimes, 13 half-dimes. 78. IGy^y minutes after 6. 
 
 79. i)-fy^ minutes, or SS^ minutes after 10. 
 
 80. First kind, £^(£^1*); second, ^(a^lA. 
 
 a—b a—b 
 
 81. 10 A.M. 82. 27^ minutes after 4. 
 
 83. 48. 85. Horse, $180 ; carriage, $95. 
 
 84. $6480. 86. A, 52 miles ; B, 55 miles. 
 
 87. ^^^ + ^^ + ^ cents. 
 
 a-\-b-^c 
 
 88. A, 11 days ; B, 22 days ; C, 33 days. 
 
 89. A, $750 ; B, $500. 90. 48 minutes after 9. 
 
 91. A, 3 days ; B, 4 days ; C, 5 days. 
 
 92. 18 yards, 15 yards. 93. ^±^. 
 
 a — c 
 
 94. $2000. 95. $1400. 96. $15,000. 
 
 97. 16^ minutes after 8. 98. Greyhound, 72 ; fox, 108. 
 
 Art. 184 ; pages 134 and 135. 
 
 3. 
 
 x= 6, 
 
 7. 
 
 x = -S, 
 
 11. 
 
 a; = -4. 
 
 15. 
 
 x= 8, 
 
 
 ^ = -2. 
 
 
 2/ = 6. 
 
 
 y = -5. 
 
 
 2/ = 10. 
 
 
 
 8. 
 
 i»= 7, 
 
 12. 
 
 x = -2, 
 
 
 
 4. 
 
 x = 4, 
 y = Z. 
 
 9. 
 
 y = -l. 
 
 1 
 x = -. 
 
 13. 
 
 y = 4. 
 
 16. 
 
 x = — l, 
 9 
 
 5. 
 
 x = 6, 
 
 
 y = -s. 
 
 
 y = -2. 
 
 17. 
 
 x = -|, 
 
 
 y = 2. 
 
 10. 
 
 3 
 
 14. 
 
 2 
 
 
 5 
 
 6. 
 
 a;=12. 
 
 
 '=!■ 
 
 
 -1 
 
 
 \ 
 
822 ALGEBRA. 
 
 Art. 185; page 136. 
 
 2. a; =5, 5. a?=l, 8. a; = — 1, -^ 9 
 
 i/ = 2. y = -t. 2/ = - 2. o 
 
 ^2 3 
 
 y = — 
 
 1 o 1 5 
 
 3. x=3, 6. «' = -^, 9- ^ = 5' ^ 
 
 4. a; = -2, 7. a^ = -3, ^^- ^=2' 13. aj = -2, 
 2/ = -2. y = -2. y = 2. 2/ = 19- 
 
 Art. 186; page 137, 
 
 2. a;=2, 6. aj=12, 9. x = -3, 12. ic=l, 
 
 2/ = 3. 2/ = -3. ^ = -11 2,^_i. 
 
 3. aj = -l, 1^ 1 ^ 
 
 2 w ^ 10. aj = , ^ 
 
 2/ = -l 
 
 y = 4. 2 ^ 
 
 2/ = -2. ^ ^~2 
 
 8. a; = 4, 11 _ 1 
 5. a; = -2, 2 ^^- ^-~I' 
 
 « V = — ~ 
 
 2/ = 3. 3 2/ = -2. 
 
 Art. 187; pages 138 to 143. 
 
 2. x=lQ, 6. ic=2, 11. x = -2, 15. a; = 3, 
 2/= 5. 2/ = 3. y = -4:. 2/ = -4. 
 
 3. .T = 24, 7. x = -.S, -p 1 
 ^ = _18.. 2/=. 08. ^^- "-J' 16. .=.-, 
 
 4. x = -16. 8. 0^=18, 2/-- ^^_3. 
 
 __^,2 ?/= 6. ^ 
 
 9. a; = 4, ' ■^'- ^-^' 
 
 5. a^ = |, 2/ = 9. 2/ = 40. • ^ = _2. 
 
 ^^_4 10. .T = -2, 14. a;=l, 18. x = -S, 
 
 ^ 3' ?/ = 3. y = -2. y=n. 
 

 
 ANSWERS. 
 
 
 32:1 
 
 19. .-*, 
 
 21. 
 22. 
 
 x=l, 23. x=2, 
 2/ = 5. y = -'2. 
 
 x=-6, 24. a;=13, 
 
 25. 
 
 
 20. a; = 12, 
 
 
 2/ = -5. y= 3. 
 
 
 
 2/= 6. 
 
 
 
 
 
 27. ^-^^ 
 
 4-35 
 
 29. a;-^-^^, 31. 
 
 a; = 
 
 dm + &w 
 
 • J ■ 7 . •» 
 
 28. 
 
 17 a — 6 ad-\-bc 
 
 2b — Sa _ c — ad y^^IILlZ^. 
 
 ^~" 17 ^~ a — b' ttd-\-bc' 
 
 ^^dm-bn ^Q ^^ bp ^ 32^ ^ = i^5Zz:^, 
 
 od — be an + bm mn'—m'n 
 
 an -cm ajJ y^ 'in'p-mp' 
 
 ^ ad — be an-{-bm, mn'—m'n 
 
 no _ ae(bm -\- dn) __ bd(cn — am) 
 
 "" ad -f- 6c ' ad-\-bc 
 
 34. 
 
 X = a6, 
 
 40. 
 
 a' 
 
 
 47. 
 
 x-^ 
 
 
 y = a + b. 
 
 m + 
 
 1 
 71 
 
 "" 2' 
 
 
 .=i, 
 
 
 .._m2- 
 
 n"- 
 
 
 ,=-1. 
 
 35. 
 
 
 ^ a 
 
 
 
 
 a 
 1 
 
 41. 
 
 1 
 
 
 48. 
 
 x=m-\-n^ 
 
 
 '-2i- 
 
 
 
 y = m-{-n. 
 
 36. 
 
 X = a^b, 
 
 
 '-h 
 
 
 49. 
 
 be — ad 
 bn — dm 
 
 
 y = a62. 
 
 42. 
 
 x = {a + b)\ 
 
 
 y-bc-ad 
 
 37. 
 
 ^ = -.^ 
 
 
 y = {a- 
 
 by. 
 
 
 cm — an 
 
 
 n 
 
 44. 
 
 » = 4, 
 
 
 50. 
 
 2 
 
 
 ^=™^- 
 
 
 y = 2. 
 
 
 
 -!• 
 
 38. 
 
 .r = a + 6, 
 
 45. 
 
 x = — 5, 
 
 
 
 
 
 7/ = a — b. • 
 
 
 y=Z. 
 
 
 51. 
 
 n 
 
 39. 
 
 x = a, 
 
 y = h. 
 
 46. 
 
 y = -i- 
 
 
 
 m 
 
32^ 
 
 1 
 
 ALGEBRA. 
 
 
 
 
 Art. 189 ; pages 145 to 147. 
 
 
 3. 
 
 a^ = 3, 
 
 10. x = -7, 
 
 16. 
 
 a; =10, 
 
 21. aj = -i. 
 
 
 ^ = -1, 
 
 2/ = -2, 
 
 
 ^= 2, 
 
 3 
 
 4. 
 
 
 2=1. 
 
 11. a; = 8, 
 
 17. 
 
 2=3. 
 
 a; = -24. 
 
 1 
 
 
 ^ = 3, 
 
 ^ = -3, 
 
 
 y = -48, 
 
 2 =-. 
 
 4 
 
 
 = 1. 
 
 ;s = — 4. 
 
 
 2 = 60. 
 
 22. a^ = l, 
 
 
 
 
 
 
 5. 
 
 2/ = 2, 
 
 12. a. = -|, 
 
 18. 
 
 "=r 
 
 
 
 2 = -4. 
 
 ^ = 1 
 
 
 ..-1, 
 
 2- -^. 
 
 6. 
 
 0^ = 2, 
 
 2 
 
 
 1 
 
 a^* 
 
 
 2/ = -l, 
 
 z=-2. 
 
 2 
 
 
 2=-L. 
 
 23. ii = 4, 
 
 a; = 5, 
 
 7. 
 
 a? = 23, 
 
 13. cc = -5, 
 
 
 10 
 
 2/ = 6, 
 
 
 2/= 6, 
 
 2/ = -5, 
 
 19. 
 
 ^^ = -7, 
 
 2 = 7. 
 
 
 ;s = 24. 
 
 ;2=-5. 
 
 
 a; = 3, 
 
 24. a^ = i, 
 
 8. 
 
 x = -2, 
 
 14. a; = 3, 
 
 
 ^ = -5, 
 
 2=1. 
 
 2' 
 1 
 
 
 2/ = 3, 
 
 ^ = 4, 
 
 
 
 2 =—1. 
 
 
 5; = 7. 
 
 2=6. 
 
 20. 
 
 - = ^, 
 
 
 
 
 
 3 
 
 
 9. 
 
 a^ = -4, 
 
 15. a; = 4, 
 
 
 y = ^, 
 
 25. x = l, 
 
 
 ^ = 2, 
 
 2/ = 6, 
 
 
 _4 
 
 y = -^ 
 
 
 z = -5. 
 
 2=2. 
 
 
 5 
 
 2 =-5. 
 
 
 26. a; = 
 
 b' + c-^ - a^ 
 
 
 28. a; = 
 
 = 3, 
 
 
 
 26c ' 
 
 
 y = 
 
 = -2, 
 
 
 
 c2-^a^_52 
 
 
 z = 
 
 = — 1. 
 
 
 y = 
 
 2ca ' 
 
 
 
 
 
 z = 
 
 a^^b'-c" 
 
 
 29. a^ = 
 
 :a, 
 
 
 2ab 
 
 
 y = 
 
 :a, 
 
 
 
 
 
 z — 
 
 :1. 
 
 
 27. a; = 
 
 = 1|. 
 
 
 
 
 
 y = 
 
 = 1, 
 
 
 30. a: = 
 
 a6c, 
 
 
 z = 
 
 1 
 
 
 y = 
 
 ab + bc-\- ca^ 
 
 
 
 '2 
 
 
 2 = 
 
 « + 6 + c. 
 
ANSWERS. 325 
 
 Art. 190 ; pages 149 to 157. 
 
 3. 32, 18. 5. A, 30 ; B, 20. 7. 24 and 18. 
 
 4. 14,7. 6. — • 8. A, $96; B, $48. 
 
 15 
 
 9. A, 48 ; B, 18. 10. — • 11. A, 16 days ; B, 26f days. 
 i. y 
 
 12. 38, 13. 13.-- 14. '''■^'* 
 
 a^ - a -h 1 
 
 15. Better horse, $160 ; poorer horse, $100 ; harness, $40. 
 
 16. First, 8 cents ; second, 7 cents ; thu*d, 4 cents. 
 
 18. 30 cents ; 15 oranges. 
 
 19. 13J bushels at 60 cents ; 26| at 90 cents. 
 
 20. Income tax, $20 ; assessed tax, $30. 
 
 21. 120, at 7 cents each. 24. 10, 22, 26. 
 
 22. Length, 80 ft. ; width, 60 ft. 25. 42, 38, 32, 24. 
 
 23. A, $45; B, $55. 27. 74. 28. 326. 
 
 29. A, 9f days ; B, 16 days ; C, 48 days. 
 
 30. Length, 30 rods; width, 20 rods; area, 600 sq. rods. 
 gi b-j- c — a c-\-a — b a-\'b — c oo 946 
 
 2 ' 2 ' 2 * ' * 
 
 33. Number of persons, ^"^"^^^'^'^ ; 
 
 6m — an 
 
 each received — ^ — ^ — ^ dollars. 
 
 bm — an 
 
 34. Whole sum, $1200; eldest, $400: second, $300; 
 
 third, $240; fourth, $260. 
 
 35. 30 at 2 for 5 cents ; 36 at 3 for 8 cents. 
 
 36. 42. 37. 38. 
 
 38. A, ^J!^ days; B, ^J!^ ; 
 
 mn -\-np — mp mp -\- np — mn 
 
 C, 
 
 2 mnp 
 
 mp-^ mn —7ip 
 
326 ALGEBRA. 
 
 40. Current, — — — miles an hour ; crew, — ~ — 
 
 ' 2bd ' ' 2bd 
 
 41. Going, 4 hours ; returning, 6 hours. 
 
 42. 759. 43. First, 22 ; second, 10. 
 
 44. 65. 45. First rate, 6 p. c. ; second, 5 p. c. 
 
 46. $120 at 5 per cent. 
 
 47. dollars at ^ '- per cent. 
 
 m — n hin — an 
 
 49. 15 miles ; b\ miles an hour. 50. 43. 
 
 51. 40 miles an hour. 53. A, 8 ; B, 6. 
 
 52. A, $13; B, $7; C, $4. 54. 58, 43, 14. 
 
 55. A, $7; B, $22; C, $21; D, $16. 
 
 56. A, $78; B, $42; C, $24. 
 
 57. A, 8 hrs. ; B, 9 hrs. ; C, 12 hrs. 
 
 58. $2000 in ^ per cents ; $1600 in four per cents. 
 
 59. A, 16 ; B, 12. 60. Fore-wheel, 8 ft. ; hind-wheel, 12 ft. 
 61. A, 8 days : B, 12 days. 62. A, 6 ; B, 5. 
 
 Art. 194; page 160. 
 
 4. 4aj4+4a^_}_5aj2_f.2;r+l. 6. a;^+ 8ar'-f-12a.-2-16a;+4. 
 
 5. x"^ -(Sx^-^Wx^-Qx^l, 7. 4a;*-4a^-lla;24.^a;+9. 
 
 8. 9a^-30a=^ + 49a2-40a + 16. 
 
 9. 4a;^-f 20ar^-3ic2-70a^ + 49. 
 
 11. a;«-4a;*H-10a^ + 4a^-20a;4-25. 
 
 12. 4a5«-f 12ar'-f-9a;* + 4fl;3 + 6a.'2 + l, 
 
 13. 9 a^ - 12 a^6 - 26 a-62-f- 20 a63 + 25 6*. 
 
 14. IQm^-^S m"n'^ — 23 m V — 6 mn^' + 9 n^. 
 
 17. l+2a'H-3a^H-4a^-H3a;*4-2a^ + aj'. 
 
 18. 9i»«-12a:^-2a;* + 28ar'^-15a^-8.r4-16. 
 
 19. a/* - 8a,'5 4- 12a;^ + 10ar»-j- 28 a^-f 12a; + 9. 
 
ANSWERS. 327 
 
 Art. 195; page 162 
 
 3. ar^ + 9a^-}-27a; + 27. 6. a^ + 12 a'b-^ 48 ab^ -\- 64 b\ 
 
 4. 8ar^-12a^4-6a;-l. 7. 27 7n^ - 27 m'^ -t d m^ - 1 . 
 
 9. a-^ + 15a26+75a62+l256^ 
 
 10. S2(^-GOx^y-{-loOxy"-12of. 
 
 11. 8a;«-36a;^4-54af-27a7\ 
 
 12. 216a^+108a;^?/+18a^/ + ary. 
 
 13. 27 m3 + 135 mrn + 225 rmv" + 125 n^ 
 
 14. 27ary-108aV/+144«^a:y-64a«. 
 16. a^-3a^4-5a^-3a;-l. 
 
 17. a3-3a26 + 3a2 + 3a62_6a6 + 3a-6' + 362-36 + l. 
 
 18. a^+3a^b-3a^c-h^ab^-6abc-]-Sa(^-\-b^-3b^c+3bc'-(^. 
 
 19. a;«-6a;^+18a.'^-32a:8 + 36a^-24a;-h8. 
 
 20. a;«4-9ar' + 30a;*H-45a^ + 30a^+9a;+l. 
 
 21. 8ic«-36ar'-f-42aJ* + 9a^-21ii'2-9a;-l. 
 
 Art. 196 ; page 164. 
 
 10. lGH-32a + 24a^H-8aT^ + a;^ 
 
 11. x'-lGx'-\-dQx^-2D6x + 2o6. 
 
 12. a^-15a^ + 90a3-270a'^ + 405a-243. 
 
 13. a^+10a^ + 40a3+80a2 4-80a + 32. 
 
 14. a:»-12ar^+60a;^-160ic»+240ar^-192a:4-64. 
 
 16. a^ - 15a*x + 90 aV - 270aV -}- 405 aa;* - 24Sa^, 
 
 17. 81 + 2166 + 2166^+9663+166^ 
 
 19. a;^2_i(3a^_|_96a:6_256a^ + 256. 
 
 20. 64 a^^ + 192 ai«6 + 240 a^b^-\- 1 60 a%^+ 60 a^6^+ 1 2 a-b'+b'''. 
 
 21. 16mi2_96^9^^2_|.2i6mV-216mV + 81ii». * 
 
 Art. 204 ; page 169. 
 
 3. a2-2a + l. 6. 5 + 3a; + a;2. 9. 2a2_5a6 + 862. 
 
 4. 2a^-a;-l. 7. 3a^-4iB-5. 10. l-7a;-2a^. 
 
 5. 3 -2a; + .1-2. 8. 7«+l-i. 11. a-b-c. 
 
 m 
 
328 
 
 
 
 ALGEBRA. 
 
 
 
 12. 
 
 a;-2 
 
 .y + 3^. 
 
 13. 3a; 
 
 "+5ar^-7. 14. 
 
 4c3- 
 
 -5c-3. 
 
 15. 
 16. 
 17. 
 18. 
 19. 
 
 2ar^- 
 2a;3- 
 
 3 
 305^- 
 
 ► a2 + 5a + 3. 
 
 2iB2?/ + a;/-2?/3^ 
 
 23. '2x+y-- 
 2x 
 
 20. 14-^-^ + ^ 
 
 2 8^16 
 
 21. i-a-^-^ 
 
 2 2 
 
 22. a 2b ^^' ■ 
 
 2a 
 
 f , f. ... 
 16ar^ 64.x^ 
 
 a' 
 
 
 
 Art. 207 ; pages 172 and 173. 
 
 
 
 2. 
 
 214. 
 
 10. 
 
 21.12. 
 
 19. 5.5678. 
 
 27. 
 
 1.3229. 
 
 3. 
 
 523. 
 
 11. 
 
 .04738. 
 
 20. 4.1593. 
 
 28. 
 
 .43301. 
 
 4. 
 
 809. 
 
 12. 
 
 900.8. 
 
 21. .83666. 
 
 29. 
 
 1.0541. 
 
 5. 
 
 5.76. 
 
 13. 
 
 .8253. 
 
 22. .28284. 
 
 30. 
 
 .44721. 
 
 6. 
 
 .497. 
 
 15. 
 
 1.4142. 
 
 23. .37947. 
 
 31. 
 
 1.1547. 
 
 7. 
 
 .286. 
 
 16. 
 
 1.7321. 
 
 24. .031623. 
 
 32. 
 
 .64550. 
 
 8. 
 
 .722. 
 
 17. 
 
 2.2361. 
 
 25. .079057. 
 
 33. 
 
 1.1726. 
 
 9. 
 
 1.082 
 
 18. 
 35. 
 
 3.3166. 
 .62261. 
 
 Art. 209 
 
 26. 1.4444. 
 
 36. .42492. 
 
 ; page 175. 
 
 34. 
 
 .94868. 
 
 7. 
 
 a^ + 2 
 
 laj — 4. 
 
 10. 2x^- 
 
 -3x-l. 13. x' 
 
 + xy 
 
 ~2f. 
 
 8. 
 
 r-2/ 
 
 — 1. 
 
 11. 2a?- 
 
 -a-5. 14.- Sa'-2ab~bK 
 
 9. 
 
 0^2-2 
 
 x+1. 
 
 12. 2-; 
 
 x-^2x^. 
 
 
 
 
 
 Art. 213 ; pages 178 and 179. 
 
 
 
 2. 
 
 31.. 
 
 8. 
 
 2.02. 
 
 14. .898. 
 
 20. 
 
 .3107. 
 
 3. 
 
 4.6. 
 
 9. 
 
 372. 
 
 15. 101.3. 
 
 21. 
 
 .7211. 
 
 4. 
 
 .88. 
 
 10. 
 
 21.6. 
 
 16. .0534. 
 
 22. 
 
 1.077. 
 
 5. 
 
 123. 
 
 11. 
 
 .803. 
 
 17. 1.260. 
 
 23. 
 
 .6376. 
 
 6. 
 
 1.14. 
 
 12. 
 
 4.89. 
 
 18. 1.817. 
 
 24. 
 
 .8736. 
 
 7. 
 
 .098. 
 
 13. 
 
 .317. 
 
 19. 1.931. 
 
 
 
ANSWERS. 329 
 
 Art. 214 ; page 179. 
 1. 2a;-32/. 2. itr-x-\-l. 3. a^-2x-2, 
 
 4. a^-l. 5. 2a; + l. 
 
 Art 219; page 182. 
 
 21. 243. 23. 216. 25. -243. 27. 128. 
 
 22. 81. 24. 32. 26. 16. 28. 1296. 
 
 Art. 224 ; pages 184 and 185. 
 
 10. 4 mi 11. Qac^. 14. 5a;"i 15. dah^. 
 
 17. a4_2 + a ^ 18. a^-ic^^ 19. a;-^-l. 
 
 20. x-'-Sx-^-4:X \ 21. 18 a'b^-\- 10 -\- 2 a 'b-\ 
 
 22. 6a.'2-7a;^-19ic^-h5a;-f-9aj^-2a;i 
 
 23. 2a;-^2/-10aJt/ i + 8ar^2/ ^- 24. 2 -4ar^a;2 + 2a~^aj». 
 
 26. 18a6-2-234-«""& + 6a-i6^ 
 
 Art. 225 ; pages 185 and 186. 
 
 11. a^-\-a^b^ + b^. 
 9 -10a;. 14. a;"^ + l. 
 17. a^^ah^-\-bK 
 16. x~^y~^ — x~^y~^ — x~^y~*. 18. m~^ —n~^-{^m^n~^. 
 
 Art. 227 ; page 187. 
 10. f. 13. c'K 14. 71 1. 15. a;-i 16. ak 
 
 Art. 229; pages 188 and 189. 
 
 8. Sx-^-2x-^-l. 16. x^-Sx^-^2xK 21. a^. 
 
 9. 2a;^ + a;-4a;*. 18. ic^'"-'". 22. x. 
 10. ah-^-2-\-arhK 19. a;-^-*. „„ a;^ 
 15. 2y-^-y~K 20. a;«-». 2a^ 
 
 5. 
 
 3c-l 
 
 6. m^. 
 
 
 7. 
 
 xH 
 
 12. 
 
 a-^- 
 
 -a"^ + l. 
 
 13. 
 
 a; 
 
 -1 
 
 15. 
 
 m^- 
 
 2m^w^+w^. 
 
 
 
 
330 
 
 ALGEBRA. 
 
 24. a^h^-a^h' 
 
 25. l+x. 
 
 26. 
 
 la? 
 
 (^ + 1) 
 
 28. 
 
 {m-\-x)^ 
 
 27. {l-dx + x'y^ 29. (l^-.^•2): 
 
 9. 
 
 Art. 235; page 191. 
 
 "5^. 10. V27t. 11. V7m^. 
 
 12. V5«6^. 13. Vo^i^. 
 
 « 
 
 Art. 236; page 192. 
 
 15. (a;-3)V«. 
 
 16. (2a? + 3)V5. 
 y. 17. (m — 9n)V3m. 
 
 12. 5xy\/xy^— 2x^y. 
 
 13. 3a&V2a6=^ + 5&. 
 
 14. (a; + ?/) Vo; 
 
 19. 1^6. 21. iv21. 23. l<^. 25. |,V7. 
 
 20. -->/30. 22. ^\/.3. 24. l^Vl.5. 26. ^^ 
 27. 2/ 
 
 9 
 
 Vl06cd. 
 
 4:W 
 
 gVl^O^. 
 
 go bVa^ -\-ax 
 2{a + x) 
 
 29. 
 
 a Vabc 
 
 b'\a-{-b) 
 
 Art. 237; page 193. 
 
 10. V^^^. 11. vT=r^. 
 
 12. \ 
 
 1+a 
 
 I -a 
 
 13. V4-4i»^. 
 
 3. 5V3. 
 
 4. 7V6. 
 
 5. 3V5- 
 
 6. -^'6. 
 
 7. 20 V2. 
 o Vl5 
 
 15 
 
 Art. 238; pages 194 and 195. 
 
 9. (2a+36)V&. 14. -^24-rv'lS- 
 10. 9V3-7V5. 15. J^3. ^ 
 
 11. 9^2. 
 
 12. |V5. 
 
 13. — V6. 
 
 16. 6aV3a. 
 
 17. (3a2-263)V^rF26. 
 
 18. 10^2-2-^3. 
 
 19. {a-4:)V7x. 
 
 20. 2Va^-/. 
 
ANSWERS. 331 
 
 Art. 239 ; pages 195 and 196. 
 
 2. -^4, -^27. 4. ^^^625, ^21G, ^4iL 
 
 3. ^'125, </l6, ^/9. 5. v'32^, a/2777\ vOI?'. 
 
 6. %'¥^\ '</y¥, K/¥^. 
 
 7. '-\/a'+2ab-^b\ ^d" -Sa'b -^Sab' -b\ 
 
 8. ^2. 9. i/5. 10. V3, v/7, ^4. 
 
 Art. 240 ; pages 196 to 198. 
 
 3. 6V7. 7. ^3. 11. 12v^288. 
 
 4. 75V6. 8. ^432. 12. V^. 
 
 5. 30a;V5. 9. v'^ftV. 13. ^3. 
 
 6. a\/6V. 10. 2a^v^. 14. v^72^. 
 
 17. x-\-^x-6. 23. 4 + 2V10. 
 
 18. 4-5V10. 24. 28 + 8V42. 
 
 19. 2a;-7V3^-12. 25. 11 -20 Vl5 + 5 VlO. 
 
 20. 8a-10Va6-^6. 26. 21-12V3. 
 
 21. x-y-{-2-\/yz-z. 27. 147H-60Vr>. 
 
 22. 2-SVx'-\-x, 28. l + 2aVl-(i'. 
 
 29. 2a-2\/a^-b'K 30. 1. 31. 2. 32. Qx-\- 
 
 Art 241 ; page 199. 
 2. 3v2. 6. a'II 8. ^54. 10. #^. 
 
 5. ^24. 7. ^1^. 9. ^32^. 11. !^/l8fz. 
 
 Art. 242 ; pages 199 and 200. 
 
 6. a^xVa^. 8. 192a; V3«. 10. 8\a%xVbx. 
 
 6. 3V2. 9. 2</2. 11. ISabVSab". 
 
 17. </3^. 19. ^^^^. 20. V3. 21. a/^. 22. V2. 
 
332 
 
 ALGEBRA. 
 
 6y 
 
 ^bxy. 6. 
 
 Art. 243 ; page 201. 
 
 7. f^^^S^^ 
 
 12-4V2 
 
 Art. 244; page 202. 
 
 a — 1 — Va + 1 
 
 7 
 
 5 + 2V3. 
 
 2V6-5. 
 
 /J a + 2 -yjdb + & 
 
 b. r 
 
 a — h 
 „ 16 + 7V10 
 
 9. 
 
 a 
 
 10. a;+l-Va^ + 2a;. 
 
 11. 2a2_i_2aV^^^n' 
 
 12. 
 
 aj2-2 + a;Var^-4 
 
 13 
 
 a^ — 2 (t Va? + a; 
 
 13. 
 
 a 4- Va^ — cc^ 
 
 a^ 
 
 14. Va*-1 
 
 2. 7.243. 
 
 Art. 245; page 203. 
 3. 3.365. 4. .101 
 
 .269. 
 
 3. -8V6. 
 
 4. — ax. 6, 
 
 11. i-4.V^^. 
 
 12. -11-4V6. 
 
 13. 2. 
 
 4. V5-2. 
 
 Art. 249; page 205. 
 5. 12Va6. 7. 5-5V^. 9. 12. 
 
 5. 3+V7. 
 
 6. v^-V^- 
 
 14. 3V5-V2. 
 
 15. 7-3V2. 
 
 -2V-15. 8. 46. 
 
 14. a' + h. 
 
 15. y^ — x^. 
 
 16. 0. 
 
 Art. 256; page 208. 
 
 7. 2V2 4-V7- 
 
 8. 3-V3. 
 
 9. V15-V5- 
 
 10. 60. 
 
 19. 2V2. 
 
 20. V2- 
 
 21. V3. 
 
 10. 3+V^- 
 
 11. 3V2 + V5' 
 13. 5 + 2V2. 
 
 16. Vm+7i — Vm — n. 
 
 17. Va + ic-l-Va. 
 

 
 
 ANSWERS 
 
 • 
 
 
 
 33 
 
 
 
 Art. 257; 
 
 ; pages 209 
 
 and 
 
 210. 
 
 
 
 3. 
 
 2. 
 
 •I- 
 
 13. -3. 
 
 18. 
 
 -3. 
 
 23. 
 
 13 
 4* 
 
 4. 
 
 4. 
 
 9. 4. 
 
 14. 2 a. 
 
 19. 
 
 25. 
 
 24. 
 
 ah 
 
 a + 6 
 
 5. 
 
 6. 
 
 10. 81. 
 
 15. -1. 
 
 20. 
 
 12. 
 
 25. 
 
 -6. 
 
 6. 
 
 5. 
 
 11. 4. 
 
 16. 4. 
 
 21. 
 
 3. 
 
 26. 
 
 3. 
 
 7. 
 
 -2. 
 
 12. 8. 
 
 17. 5. 
 
 22. 
 
 ^. 
 
 27. 
 
 3a-l 
 
 Art. 259; page 212. 
 
 h 
 
 3. ±3. 7. ±^^' 11- ±3- 15- ±-2 
 
 4. ±5. 8. ±1. 12. ±2. 16. ±2. 
 
 5. ±-' 9. ±V19- 13. ±V^W^. 17. ±1. 
 
 6. ±2V^nr. 10. ±2. 14. ±V2. 18. ±^. 
 
 Art. 262 ; page 215. 
 
 3.1,-5. 5.4,3. 7. 2, -|. 9. |, i. 11.1,-?- 
 
 3 2 2 2 
 
 4. 4,1. 6. 2,-3.8. -2, -J. 10. 3,-i. 12.|,-| 
 
 Art. 263 ; page 216. 
 
 2. -2,-. 4. ?,i. 6. -1,-—. 8. -, --. 
 
 '4 5' 5 '2 4' 9 
 
 3. 1,-li. 5. 1,5. 7. 2, -i^. 9. K-' 
 
 9 4 8 2 3 
 
 10 i -I. 11 5 --L. 
 
 ^"- 4' 8 "• 4' 12 
 
SS4 
 
 L 
 
 
 
 ALGEBKA. 
 
 
 
 
 
 
 
 Art. 265 ; page 218 
 
 
 
 
 3. 
 
 -a,i. 
 
 6. 
 
 -2, 
 
 |. ,.., 
 
 2 
 
 7* 
 
 
 -i-^- 
 
 4. 
 
 '-!• 
 
 7. 
 
 1 
 2'" 
 
 .5. ,.._ 
 
 4, 
 
 5 
 3* 
 
 "■ -j -i 
 
 5. 
 
 6, -3. 
 
 8. 
 
 5, - 
 
 1 U. 3, 
 
 — 
 
 1 
 5* 
 
 »l'-r 
 
 
 
 15. 
 
 1 1 
 3' 5* 
 
 "I- 
 
 2 
 5* 
 
 
 
 
 
 Art. 266 ; pages 218 to 221. 
 
 
 1. 
 
 1 1 
 
 2' 6* 
 
 
 13. 
 
 -10±V78. 
 
 
 25. 
 
 4,0. 
 
 2. 
 
 ^1, -4. 
 
 
 14. 
 
 ■■B- 
 
 
 26. 
 
 -2 11. 
 '65 
 
 3. 
 
 '-1 
 
 
 15. 
 
 ^•r 
 
 
 27. 
 
 '•I 
 
 4. 
 
 5 15 
 2'T 
 
 
 16. 
 
 '■-1 
 
 
 28. 
 
 *-f 
 
 5. 
 
 13, -2. 
 
 
 17. 
 
 7 5 
 2' 2 
 
 
 29. 
 
 3, -2. 
 
 6. 
 
 1 1 
 2' 14' 
 
 
 18. 
 
 -f 
 
 
 30. 
 
 .,-|. 
 
 7. 
 
 ■•?• 
 
 
 19. 
 
 '•[ 
 
 
 33. 
 
 a-}-5, a — b. 
 
 8. 
 
 3 1 
 2' 2' 
 
 
 20. 
 
 ■•I- 
 
 
 34. 
 
 a, —6. 
 
 9. 
 
 -'■-^ 
 
 • 
 
 21. 
 
 '23 
 
 
 35. 
 
 l,a. 
 
 10. 
 
 5, -3. 
 
 
 22. 
 
 1, -18. 
 
 
 36. 
 
 -16, -2c. 
 
 11. 
 
 4 ± 2 V3. 
 
 
 23. 
 
 3 ^1. 
 
 
 37. 
 
 m-, — m^ 
 
 12. 
 
 .,-1 
 
 
 24. 
 
 2. 
 
 
 38. 
 
 6 d 
 a c 
 
ANSWERS. 335 
 
 39. 2p,-6p. 41. -,-. 43. -, -• 
 
 40. -i^, -5. 42. m. ^. 44. a, L 
 
 2 3 m -f 1 a 
 
 45. a + ^ 46. 50 "'^^ "" 
 
 c a + b 
 
 46. (a + 6)2, -(a -6)2. 51. -a, -6. 
 
 47. 9??i, — m. 52. a — 6, —a — c. 
 
 >io o Ko2« — ^ 3 a + 26 
 
 48. a, —2a. 53. ^ 
 
 49. -a, —2a. 64. 
 
 oc 6c 
 
 a + 6 g — 6 
 a — 6 a + 6 
 
 Art. 267 ; page 222. 
 
 3. 2, --. 5. 5, 2. 7. -1, -i- 
 
 2 6 
 
 4. 5, _i. 6. -2,5. 8.2,1. 
 
 9. 5, _1. 11. ?, 2. 
 
 .22 33 
 
 10. -I, _?. 12. 5, _§. 
 
 2 3 2' 5 
 
 Art. 268 ; pages 224 to 227. 
 
 3. 10 barrels, at $17.50. 5. 21,6. 
 
 4. 9, 6. 6. Length, 125 ; breadth, 50. 
 
 7. 14, 5. 12. 16. 17. 9. 
 
 8. 7,8. 13. 16 barrels, at $6. 18. 15,9. 
 
 9. 16, 10. 14. 13, 6. 19. 3712. 
 
 10. 5,3. 15. 3 inches. 20. $80 or $20. 
 
 11. 7, 8, 9. 16. $30. 21. 20. 
 
 22. Area of court, 529 sq. yds. ; width of walk, 4 yds. 
 
 23. 36 bushels, at $1.40. 24. 84. 
 
336 ALGEBRA. 
 
 25. Larger pipe, 5 hours ; smaller, 7 hours. 
 
 .26. $2000. 27. 5. 28. 6. 29. 343 cubic inches. 
 
 30. First, 14,400 ; second, 625 ; or, first, 8464 ; second, 6561. 
 
 31. 38 or 266 miles. 32. 70 miles. 
 33. 100, at $15 each. 
 
 Art. 270 ; pages 229 and 230. 
 
 4. ±3, ±4. 7. ±h ±1^5. 10. ±1, ±2. 
 
 2 5 
 
 5. -^3, --^23. 8. ±1, ±^- 11. 2, -3. 
 
 6. 
 
 1, -2. 
 
 9. 
 
 '■-1 
 
 
 12. 4, ^. 
 
 13. 
 
 243, -28 
 
 -^784. 
 
 14. 
 
 (± 
 
 .)■. {-'ij 
 
 15. 
 
 «• m- 
 
 17. 
 
 1 /2\f 
 4' [sj ' 
 
 
 «. .. f 
 
 16. 
 
 "• (!)'■ 
 
 18. 
 21. 
 
 25, A. 
 ' 16 
 
 -32,. 2-i 
 
 
 20. ± 8, ( 
 
 (-¥)*■ 
 
 Art. 271 ; page 232. 
 
 4. 2, -2, 3, 7. 11. 8, -2, 3±V110. 
 
 5. 2, -2, -3, -7. 12. 6, -9. 
 
 6 
 
 1 9 5±'> ^^ 13 ^ ^ -3±2V3 
 
 w. 
 
 1, y, oiaizyz. lo. ^, ^, ^ 
 
 7. 
 
 ±2, ±V11- 14. -2, --^15. 
 
 8. 
 
 9 -3±V-55 IK o 7 9 
 3, -, ^ . 15. 0, 2, -, -. 
 
 9. 
 
 2, -3, 4, -5. 16. 1, -1, -6, -8. 
 
 10. 
 
 1, 2, -5, 8. 17. 0, -5, i, -^. 
 
 
 18. a + 6^ a + 3A/36«. 
 
2/=-«, V. 
 
 ANSWERS. 887 
 
 Art. 274; page 234. 
 
 Note. In this, and the three following articles, the answers are 
 arranged in the order in which they are to be taken ; thus, in Ex. 2, 
 
 the value a: = 1 is to be taken with m = — 2, and x = with y = —. 
 
 •^ 25 -^ 25 
 
 2. 0^ = 1, -^; 6. a^ = 3, -4; 11. x = ^, -1|; 
 
 = _2 i5 ^ = ^'-^- ^2 62 
 
 ^ '25* 7. a:=2,3; ^ ' 13* 
 
 3. if=7, -8; 2/ = -3, -2. 12. x = i, -1; 
 
 8. a; = 5, -2; 2' = 2> - 1- 
 
 4. . = 9,-6; , = _2,5. ^g. ,^3,^, 
 y = 6, -9. 3 
 
 5. ^=2,-1; 2/-2, - — 
 
 _ 5 10. a;=6, -4; 14. a; = 4, -2; 
 
 ^~ '3* 2/ = -4, 6. 2/ = 8, -1. 
 
 Art. 275 ; page 237. 
 
 4. a; = 3, -2; 10. a; = 5, -3; 15. a;=7, -12; 
 
 2/ = -2,3. 2/=3, -5. 2/ = -12, 7. 
 
 6. ^'=^'-^' 11. x=2, 1; 16. a;=15, -3; 
 
 y-^^-^' 2^=1,2. 2/ = -3, 15. 
 
 6.a; = -3, -7; 
 
 2/ = 7, 3. 12. a^ = 8, -^; 17. a; = - 1, - 4 ; 
 
 7. a;=2, -3; 
 2/ = -3, 2. 
 
 8. a;= ±4, ±3; 
 2/= ±3, ±4. 
 
 15 
 
 ^ = 4, 1. 
 
 13. a^=±7, ±6; l^' ^=6' "^^^ 
 2/=±6, ±7. 2/=ll, -6. 
 
 9. a; = 3, -7; 14. a; = 3, -7; 19. a; = 7, - 9 ; 
 
 ?/ = -7, 3. 2/ = 7, -3. 2/ = -9, 7. 
 
338 ALGEBRA. 
 
 Art. 276; page 239. 
 2. x=±7, ±^V2; 7. x = ±3, ±|; 
 
 2/ = ±2, :p|v2. 2/ = ±5, ±^. 
 
 3 
 2/ = Tl, ^iv3. 
 
 4. a;=±2, ±l-y/5; 
 
 9. aj=±2, ±^V31; 
 2/ = ±3, T^V31. 
 
 
 5. a;=±2, ±-V10; 
 5 
 
 10. x = ±2, ±YjVii; 
 
 1 2 .. . = :f1,±^V11. 
 
 ,= ±-,^-Vio. 
 
 11. a;=±l, ±2; 
 
 6. aj=±3, ±2V2; 3 ^9 
 2/ = ±l, ±V2. ^ 2' 8 
 
 Art. 277; pages 242 and 243. 
 
 3 
 
 5. aj = — 5, ; 10. x=2m, — m; 
 
 . 2/ = m, -2m. 
 ■ 2/=l, -^• 
 
 '^ 11. a; = 3,2, -3±V3; 
 
 ^- ^=1'^; 2/=2,3, -3q:V3. 
 2/ = 8, 1. 
 
 7. 0^ = 4, -4; 12. a^=±2, i^V^; 
 
 2/ = ±o, ±5. 
 
 5 
 
 2/ = =f3, T-a/S. 
 8. ic = 2, 3; ^ -^ ' +^V 
 
 2/ = 3, 2. 13. a; = 8, 4; 
 
 = -2, 
 
 = -5, 
 
 11 
 
 8 14. a; =±2, ±14; 
 
 11 2/ = t3, :f5. 
 

 ANSWERS. 33! 
 
 16. 
 
 x = K'-; 
 
 18. 
 
 --7'^; 
 
 
 2' 9' 
 
 
 4 7 
 
 
 1 1 
 
 
 1 1 
 
 
 ^ = 9' 2- 
 
 x = 2a, —a — b; 
 
 19. 
 
 ^ = -7'-4- 
 
 16. 
 
 07=3,4, _4±V- 11 
 
 
 y = a + b, -2a. 
 
 
 y = 4,3, -4^V-11 
 
 17. 
 
 »= = |,-3; 
 
 20. 
 
 07 = ±3, ±2; 
 
 
 y = ±\, ±V3- 
 
 
 2/ = q:2, T3. 
 
 21. 
 
 x=\, -3, 1±V^^; y 
 
 = -2 
 
 3, 1, lq:V-2. 
 
 22. 
 
 .-1,2,''^^^-''; y- 
 
 , -3±V-55 
 ' ^' 2 • 
 
 23. 
 
 07= 2 a, —a; 
 
 32. 
 
 07=2,1, 3±V-19 
 
 
 y = a, -2a. 
 
 
 2 
 
 24. 
 
 a; = 4,9; 
 2/ = 9,4. 
 
 
 4 
 
 25. 
 
 a;=a — &, b — a; 
 
 33. 
 
 
 2/ = a + 6, 2a. 
 
 
 2/ = -l, 2. 
 
 26. 
 
 05=2,3; 
 
 34. 
 
 07 = 4, -; 
 
 
 2/ = 3, 2. 
 
 
 9 
 o 22 
 
 27. 
 
 x = b ± a; 
 
 
 y=2,-; 
 
 o 59 
 
 
 y = — b ± a. 
 
 
 
 
 
 2=3, 
 
 28. 
 
 a; = 4, 16, -12±V5^; 
 
 
 9 
 
 
 2/ = 5, -7, -lipvoB. 
 
 35. 
 
 ^1 ^59 
 07 = ±1, ±---; 
 
 29. 
 
 07 = 9, 5^2_o^; 
 
 
 4 
 o . 31 
 
 
 A 20 
 
 
 2/=if3, ±^. 
 
 
 2/ = 4, 
 
 
 8 
 
 
 ^ 117 
 
 a7=±2, ±1^7; 
 
 36. 
 
 07 = 3, -7; 
 
 30. 
 
 
 y = -l, -21. 
 
 
 2/ = ±5, TyV7. 
 
 37. 
 
 ^ ^ o -9±V309. 
 
 07 — O, Z, , 
 
 31. 
 
 a:=2a — 6, a-26; 
 
 
 2,= 2,3,-^^^^«^. 
 
 
 y = a — 2b, 2a — b. 
 
 
 ^ ' ' 12 
 
340 ALGEBllA. 
 
 Art. 278 ; pages 243 to 245. 
 
 1. 9, 5. 3. 900 square rods, 400 square rods. 
 
 2. 13, 6. 4. 3, 4. 
 
 5. Length, 10 rods ; breadth, 6 rods. 6. 7, 4. 
 
 7. Duck, $1.75; turkey, $2.25. g 3 -16 
 
 8. 21 or 12. * 5 24 ' 
 
 10. Length, 150 feet; breadth, 100 feet. 
 
 11. Length, 16 rods ; breadth, 10 rods ; or, length, 13^ rods ; 
 
 breadth, 12 rods. 
 
 12. Rate of the boatman, 4 miles an hour ; of the stream, 
 
 2 miles an hour. 
 
 13. A, 40 acres at $8 ; B, 64 acres at $5. 14. 7, 5. 
 15. 5, 2. 16. Hind-wheel, 12 feet; fore-wheel, 9 feet. 
 
 17. First rate, 7 per cent ; second, 6 per cent. 
 
 18. Length, 16 yards ; width, 2 yards. 
 
 Art. 281 ; page 247. 
 
 9. x^-9x = -20. 15. 6a;^ + 31a.- = -35. 
 
 10. x' + 2x = S. 16. 30^2+170^=0. 
 
 11. 5a^-12a;=9. 17. x'-2ax-bx=-a'-aJ) + 2b\ 
 
 12. 3i^_2a;=133. 18. x^ -2vix = m'^-m^ 
 
 13. 12a^-17« = -6. 19. x^-4.x = -l. 
 
 14. 21ic2 + 44a; = 32. 20. 4a^ — 4:mx = n-m\ 
 
 Art. 282 ; page 249. 
 
 6. 0, I' 10. 0, ±3. 
 
 16. 
 
 -1, 2, ±3, ±4. 
 
 9. 2, -4, -5. 12. 0,-4. 
 
 17. 
 18. 
 
 , l±V-3 
 "'' 2 * 
 
 13. ±a,«±^«^ + ^^ 
 
 2 
 
 _^, ±l±V-3 
 =^^' 2 ' 
 
 ... ., i. -|. 
 
 19. 
 
 1, ±3. 20. ±1, 
 
ANSVv^ERS. 341 
 
 Art 283 ; page 251. 
 
 3. (a; + 5)(a; + 8). 7. (a; -3) (4a; -3). 
 
 4. (a;-2)(a;-9). 8. (5a;+ 1) (a;+ 7). 
 
 5. (a; -10) (a; + 6). 9. (a; + 4) (4a;- 1). 
 
 6. (a; -5) (2a; + 3). 10. (13 + a;) (3 -a;). 
 
 11. (2-3a;)(H-2a;). 
 
 12. (a;-2 + V3)(^-2-V3). 
 
 13. (3x-l + -y/5){Sx-l-^o). 
 
 14. (4a; -3) (2a; -3). 16. (7a;-i-l)(a; + 7). 
 
 15. (2 + a;)(3-2a;). 17. (a;- 3a) (6a; + 5a). 
 
 18. (5-6a;)(l+2a;). 
 
 19. (3a;-2 + V3)(3a;-2-V3). 
 
 20. (3a; + 22/) (4a; -52/). 22. (10a;- 3) (a; -2). 
 
 21. (4a;-2/)(2a;-|-52/). 23. (4a;4-5m) (5.^• + 4m). 
 
 24. (8a; -5) (2a; -3). 
 
 25. (V17 + 4 + .r)(V17-4-a;). 
 
 26. (36-2a;)(56 + 12a;). 
 
 27. (7a; + 3mw)(3a; + 7mri). 
 
 28. (5a;-2 + V6)(5a;-2-V6). 
 
 Art. 284 ; page 252. 
 
 4. (x'-\-x-{-l){3cr-x-hl). 
 
 6. (a;2-h3a;H- l)(ar'-3a;+l). 
 
 6. {2a'-^2ab-b-){2a^-2ab-h^). 
 
 7. (m- + 4 mn + n^) (m^ — 4 mn -\- n-) . 
 
 8. {l-^Sb-2b'){l-Sb-2b'). 
 
 9. (ar^4-4.T2/ + 2?/2)(ar'-4a;2/ + 22/0. 
 
 10. (2a2 + 2a4-3)(2a2_2« + 3). 
 
 11. {27rv' + 2m-5){2m^-2m-5). 
 
 12. (a2 + aa;V3-a;')(a'-aa;V3-a;'). 
 
 13. '(a^ + a:V2 + l)(.'^-a;V2 + l). 
 
342 ALGEBRA. 
 
 14. (2a2 4-a6 + 462)(2a2_a&4.4&2). 
 
 15. (4 a^ H- 5 mx — 3 m-) (4i«^ — 5 mx — 3 m^) . 
 
 16. {3x- + Sx-^2-\-2){3x--Sx^'2-^2). 
 
 17. (3a- + 4am + om^) (Sa^ - 4am + om^). 
 
 18. (2 + 2w- 7^2) (2 -271 -7^2). 
 
 19. (4a^4-3aJ3/-52/2)(4a^-3a:?/-5/). 
 
 Art. 309 ; pages 262 and 263. 
 
 3. 
 
 4. 
 
 5. A. 
 
 35 
 
 7. 
 
 12. 9. |V3. 
 
 11. 
 
 2, -3, 
 
 4. 
 
 11. 
 
 6. ?5. 
 
 27 
 
 8. 
 
 14V3. 10. 3, -1-. 
 
 12. 
 
 -!■ 
 
 13. 
 
 14. 
 
 x = 
 
 y = 
 
 x = 
 
 y = 
 
 :±6, 
 :±10. 
 
 : ± a% 
 
 
 15. 25, 20. 
 
 16. 23, 27. 
 
 17. 9, 3. 
 
 18. 
 19. 
 24. 
 
 8, 18. 
 26, 14 
 17, 12 
 
 25. 12, 8. 26. A, $105; B, $189; C, $270. 27. 8:7. 
 
 28. First, 1:2; second, 2:1. 29. ^^, ^^. 
 
 . 2 ' 2 
 
 Art. 314; page 265. 
 
 2. Z=71, 
 
 6. ^ = f, 
 
 JS = 6A0. 
 
 ^_315. 
 
 3. Z = -69, 
 
 2 
 
 JS = - 620. 
 
 7 ?--2 
 
 7. (- ^, 
 
 4. Z=57, 
 
 s = o. 
 
 S = 652. 
 
 R 7 23 
 8. l-~, 
 
 5. Z = -145, 
 /S = -2175. 
 
 r, 62 
 
 9. 
 
 1 = 
 
 ._ 5 
 11 
 
 
 S = 
 
 .1 
 2* 
 
 10. 
 
 1 = 
 
 = 5, 
 
 
 s = 
 
 :17. 
 
 11. 
 
 1= 
 
 137 
 15' 
 
 
 s = 
 
 917 
 
 15 
 
ANSWERS. 343 
 
 Art. 315; pages 267 to 269. 
 
 4. 
 
 a = 3, 
 ^' = 741. 
 
 
 10. 
 
 n= 16, 
 Z = -43. 
 
 16. 
 
 H 
 
 6. 
 
 a =11, 
 Z=-12i 
 
 
 11. 
 
 n=18, 
 .^ = 411. 
 
 
 -f 
 
 
 
 
 
 17. 
 
 a = 4, -5; 
 
 6. 
 
 a-i. 
 
 
 12. 
 
 a = 3, 
 
 
 n = 52, 43. 
 
 
 S = 39. 
 
 
 13. 
 
 1 1 
 
 18. 
 
 71=14, 
 
 7. 
 
 cz = -JL, 
 
 
 " = 2'-6^ 
 
 
 Z = -15J. 
 
 
 12' 
 
 
 
 ?i=10, 12. 
 
 
 
 
 ? = -lJ. 
 
 
 14. 
 
 a = -i, 
 
 20. 
 
 71-1 
 
 8. 
 
 d = -2J, 
 
 
 
 2' 
 
 
 
 
 n = 13. 
 
 
 
 cZ=2. 
 
 21. 
 
 ^j 2(s-a7i) 
 
 9. 
 
 a = 5, 
 
 
 15. 
 
 15' 
 
 
 n(n-l)' 
 J 2 s — an 
 n 
 
 
 d = -3. 
 
 
 
 71=16. 
 
 
 22. 
 
 2S- 
 a = 
 
 n(n- 
 2n 
 
 -l)d 
 
 > 24. a: 
 
 = l-{r, 
 
 i-l)d, 
 
 
 l_2S + 
 
 n(n~ 
 
 -l)d 
 
 ; >S: 
 
 = 'il^i 
 
 -(n-l)d-]. 
 
 
 
 2n 
 
 
 
 
 23. 
 
 d 
 
 f 1, 
 
 
 25. a: 
 
 2S- 
 n 
 
 nl 
 
 
 S_{l + a 
 
 )(l- 
 
 -a + d) ^^ 
 
 _2{nl- 
 
 zSl. 
 
 2d n{n-l) 
 
 26 I = -<^±V^^^'»"+(2a-(Z)^ 
 2 
 
 27 d= ^'~^' 28 ct^ ^^V(^^ + ^0'-^^^ 
 
 2/S'-a-^' ' 2 
 
 
 71 
 
 _2Z + ciq:V( 
 
 ;2Z + d)2_8d>S 
 2cZ 
 
 
 Art. 316 ; 
 
 page 270. 
 
 
 
 .. ..!. 
 
 .. .-t 
 
 .. .-!. 
 
 
 8. 2ab. 
 
 e^ .=!. 
 
 .. .-|: 
 
 'il- 
 
 
 9 a^+6^ 
 
344 ALGEBRA. 
 
 Art. 317 ; pages 271 and 272. 
 
 3. 2500. 6. 11. g 23a^ 
 
 4. -43. 7. Z=10m-277i, * 3 
 
 5. 4,11,18,25. S=65m-135n. 9. 62,750. 
 10. 2, 6, 10, 14 ; or, - 2, - 6, - 10, - 14. 11. 22. 
 
 12. -4, -1,2, 5,8; or, ^, i^, A, _il _Z. 
 
 14' 7 14' 7 2 
 
 13. After 9 days, at a distance of 90 leagues. 
 
 14. 41171 feet. 15. 3, 7, 11 ; or, 4i, 7i, 10|-, 
 16. 8. 17. $2950. 18. 852. 
 
 Art. 321 ; pages 274 and 275. 
 
 3. 1 = 256, 
 >^ = 511. 
 
 7 Z- 1 
 
 2048' 
 
 10. 
 
 324' 
 
 4. l='\ 
 243' 
 
 2048 
 
 
 o 91 
 ^=162* 
 
 ^ _ 2059 
 243 
 
 ft 7 729 
 8- ^= 64' 
 
 11. 
 
 Z=192, 
 
 5. Z=2048, 
 >S'=1638. 
 
 192 
 
 
 >S=129. 
 
 6. Z = — i-, 
 256' 
 
 »■ -h 
 
 12. 
 
 1= ' , 
 
 768 
 
 256 
 
 32 
 
 
 256 
 
 
 Art. 322 ; pages 276 and 277. 
 
 
 3. a = i. 
 
 5. r = 3, -3; 
 
 7. 
 
 «=| 
 
 2 
 , 1023 
 
 ^= 2 • 
 
 /S =2186, 1094. 
 
 
 3 
 
 6561 
 
 4. „ = -|, 
 Z = 48. 
 
 6. n = 5, 
 aS'=121. 
 
 8. 
 
 1 
 '" = -4' 
 c, 2457 
 '^-1024- 
 
9. 
 
 n = 
 
 = 7, 
 
 
 
 r = 
 
 1 
 ''2 
 
 
 10. 
 
 n = 
 
 6, 
 
 
 
 1 = 
 
 — 
 
 243 
 2 
 
 ANSWERS. 345 
 
 11. a = -l, 14. a = rl-{r-l)S. 
 n = 8. 
 
 12. 1 = ^^'-'^^ . 15. «=-!-, 
 r r^-i 
 
 lZ.r = ^^. S= ^C-"-!) ■ 
 
 S-l r»-'(r-l) 
 
 16. a = -fc^lM, ^^'^ V-1)^. 
 r" — 1 r" — 1 
 
 
 
 
 X f . / 
 
 Ma' ^~ '*-^^-"-> 
 
 
 _ 
 
 
 
 
 Art. 323; page 278. 
 
 
 2. 
 
 4. 
 
 
 4. 
 
 3 ^9 
 -4* ^' I' 
 
 fi 160 
 ®- 19" 
 
 3. 
 
 8 
 3* 
 
 
 5. 
 
 15 - 30 
 4* * 11* 
 
 Art. 324 ; page 279. 
 
 9. ^ . 
 
 2. 
 
 8 
 11* 
 
 3. 
 
 11 
 27* 
 
 4. 1^ 5. ^^ 6. ^^ 
 15 165 825 
 
 7 237 
 * 1100' 
 
 Art. 325 ; page 279. 
 
 g g 4 8 16 32 64 
 
 ' 3' 9' 27' 81' 243* 
 « . 3 9 .27 81 .243 
 
 **• "^2' 2' "*"¥' y "^1"* 
 
 4. -6, -18, -54, -162, -486, -1458. 
 
 . ± -, -, ± — , — , ± — , , ± 
 
 4 8 16 32 64 128 256 
 
 6. ±4, -8, ±16, -32, ±64. 
 
 - 9 27 81 243 ^ . o ^ •> n •' in « 
 
 7. , — , , 0.5. 9. 4ar— 9?/-. 10.— 
 
 4 16 64 256 ^ h 
 
346 ALGEBRA. 
 
 Art. 326 ; pages 280 and 281. 
 2. 3. 3. 5, 10, 20, 40; or, -15, 30, -GO, 120. 
 
 4. 5, 15, 45 ; or, 40, - 20, 10. 5. ± 4. 6. $64. 
 
 n/ omn* 4. Q o A Q -in 810 540 360 240 
 
 7. 3100 feet. 8. 2, 4, 8, 16 ; or, — , - — , -_,___. 
 
 ^ _81_^ 10. 3, 9, 27. 12. 1, 2, 4. 
 
 8192* 11. 2, 4, 8 ; or, - 2, 4, - 8. 
 
 Art. 332 ; pages 284 and 285. 
 
 2. c^ + 4 c^d" 4 -f 6 c^d~ ' + 4 c^cT^ + d-^ 
 
 3. m~^ — 5m~'^n^ + 10m~->i* — lOm^W + bm'^n^ — n^^. 
 
 4. a^?/~^ — 9ic?/~^ + 27a;~^2/~ 27a;~^2/^. 
 
 5. 0;^"*+ 10a;^'"2/"+ 40ic3'"/"4-80a;2m^n_|_ 80a;"'/'^+ 322/^\ 
 
 6. a^2_f.i2a9aj* + 54a«a;+108aV'+81aj2. 
 
 7 . m^w~ ^ — 5 m ^ ti" ^ _j_ x m*?i~ ^—10 7?i ^ ^i" 2 _^ 5 ^3^ _ ^^^2 ^ 
 
 9. m^ — 2 wPv? H 
 
 2 2 16 
 
 10. a^6"^-5a^6-i+10a*6~3_i0a-^6i+5a"^'6-a"'63. 
 
 11. a«-12a'^' + 54a"3-_l08a^ + 81al 
 
 12. \^x'^y'^ + Ux^y~^-\-Qxy + x~^yi + ^^—^' 
 
 16 
 
 13 a-12 2a-^"a:^ I ^^'^^ 20a-V'' 5a-V 2a V^ a?^ 
 3 27 27 81 729 
 
 14. ar''+15a;^V^+90a;V'4-270a;%~"'-f405i»V^+243?/-2. 
 
 IK a"lfix~^^ 3 5^0?"* , ^ •?!,-** o -61.-^ 
 15. h6a"^6 ^o;'^ — 8a ^6 -a;. 
 
 8 2 
 
 16. 81 a-362_io8a-26 + 54a-^ -126-1 + a6-2. 
 
ANSWERS. 347 
 
 -^ a^ , 12a2 60a ,..^, 2406 ,19252 G463 
 b^ b^ b a a^ a^ 
 
 18. l-4:X + 2x' + S2tf-5x*-8a^-^2x^-\-4x^-{-x^. 
 
 19. o^-{-4x^-\-2xl^-8x^-bx'-{-8x^-\-2xr-4.x+l. 
 
 20. l-\-8x-\-20x'-{-8x^-26x'-Sa^-^20a^-8af-\-a^. 
 
 21. l-5a;+15a^-30aT^ + 45a;*-51ar' + 45aj«-30af 
 
 Art. 333 ; page 286. 
 2. 4G2aV. 6. ^i^'- 9. USGOx-^vK 
 
 7. 715a;". 10. -Uo5a^x-\ 
 
 3. 252 m^ 
 
 4. -792(rW. 
 
 5. 1001 a8. 8. -^1^?—^. 11.126720. 
 
 16 
 
 Art. 343 ; page 290. 
 
 2. .7781. 7. 1.3222. 12. 1.9912. 17. 2.1303. 
 
 3. 1.14G1. 8. 1.7993. 13. 2.0212. 18. 2.2252. 
 
 4. .9030. 9. 1.7481. 14. 2.0491'. 19. 2.1673. 
 
 5. 1.0791. 10. 1.9242. 15. 2.1582. 20. 2.5741. 
 
 6. 1.1761. 11. 1.6532. 16. 2.3343. 21. 2.5353. 
 
 Art. 345 ; page 291. 
 
 2. .3680. 5. 1.5441. 8. .2252. 11. .8539. 
 
 3. .1549. 6. .1182. 9. 2.2431. 12. .7660. 
 
 4. .5229. 7. 2.0970. 10. 1.0458. 13. .7360. 
 
 Art. 348 ; page 292. 
 
 3. .2863. 6. .1398. 9. 4.5844. 12. .4225. 
 
 4. 2.7090. 7. .7194. 10. 3.2620. 13. .1590. 
 
 5. 4.2255. 8. .6611. 11. .9801. 14. .0430. 
 
348 ALGEBRA. 
 
 15. .1165. 18. .1750. 22. .2601. 25. .2215. 
 
 16. .3860. 20. 2.6145. 23. .6884. 26. .2494. 
 
 17. .2212. 21. .1678. 24. .1840. 27. .1449. 
 
 Art. 350 ; page 293. 
 
 2. .2552. 7. 7.7323 -IC. 12. 2.4804. 
 
 3. 1.3522. 8. 6.4983-10. 13. 8.7905-10. 
 
 4. 9.2922-10. 9. 3.8663. 14. 6.3588. 
 
 5. 8.6811-10. 10. .6074. 15. .1964. 
 
 6. 1.5841. 11. 9.6511-10. 16. .0698. 
 
 17. .1688. 
 
 Art. 355 ; page 297. 
 
 6. 9.8878-10. 10. 1.3028. 14. 0.7144. 
 
 7. 3.0237. 11. 4.9659. 15. 3.0155. 
 
 8. 0.5177. 12. 9.6055-10. 16. 8.9379-10. 
 
 9. 8.7164-10. 13. 7.8560-10. 17. 9.0610-10. 
 
 Art. 357 ; page 299. 
 
 4. 1.646. 8. .003318. 11. .2079. 14. 63329. 
 
 5. 8886. 9. 10221. 12. 44.48. 15. .01301. 
 7. .01461. 10. 9.492. 13. .1109. 16. 502.9. 
 
 Art. 361 ; pages 302 to 304. 
 
 5. .01157. 9. -3.908. 
 
 6. -.7032. 10. 1782. 
 
 7. 7.672. 11. .3500. 
 
 8. .6688. 12. -.4748. 
 
 1. 
 
 8.454. 
 
 2. 
 
 10.73. 
 
 3. 
 
 - 2202 
 
 4. 
 
 .2179. 
 
ANSWERS. 
 
 349 
 
 13. .4127. 
 
 14. -4.671. 
 
 15. .2415. 
 
 16. -.0725. 
 
 17. 13587. 
 
 18. .006415. 
 
 19. -1.184. 
 
 20. .000007038. 
 
 21. 2.924. 
 
 22. .9146. 
 
 23. 4.638. 
 
 24. .0000639. 
 
 25. 1.414. 
 
 26. 1.495. 
 
 27. -1.246. 
 
 28. .6553. 
 
 29. .2846. 
 
 30. 2.372. 
 
 31. -.5142. 
 
 32. .1588. 
 36. 5.883. 
 
 36. .7885. 
 
 37. 1.195. 
 
 38. .6803. 
 
 39. .6443. 
 
 40. .5010. 
 
 41. 1.062. 
 
 42. -.9102. 
 57. .03344. 
 
 43. 
 
 1.093. 
 
 44. 
 
 .7035. 
 
 45. 
 
 .5807. 
 
 46. 
 
 -.6313. 
 
 47. 
 
 24.62. 
 
 48. 
 
 .2979. 
 
 49. 
 
 98.50. 
 
 60. 
 
 1.660. 
 
 61. 
 
 3.076. 
 
 62. 
 
 .8678. 
 
 63. 
 
 1.134. 
 
 64. 
 
 .5881. 
 
 56. 
 
 1.805. 
 
 66. 
 
 .003229. 
 
 3. .4581, 
 
 8. 
 
 Art. 363 ; page 305. 
 4. .1853. 5. -.4949. 6. -.2601. 
 
 y mlog& + 7ilogc Q ^^^ log^-logc^ I I 
 
 log a log r 
 
 log n — log m log r 
 
 11. w = 
 
 log I — log a 
 
 \og{S-a)-\og{S-l) 
 
 + 1. 
 
 12. ,,_ log?-lo^rr^-(r-l) ^] ^ ^ 
 logr 
 
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