r 
 
 ev5 
 
 LIBRARY 
 
 
 UNIVERSITY OF CALIFORNIA. 
 
 GIKT OK 
 
 Received K^^^a^^S^^iSS^ 
 
 Accessions No. ^<^^ §"/ .S"-^^// iV^. 
 
 fib 
 
ROBINSON'S MATHEMATICAL SERIES, 
 
 KEY TO ROBINSON'S 
 
 NEW 
 
 UNIVERSITY ALGEBRA. 
 
 FOR TEACHERS AND PRIVATE LEARNERS. 
 
 NEW Y O K K : 
 IVISON, PHINNEY, BLAKEMAN & CO., 
 
 CHICAGO: S. C. GRIGGS & CO. 
 
 180 7. 
 
ROB INSON'S 
 
 27i€ most COMPLBTB, most Practical, and most Scientific Series of 
 Mathematical Text-Books ever issued in this country 
 
 {IN TWKNTY-X^VO VOLTJMiEJS.) 
 
 1 Robinson's Proan^essive Table Book, • 
 
 II. Bobinson's Progressive Primary Arithmetic,- 
 
 III. Robinson's Progressive Intellectual Arithmetic, 
 
 IV. Robinson's Rudiments of Written Arithmetic, 
 V. Robinson's Progressive Practical Arithmetic, 
 
 VI. Robinson's Key to Practical Arithmetic, - 
 
 VII. Robinson's Progressive Higher Arithmetic, • 
 
 VIIL Robinson's Key to Higher Arithmetic, - 
 
 IX. Robinson's New Elementary Algebra, 
 
 X. Robinson's Key to Elementary Algebra, - 
 
 XI. Robinson's University Algebra, . - . . 
 
 XIL Robinson's Key to University Algebra, - 
 
 XIII. Robinson's New University Algebra, 
 
 XIV. Robinson's Key to New University Algebra, • 
 XV. Robinson's New Geometry and Trigonometry, 
 
 XVI. Robinson's Surveying and Navigation, - 
 
 XVII. Robinson's Analyt. Geometry and Conic Sections 
 
 XVIII. Robinson's Differon. and Int. Calculus, (in preparation,) 
 
 XIX. Robinson's Elementary Astronomy, - 
 
 XX. Robinson's University Astronomy, • 
 
 XXI. Robinson's Mathematical Operations, 
 
 XXII. Robinson's Key to Geometry and Trigonometry, 
 Sections and Analytical Geometry, 
 
 Conia 
 
 Entered, according to Act ot Congress, in the year 1862, by 
 
 DAKIEL W. FISH, A. M., 
 
 In the Clerk's Office of the District Court of the United States for the Korthern 
 District of New York. 
 
PREFACE. 
 
 This volume contains solutions of nearly all the examples 
 found in Robinson's New University Algebra. The greater 
 part of these solutions might have been omitted, if the sole 
 object of a Key were to aid teachers, of limited acquirements 
 and experience, in overcoming difficulties otherwise insur- 
 mountable by them. Though this may be one of the primary 
 objects of a Key, there is, nevertheless, a higher and far more 
 important use. 
 
 This higher object is purely educational, and is simply an 
 extension of the author's design in the solutions given in the 
 text-book. It is to illustrate more completely the principles 
 of the science, in their practical bearings ; and to show the 
 application of the rules and methods taught, under the 
 greatest possible variety of circumstances. Viewed in this 
 light, and used with discretion, a Key to an Algebra may be 
 serviceable to all students of the science, and especially to 
 teachers who, by reason of limited educational advantages, 
 find it necessary to apply themselves diligently to self-culture 
 in their profession. 
 
 These remarks apply more particularly to those examples 
 which are designed to tax the ingenuity, and to call forth 
 algebraic skill. In the case of examples requiring long and 
 
iv PREFACE. 
 
 tedious numerical calculations, a Key is often a great con- 
 venience in detecting mistakes, the correction of which would 
 require from the teacher, under the pressure of school-room 
 duties, a needless expenditure of time and labor. 
 
 We have given full solutions of the numerical equations 
 of higher degrees, found on the last page of the Algebra, — a 
 feature of the work which will be acceptable to all who use 
 the book. The attention of teachers is invited particularly 
 to the system of decimal contraction applied in these solu- 
 tions, and to the convenient method of marking the cor- 
 responding terms in the several columns. 
 
 September, 1862. 
 
■:■■ ■;^ 
 kSyto 
 
 ROBINSON'S 
 
 NEW UNIVEKSITY ALGEBRA. 
 
 USE OF THE PARENTHESIS. 
 (61, page 29.) 
 
 1. 3a + {2b^—a—d + m) = da + 2b^—a—d-\-m 
 
 = 2a4-2i' — d-\-m^ Ans. 
 
 2. 4a;'— y— (3a:— 7y + 5)4-2ar=:4j;'— y— 3a: + 7y— 5+2a: 
 
 = 4a;'4-6y— a;— 5, ^W5. 
 
 3. a + 2c — (4c — 3a + 2m') =a + 2c— 4c + 3a — 2m' 
 
 = 4a — 2c — 2m', ^n5. 
 
 4. 4i:'— 2a:' — [x»— (2a:' + 5a;— 7)— 6a: + l] = 4a:«— 2a:'— a:' + 
 (2a:' + 5a;— 7) +Qx—\ = 4a:'— 2a;'— a;'+ 2a:' + 5a; — 7 + 6a; — 1 = 
 
 3x* + \lx — Q^ Ans, 
 
 5. a + 2m— Jc+a;- [a— m— (c — 2a:)]j 
 = a4-2m — c— a; + [a— m— (c — 2a;)] 
 
 = a4-2m— c— aj + a— m— c4-2a; = 2a + m — 2c+a', Ans. 
 
 6. 3a;' — 4.1;— am— j.r'— a; — [3am — (2a; 4 2am) + 2a;'] — 5am j 
 = 3a:' — 4a;— «m— a:' + a: + [3am — (2a: + 2am) + 2i:'] + 5am 
 = 3a;'— 4a;— am— x' + a; + 3am — 2a;— 2om + 2a:' + 5am 
 
 = 4a;' — 5a: + 5am, Ans. 
 
 7. 3a— J2m» + [5c— 9a— (3a+m')] + 6a — (m' + 5c)J 
 = 3a — 2m' — [5c — 9a — (3a + m')] — 6a + m' + 5c. 
 
 = 3a— 2m' — 5c + 9a + 3a-4-m' — 6a + m' + 5c = 9a, Ans. 
 (29) 
 
FACTORING. 
 
 8. a;'— S5mc'— [j:'— (3c— 3r.ic') + 3c — (j-' — 2mc'— c)]j 
 = x^ — 5mc^-h[.c' — (3c-3mc')-^Sc — (.v' — 2mc^—c)] 
 = a;' — 07wc' + j;' — 3c + 3;?ic'' + 3c— x' + 2mc' + c 
 
 = x' + c, ^n*. 
 
 9. m' — m — l—\ m'' — 2m — 2 — [«i' — 3;?i — 3 — (m' — 4;/i — 4)] \ 
 = m^—m — l—m^ + 2m + 2 + [7?i'' — 3m — 3 — (m'' — 4m — 4)] 
 = m' — m — 1 — 7n' + 2 m + 2 + m' — 3 ;n — 3 - m' + 4 ??i + 4 
 
 = 2rn4-2, Ans, 
 
 10. 52»_3z' + 42-l-[22'-(32' — 22: + l)-0' + 2] 
 — 5z' — 3z^ + 42—1 — 22' + 32' — 22 + 1+2' — z 
 
 =: 42* +2, ^n*. 
 
 11. 4c'-2c' + c + l-(3c'-c'-c-T)-(c'-4c' + 2c + 8) 
 = 4c'-2c' + c + l — 3c' + c' + c + Y-c' + 4c'-2c-8 
 
 12. 3a'6 — 4ccf— (3c</— 2a'6) — [a» + c— (5crf + 3a'6)-f (3a' + 2crf) 
 4-a'] z=z Sa'b — 4cd — 3cd+ 2a'b—a^—c + (5cd -f 3a'6) — (3a'+ 2cd) 
 — a' = 3a'6— 4cJ— 3rflr + 2a'6 — a' — c + 5cc? + 3a'6 — 3a' — 2crf— a* 
 
 — 8a'6 — 4c(/— 5a'— c, Ans, 
 
 13. — (4a'77i + 3mV— (7w'c? — 9a'm— n)— J5n— [wV— (2n + 
 a'm) + San'] - 5a'm | — 1 2a'm) = _ 4a'm - 37?iW + (7m'cf — 
 9a'm — n) + {on — [in^d— {2n + a'w) + 3aw']— 5a'mf + 12a'm 
 r=: —4,a^m~ Sm'^d + T/nV — 9a^m — 7i+5n — [m''d—(2n + a'm) + 
 3a'n] — 5a'w + 12«'m = — 4a';w— 3m'(/+ Im'd— 9a^m—n + 5n^ 
 m*d + 2» + a'w — 3an' — 5a'w + 1 2a'wi 
 
 = 3wV + 6n— 5a'm — 3an', ^w*. 
 
 FACTORING. 
 (95, page 51.) 
 
 5. x*—x^y-\-xy*—y* = z\x—7/) + 7/\x—7/) = {x^-}-i/)(£—i/\ Ans, 
 
 6. a*b' + 2a'b'-{-a'b' = a'b\a^ -]-2ab + b') 
 
 = a''b\ai-b)(ai-b),An8, 
 ( 29-51 ) 
 
SUBSTITUTION. 
 
 7. {x*—x)a+ {z*-{-x)(db—c) — q = ax"— ax + 36j;'+ Zhx—cx*— 
 cx—q = ax*-{-3bx*—cx^—ax + 3hx — cx — q 
 
 — (a + 3b — c)x^ — (a — Sb + c)x—q, Ans, 
 
 8. a*m— 9am'=a7n(a* — 9w') = a7?i(a'-f 3m)(a*— 3m), Ans. 
 
 9. By (89, 4), 8a*-x* = {2a-x){-iu^ + 2ax-\-x'), Ans. 
 
 10. y' + 243 is the sum of the fifth powers of y, and 3 ; hence 
 
 by (89, 1), 
 
 / + 243=:(y + 3)(/-3y« + 9y'-2'7y + 8l), ^715. 
 
 11. By (70, III.), we have 
 
 z'-y- = (i:« + y')(.«:'-/). 
 And by (89, 1 and 4), 
 (x'-}-y')(x'-i/*)=.{x + y){x'-xi/ + y'){x~-y){x' + xi/+^'),Ans, 
 
 12. a' — ab^ + 2abc—a^ = a(a^ — b' + 2bc—c') 
 
 =:a\a^-{b-cy\ 
 Or by (70, III.), = a(a + 6-c)(a— 6 + c), Ans. 
 
 SUBSTITUTION. 
 (96, page 51.) 
 
 a*=(a-by =a'-2ab + b' 
 ab={a-b)b= ab-b' 
 
 b* = 6' 
 
 a' + ab + b^ =a^—ab-[-b% Ans. 
 
 2. a*=z (.r + 2)' =x' + 4a; + 4 
 — 2a=— 2(.c+2)= — 2ar— 4 
 
 1 = 1 
 
 a" — 2a + 1 =a:' + 2a; + 1, Ans. 
 
 3. y* = {x + Sy =a;* + 12x* + 54a:' + 108^ + 81 
 — 2y*= — 2(a: + 3)= -2a:*-18r'— 54a;-54 
 
 f = {x + sy = x' + 6x + d 
 
 -6 = -6 
 
 y*_2y« + y» — 6 =a;* + 10i;' + 37j;'' + 60.c + 30, vlw*. 
 (61) 
 
6 
 
 SUBSTITUTION. 
 
 x«=(»+r)' =«' + 2sr+r» 
 ax=(s-{-r)a= as-\-ar 
 b = h 
 
 or =a 
 
 a*b=a* xa =d 
 a^b*=a^ xd*=a 
 ab*=a X a'=a 
 b* =a 
 
 a* + a'b + a*b^ + ab' + b*z= 5a\ Ans, 
 
 ax^=(Tn — l){m-\-iy = m'+ m'— m—l 
 
 a'x=(7n — iy(m-\-l)=m'— iii^— m + \ 
 
 a'=(m — 1)' =m* — 3w' + 3w — 1 
 
 «*+/ =:2a* +12?? +26* 
 
 = 2(a* + 6a'6'+6*), ^n*. 
 
 8. When a + i+c=5, ar + a + 6 + c=ar+5, and a;— a — 6^c=a; — 
 (a + 6 + c)-=a;— 5. 
 
 (a;+a + 6+c)»=(a; + s)»=:a;* + 5^*s + 10xV + 10xV + 5ar«*+5» 
 (ar— a — 6 — c)'=:(a;— 5)'=x' — 5ar*5+10xV— 10xV + 5a-s*— «• 
 
 (jr + a+6+c)' + (ar— a — 6— c)'=2^' +20a:V +10a:«* 
 
 =2(a;' + 10a?V + 5a-«*), ^n5. 
 
 x*= (y_2)''=y'-6y' + 12y-8 
 
 — 7ar=-7(y-2) = — 7y + 14 
 
 6 = 6 
 
 ar* — 7a: + 6 r=:y' — 6y' + 5y, ^n5. 
 
 (52) 
 

 GREATEST CO 
 
 10. x'= (y+l)*=/ + 5/ + 1^5yTTOF + 5y + l 
 _2.c* = — 2(y + l)*= — 2/ — 8/— 12y' — 8y + 2 
 
 3.c»= 3(i/ + iy= 3/4-9/ + 9y + 3 
 
 8x= 8(y + l)= 8y + 8 
 
 -3 = -3 
 
 x*—2x*-^dx* — 1x^ + 8x—3 — 7/'' + Si/*-{-5f/% Ans. 
 
 11. Expand before substituting, and wo have 
 2{a—by{b-cy = 2a'b'-4.ab' + 2b*-Wbc + 8ab'c-4b'c + 2a'c'- 
 
 [4abc'-\-2b^c* 
 2{a-bY(c—ay = 2a'c^ — 4abc'-^2b\*^4a'c+8i''bc — 4ab'c + 2a* — 
 
 [4a'h + 2a^b^ 
 2{b—cy{c—ay=:2b'c' — 4bc' + 2c* — 4ab'c + 8a0c' — 4ac' + 2a''b' — 
 
 [4a'6r + 2aV 
 If we add the quantities thus obtained, all the terms containing 
 three letters each will disappear. Arranging the terms containing a 
 and b according to their powers ; also the terms containing b and c 
 according to their powers ; also the terms containing a and c ac- 
 cording to their powers, — observing to separate the terras 2a*, 26*, 
 and 2c*, into parts, we have three polynomials, as follows : 
 
 a*-4a'b + Qa'b'-4ab'-\-b*= {a-by=zx* ; 
 h*-4b*c + Qb'c* -4^c' + c*= (6-c)*=/; 
 
 c*-4c'a + 6cV — 4ca'+tt*=(c-«)*=2;*j 
 
 Sum =:a;*-j-y*+2*, Ans, 
 
 GREATEST COMMON DIVISOR. 
 (lOO, page 53.) 
 
 2a' be' = 2a'bc' 
 
 6ab'c' = Sx2ab'c'' 
 
 10a'6c' = 5x2a'6c' 
 
 Hence, 
 
 Hence, 
 
 2a6c', Ans, 
 
 6x'i/'z*=5xYz' 
 Gx'i/z* =3 X2ar»y2' 
 I2x^i/z^ =2x2 xSx^'ijz' 
 
 (52-63) 
 
 x^i/z^y Ana, 
 
10 
 
 4. 
 
 GREATEST COMMON DIVISOR. 
 x^-,/ = {x + y){.t-y) 
 
 x*-2xy + y^ = 
 
 Hence, 
 
 {^-y){^-y) 
 
 lie nee, 
 
 Hence, 
 
 V. 
 
 Hence, 
 
 x—y^ Ans, 
 
 a' m — h^m = m((i — h) (a + b) 
 2'ic^m — 2bc^7nz=z2c^ X 7n(a — b) 
 
 m(a— 6), Ans. 
 
 aV — 3aV + aV= a^x{x^ — 3x+\) 
 daxz^ — ax'^z* —az'=— az'lx' — 3x + 1 ) 
 
 a(x' — 3a;Xl), Arts, 
 
 lCa;'-l = (4i:4-l){4a:-l) (by 70, III.) 
 
 l-8a;+16x'= (4a^— l)(4x-l) " " II. 
 
 4^—1, Ans, 
 
 (105, page 60.) 
 
 FIRST OPERATION. 
 
 a:* — 2x'— 4i:'+ lU- eix'-S-r' + lYar-lO 
 
 a:*_8x'H-17j:' 
 
 10.r 
 
 .r + 6 
 
 G.r' — 21^'+ 2U— 6 
 
 Or' — 48j;' + 102j- — 60 
 
 27x'~ 81a: + 54, 1st remainder. 
 Dividing iliis remainder by 27, we have a;' — 3j;4-2 for the next 
 divisor. 
 
 SECOND OPERATION. 
 
 a:' — 8j:' + 17j;— 10 
 .r' — 3jr'+ 2x 
 
 j:'_3ar4.2 
 
 a;— 5 
 
 — 5a;' + 15ar— 10 
 
 — 5.r* + 15.r— 10 
 
 'WTience, a:' — 3a; +2, Ans. 
 
 2. No pteparation is necessary for the first operation, and as a; is 
 involved to the same power in both of the polynominals, it is imma- 
 terial which is taken as a divisor. 
 
 1st. 
 6x'+ a:' — 44ar + 21l6a-' — 26a:'+46a:— 42 
 
 6a;' — 26a;' + 46a;— 42 1 1 
 27a;' — 902; + 63 
 
 (53-60) 
 
GREATEST COMMON DIVISOR. 11 
 
 Dividing this remainder by 9, we have dx* — 10x-\-^ for the next 
 divisor. 
 
 2d. 
 
 6a;'— 26j;' + 46a;-42 
 6a;'— 20a:' + 14a; 
 
 3a:'-10a; + V 
 
 2a-— 2 
 
 — 6a;' + 32a;-42 
 
 — 6a;' 4- 20a;— 14 
 
 12a;-28 
 Dividing this last remainder by 4, wo have 3a;— 7 for the next 
 divisor. 
 
 3d. 
 
 3a;'— 10a; + 7 
 
 3a;- 7 
 
 3a;'- 1x 
 
 a;-l 
 
 - 3.C + 7 
 
 — 3a; + Va; 
 
 
 Hence, 3a:— 7, Ans» 
 
 3. The greater polynomial must be multiplied by 3a, to render its 
 first term divisible by the first term of the less polynomial? 
 
 1st. 
 3aa;' — ISa'a;' + 30a'a; — 9a*[i3aa;' — 14a'a; + 1 5a' 
 
 3ax' — 14a'a;' + 15a'a; [ r, —4a 
 
 — 4aV + 15a''a:-9a* 
 
 — 12a'a;' + 45a'a; — 27a*, new prepared dividend. 
 
 — 12a'x' + 56a*a;-60«* 
 
 — lla'a; + 33tt* 
 
 a;— 3a, next divisor. 
 
 In the above operation the first remainder is multiplied by 3, to 
 render its first term divisible by the first term of the divisor. The 
 last remainder is divided by — 11a' for the next divisor. 
 
 ad. 
 
 3aa;' — 14a'a; + 15tt' 
 3aa;'— 9a'a; 
 
 ar— 3a 
 
 3aa; — 5a' 
 
 — 6a'a;+15a' 
 
 — 5a'a;+15a' 
 
 Hence, a;— 3a, Ans. 
 (60) 
 
12 
 
 GREATEST COMMON DIVISOR. 
 
 1st 
 
 x*—8x' + l9x^ — i4 z \x 
 — 5x' + 30.r — 40 
 
 Mviding by —5, we 
 
 have X*- 
 
 -62*4-8 for the second divisor. 
 
 2d. 
 
 
 
 
 3d. 
 
 a:'-8x'4-19j; — 14 
 
 x'- 
 
 X- 
 
 -6^4 
 
 -2 
 
 -8 
 
 a;«_6x4-8 
 x^-2x 
 
 -X4-2 
 
 'a;'-6.c' + 8.r 
 
 -X4-4 
 
 -2x''4-ll-r-14 
 
 
 
 — 4x4-8 
 
 -2:t' + 12jr-16 
 
 
 
 
 -4x4-8 
 
 -x + 2 
 
 
 
 Hence, 
 
 — X4-2, or X— 2, Ans, 
 
 1st. 
 
 
 
 
 ad. 
 
 
 a'4-5a'4-5a4-l 
 a' 4-1 
 
 a'4-1 
 
 a" 4-1 
 a*-\-a' 
 
 a 4-1 
 
 a' — a4-l 
 
 5a' 4- 5a 
 Suppress (5a), and we have a 4-1 
 for the next divisor. 
 
 6. Multiply the first polynomial by 2. 
 
 1st. 
 4a*-- 10a*6- 6a'6' 4- 14a6* 4- 66* 
 4a* -^ 2a'6-4a'6'- Sab' 
 
 -a' 4-1 
 
 a + 1 
 
 a + l 
 
 Hence, a 4- 1, Ans, 
 
 4a*-2a'b-4ab'-3b* 
 
 a —2b 
 
 — 8a"6 - 2a'6'' + 1 lab' + 6b* 
 
 — 8a'64-4a'6'4- 8a6'4-6ft* 
 
 Suppress — 3a6% 
 and we have, 
 
 — 6a'6''4-9a6' 
 
 2a — 36 for the next divisor. 
 
 ad. 
 
 4a' — 2a'6 — 4a6« — 36»j2a— 36 
 4a' 
 
 6a'6 
 
 2a"4-2a64-6* 
 
 4a'6— 4a6' 
 
 4a'6-6a6' 
 
 2^ 
 
 2a6'. 
 
 36' 
 36' 
 
 Hence, 2a— 36, Ans, 
 
 (60) 
 
GREATEST COMMON DIVISOB, 13 
 
 7. Multiply the greater polynomial by 4, to render its first term 
 divisible by the first term of the other polynomial. 
 
 1st. 
 1 2^' — 1 6 j: V + 1 2.ry' - 8y' 
 12x' — 21arV+ 9.r/ 
 
 4.r' — 7ary + 3y' 
 
 Multiply by 4. bx'y + ^xy^— 8y' 
 
 20j:'y + 12a:y' — 32y' New prepared dividend. 
 20a;V— 35 a;y' + 15y' 
 47x/ — 47^' 
 Dividing by 47y', we have a:— y for a new divisor. 
 
 3d. 
 4a;' — 7ary + 3/ 
 4a;' — 4.ry 
 
 4a:-3y 
 
 — 3a:y + 3y' 
 — 3a;y + 3y' 
 
 4ar*— 2a;* + 4a;'— 27^'+ 4a;— 7 
 4a;'H-12a:* — 38.t'+ 8a;' — 10a; 
 
 Hence, x—y^ Ans, 
 
 2ir* + 6a;' — 19a;' + 4a;— 5 
 
 2a;- 7 
 
 — 14x*4-42a;'— 35a;' + 14a:- 7 
 
 — 14j* — 42a;'4-133a;' — 28a;+35 
 
 84a;'^— 168a;' + 42a;— 42. 
 
 Dividing the remainder by 42, we have 2a;'— 4a;'+a;— 1 for the 
 next divisor. 
 
 ad. 
 2a;* + 6a;' — 19a;' + 4a;— 5| 2a;'-4a;'+a;— 1 
 2.r*— 4a;*- t- x*—x | a;+5 
 10a;' — 20a;' + 6a; — 5 
 10a;' — 20a;' + 5a;— 5 
 
 Hence, 2a;'— 4a;'+a;— 1, Ans. 
 
 9. The first polynomial contains the monomial factor ac^ and the 
 second contains the monomial factor c'. Suppress these factors, and 
 set aside €, which is common to both, as one factor of the greatest 
 common divisor; then apply the process of division to the resulting 
 polynomials. 
 
 (60) 
 
14 
 
 GREATEST COMMON DIVISOR. 
 
 1st. 
 
 
 a* — ea*m-h5m* 
 
 Dividing tho remainder by 2wi, we have a*—m for the next 
 
 divisor. 
 
 ad. 
 
 6a'm + 5m' «' — m 
 
 a 
 
 o.'— a'm 
 
 6m 
 
 — 5a'm4-6m' 
 — 5a'm4-5m' 
 
 Hence, c(a* — m), Ans, 
 
 10. Multiply the greater polynomial by 2, to render its first term 
 divisible by the first term of the other polynomial. 
 
 1st. 
 2x*-' 8x'— 32a;' + 14j:-f 48 2^' — 15ar'4-0ar + 40 
 2x*— 15x'+ 9j;'4- 40.C 
 
 X, +7 
 
 7x'— 41.C'— 26x+ 48 
 14x»— 82a:'— 52x+ 96 
 14j:' — 105x'4- 63j' + 280 
 
 Kew prepared dividend. 
 
 23j;' — 115a;— 184 
 Divide by 23, and we have a;' — 5a;— 8 for the next divisor. 
 
 «d. 
 
 2a;' — 15r'+ 9.i; + 40 
 2a;' — lOa;' — 16a; 
 
 5a;— 8 
 
 2a;— 5 
 
 11. 
 
 — 5a;' + 2 5a; 4- 40 
 
 — 5a;' + 25a; + 40 Hence, a;' — 5a;— 8, ^n*. 
 
 1st 
 
 15a;* + 7la;*+ 60a;'— 66i3a;" — l7a;*-20a;' + 84 
 15a;" — 85a;* — 100a;' + 420 
 
 Suppress 4, 156a;* + 160a;" — 476 
 
 and we have 39a;* + 40a;' — 119 for the next divisor. 
 
 Multiply the first divisor by 13 to render division possible, and 
 proceed as follows : 
 
 (60) 
 
GREATEST COMMON DIVISOR. 
 
 15 
 
 ad. 
 
 Z9x' — 22lz*~- 200ar' + 1092 
 39a:''+ 40x*— 119^' 
 
 39a?* + 40ar'-119 
 
 x\ +29 
 
 Suppress —3, —261a;*— 14 It' +1092 
 
 and we have Slx*-{- 47a;'— 364 
 
 Multiplymgbyl3,wehave 1131a;* + 61 la;' — 4732 New prepared dividend. 
 1131a;* + 1160.g' — 3451 
 -549.t' — 1281 
 Suppress the factor —183, and we have 3x' + 7 for the next divisor. 
 
 3d. 
 
 39a;* + 40a;' — 119 
 39a;* + 9 la;' 
 
 3a;' + 7 
 
 I3a;'-17 
 
 — 51a;'— 119 
 
 — 51a;'-119 
 
 Whence, 3a;' + 7, Ans. 
 
 12. We will first find the greatest common divisor of the first 
 two polynomials, using the second as dividend, and the first as 
 divisor. 
 
 1st. 
 
 6a* — 14a'm'+ 4m\ 
 6a* + 28o'm' — 10m* 
 
 3rt* + 14a'»i'--6m* 
 
 — 42a'm' + 14m* 
 Suppressing— 14m', we have 3a' — m' for the next divisor. 
 
 2d. 
 
 3a* + 14a'm' — 57?^*■3a•• 
 3a*— a^rn* 
 
 m' 
 
 I a' + 5m' 
 
 + 15a'm' — 5m*- 
 ISa'm' — 5m* 
 
 Hence, 3a'— m' is the greatest common divisor of the first two poly- 
 nomials. We must now find the greatest common divisor of this 
 result and the third polynomial, which will be the greatest common 
 divisor required. 
 
 (60) 
 
16 GREATEST COMMON DIVISOR. 
 
 3a*— a^m* 
 
 3a' 
 
 7m» 
 
 -21a'/w' + Vm* 
 
 Whence, 3a' — m', ^na. 
 
 13. The second polynominal contains the niononiinal factor bi/. 
 Suppressing this factor, we use the resulting polynominal as the first 
 divisor. 
 
 1st. 
 
 2aV— 2tt'teV+ oh'xy*- 6'/ 
 2a V — Aa^hx'y + 2a6'a-y' 
 
 a^x* — 2ahxi/-{-b^y* 
 
 2ffa?-f 2Ay 
 
 2aV,x^y — 4ah\ry^ + 2bY 
 3<'b''xy^-3h'y' 
 Suppressing 36'y' we have ax— by for the next divisor. 
 
 2d. 
 a^z* — 2abxy + h^y^ax — by 
 
 a'x' — abxy lax — by 
 
 — abxy + b^y* 
 
 — abxy + b^y' 
 
 Whence, ax— by, Ans, 
 
 1st. 
 14. 9a* 4- 12a' + 10a' + 4a + l;3a* + 8a» + 14a'4-8a4-3 
 
 9a* + 24a' + 42a' -f 24a + 9 ,3 
 
 Suppressing —4, —12a' — 32a' — 20a — 8 
 
 we have 3a' + 8a' + 5a + 2 for the next divisor. 
 
 ad. 
 3a* + 8a' + 14a' + 8a+3|3a' + 8a' + 5a + 2 
 
 3a* + 8a'+ 5a' + 2a 
 
 Suppressing 3, 9a' + 6a + 3 
 
 and we have 3a' + 2a + 1 for the next divisor. 
 
 (60) 
 
Whence, 
 
 LEAST COMMON MULTIPLE. 
 
 3d. 
 
 3a' + 8a' + 5a + 2i3a' + 2a + l 
 3a' + 2tt'+ a a + 2 
 
 17 
 
 6a' + 4a + 2 
 6a'4-4a + 2 
 
 3a'4-2a + l, Ans, 
 
 2. 
 
 Hence, 
 
 3. 
 
 Hence, 
 
 Hence, 
 
 LEAST COMMON MULTIPLE. 
 
 (109, Page 62) 
 2a*hc — 2a'bc 
 
 I0ab'd=2x5ub'd 
 l5abcd = S x 5abcd 
 
 2 X 3 X 5 X a*6Vc?=30a*6V(/, Ans. 
 
 32r'y=3x'y 
 15xf/^ = 3 X 5 xry' 
 lOxyz' z^ 2 X 5 X zyz* 
 5x'y^zz=z5x'i/^z 
 
 2 X 3 X 5xYz^=^0xYz\ Ans, 
 
 x* + xr/=x{x + j/) 
 
 {x + y)(x—y)xy—x^y—xy\Ans. 
 
 x^—a* = {x' + a'')(x-\-a)(x—a) 
 x'*—a^= (x-^-a) [x—a) 
 
 x' + a' = {x' + a') 
 2a'ar' + rt*= {x + aY{x-ay 
 
 Hence, 
 
 (x' + «') (x-\-ay{x-ay=x'-a'x*--a*x* + a*, ^n«. 
 
 6. x*-x= a:(.r + l)(:c-l) 
 
 :i:'-l=(^»+ar+l)(.r-l) 
 ar'4.1=(a;'-ar4-l)(ar+l) 
 
 Hence, x{x'+x + l)(x'-x+l)(x-^l)(x-l)=zx(x'—l)=x'^x, 
 
 Ans. 
 (62) 
 
18 LEAST COMMON MULTIPLE. 
 
 X^ + 2X+1=: (^+1)' 
 
 a:' — 2jr + l= (^-1)' 
 
 x + l= {x+1) 
 
 ar-l= (jr— 1) 
 
 Hence, (x' + iy{x -f \y{x-iy=x'-2x* + 1, ^»5. 
 
 8. 4x» + 2a'=2.r(2x' + l) 
 6j:'-4jr=2r(3a:-2) 
 6j:' + 4jr=:2.r(32: +2) 
 
 Hence, 2x{2x'-{-\)(3x — 2)(3x-\-2) = 36x^ + 2x' — 8x, Ana, 
 
 9. ar' — 4a' = (.c + 2ff)(j: — 2«) 
 (ar + 2a)' = (x+2a)' 
 (x-2ay= {x-2ay 
 
 Hence, (j: + 2a)'(j:— 2a)'=(i:' — 4a')*=x'— 12j:V-+-48xV — 64a', 
 
 10. a*-b*=^{a' + P)(a + b)(a-b) 
 
 a'-b'= la-b){a' + ab + b') 
 
 a'-b'z= (a + i)(a-6) 
 
 a-b= (a-b) 
 
 Hence, (a' + 6')(a + 6)(a-6Xa' + a6 + 6')=o'+a'6+a*6'— a'6*-a6''— 6', 
 
 Ans. 
 
 (110, page 63.) 
 
 1. The greatest common divisor of the two polynomials, ( 10«5), 
 is a:' — 2.C + 2 ; 
 
 (x* — 5x* + 8x—G)-i-(x' — 2x + 2)=x—3 ; hence, 
 lx-3){x'+x' — ix + 6)=x'-^2z* — W + 18x—l8, Ans. 
 
 2. The greatest common divisor of the given polynomials, (105), 
 is ar — 5 ; 
 
 (x'-l2x-{-35)^(x-5)=x-1; hence, 
 (x-1){x' — 2x^ — }dx + 20)=x* — 9x' — 5x^ + l53x—l40, Ans. 
 
 3. The greatest common divisor of the given polynomials, (I05), 
 is 2am' — 1 ; 
 
 (63) 
 
LEAST COMMON MULTIPLE. 19 
 
 (2a'wi* + 3aw' — 2)-7-(2ow'— l)=am* + 2 ; hence, 
 
 {am^ + 2){6a^m*—am'' — l) = Qa'm^-\-Ua'm* — Sam* — 2, Ans, 
 
 4. The greatest common divisor of the given polynomials is 
 
 {x' — 5x* + 1x — 2)^{x' — 3x-hl)=x — 2; hence, 
 {x-2){2x'-5x'—x-hl)z^2x*-9x' + 9x'i-3x-2jAns. 
 
 5. Tlie greatest common divisor of the given polynomials ie 
 3j:'-5; 
 
 (3x' + ex' — 5x— 10)-^ (3^' — 5) =a: + 2 ; hence, 
 
 lx + 2){6x* — 4x'-l0) = Qx* + 12x* — 4x* — 8x' — l0x—20, Ans. 
 
 6. The greatest common divisor of first two polynomials is a: 4- 2 ; 
 
 (^x'-2x-8)-^(x + 2) = (x-4)', 
 hence, {z—4)(x* + 7x + lO)=zx*-\-Sx* — l8x-40, the least common 
 multiple of the first two polynomials. We now proceed to find the 
 least common multiple of this result, and the third polynomial. 
 The greatest common divisor, <fec., is a;'-f ar— 20; 
 (.r'-fa;— 20)-^(J:'^-x — 20) = 1 ; hence, 
 
 {x* + dx* — l8x—40)x\=x* + 3x* — 18x—40y Ans, 
 
 *l. Tlie greatest common divisor of the first two polynomials is 
 a— 26; and 
 
 {a} — ah — 2h'')-^{a — 2h)=a-\-h\ hence, 
 
 {a-[-h){a*-3ab + 2h') = a'-2a'b-ab\+2h\ 
 which is the least common multiple of the first two polynomials. 
 
 The greatest common divisor of this result and the third poly- 
 nomial is a' — 6'; hence, 
 
 a»_2a'6— a6' + 26', Ans. 
 
 8. The greatest common divisor of the first two polynomials is 
 2.T— y, and (22:'— 5xy-|-2y')-7-{2.r--y)=a;— 2y. Hence, we have 
 
 (x-2y){2x''-1xyJc 3y')= 2x*— lla;'y+ I7.ry'— 6y', 
 which is the least common multiple of the first two polynomials. 
 
 The greatest common divisor of this result and the third poly- 
 nomial is x^—bxy + Qy'* ; hence, 
 
 2x*—\\x'y-ir\1xy^ -6y\ Ans, 
 (63) 
 
20 FRACTIONS, 
 
 FRACTIONS. 
 
 REDUCTION. 
 
 (1245 page 67.) 
 
 nx\jz _ {7xyz)x* _ X* 
 ^- 2\xy^z~{1xyz)Zy'-^y''^\'' 
 
 xy-\-y (^ + i)y y 
 
 z>-6V _ x\x^- b^) _ X* 
 
 2x'— 16j;-6_2(x' — Sir— 3)_2 
 ^* 3x*-24j:-9~3(j:'-8j:-3)~3' 
 
 8. The greatest common divisor of the numerator and denomina- 
 tor, found by (105), is 2x— 3 ; hence, 
 
 (2x' — 1x^+l4x—l2)-i-(2x—3)=zx^ — 2x + 4: 
 l4tx'—4f — l3x-{-l5)-^(2x—3) = 2x^-\-x—5 
 
 x* 2a; + 4 
 
 And we have for the reduced fraction, —^ , Ans, 
 
 2x'-\-x—5 
 
 a*c-\-2abc-}-b*c _ c{a -^ b){a -h b) _ c 
 - ^V a'+3a'6 + 3a6' + 6»~ (^+^(^6)(a + b)~^b ' * 
 
 a' — 3a'jr + 3ax'— a;' _ («— ar)'_ (a— a:)(a— ar)'_a' — 2ax + ic' 
 a'— a:' ~ a^ — x* (a—x)(a-j-x) ~ a + x * 
 
 Ans. 
 
 11. The greatest common divisor of the numerator and denomi- 
 nator is 2a^ + 3x ; hence, 
 
 (6a' + 1ax—3x^)-^(2a-^3x) = 3a—x 
 (Qa^ + Uax-\-3x^)^(2a + 3x) = 3a+x 
 
 Qa — X 
 And we have for the reduced fraction, , Ans. 
 
 3a +x 
 
 (67) 
 
KEDUCTION. 21^ 
 
 12. The greatest common divisor of the numerator and denomi- 
 nator is x'—x'—x-^-l ; hence, 
 
 ^x^^x*—x ^l)-ir{x^—x^—x+l)=x^ + l 
 
 (^3;*—X^ — X^+x)^{x^ — X^'-X-^l)=zX 
 
 x^ + 1 
 And we have for the reduced fraction, , Ans, 
 
 x 
 
 13 (•^ + yr-2:--/ _ (ar + y)>-(a:' + /) 
 • (x + yy-x'-f {x + yy-{x' + f) 
 
 J^x + y){{x-^yY-{x*-x'y + x\/--x/-iry')\ 
 
 (x^y)[{x + yy-(x'-xy + f)] 
 __5x'y-h5x^y'-^5xy* 
 ~ diy 
 
 14. Removing the parentheses by multiplication and involution, 
 "we have 
 ( Sx^ — l)(2x^ — l)—x\5x^ — 1) __ Qx* — 5x^-hl — 5x* + 1x* __ 
 (3a:'-l)' + (a:'— 3a:)» ~ 9a:* — 6ar' + \+x'—Qx' + 9a;* ~" 
 
 _^Nh2^+l (a:'-f 1)' _ 1 . 
 
 a:«4-32:* + 32:» + l ~ (a^' + l)' "" a:» + l* 
 
 ( 125, page 68.) 
 
 , 2jr* — 2y* 2(ar— y)(x' + a:yH-y') , , ,^ , 
 
 4. -^37^= ^ ^;^_y ^^^^ = 2(a:' + ary + y')>^^> 
 
 ^ 24a;»— 18a?— 6 „ ^ 2ar 6 
 
 6. =3jr— 2 ; 
 
 8a; 8 8 
 
 « 2a; + 6 
 
 = 3a;— 2 — 
 
 8 
 
 = 3a;— 2 , Ans, 
 
 4 
 
 ^ 66a;' + 126a;-140 „ ^ —14 
 8. ^ . ^, =8a;— 6 + 
 
 7a;+21 7a; + 21 
 
 2 
 
 = 8a;— 6 , Ant, 
 
 a; + 3' 
 
 (67-68) 
 
22 
 
 FRACTIONS. 
 
 10. '4i:i^=..+v+^4^ 
 
 X —y a: — y 
 
 =;.*+^/+__ y'(— y) 
 
 ( 126, page 69.) 
 
 1. l+a+- = , Alls. 
 
 2. 26 j—=z 1_, ^n*. 
 
 a6 + ar bah + ah-\-x Qah+x ^ 
 
 3. 5a H — = = — ^n«. 
 
 6 6 ' 
 
 ^ ,^ 3a 4- 6 126 + 3a + 6 136 + 3a , 
 
 4. 12+—^— = = ^Ans. 
 
 ^ ^ 2j— 5 \bx—{2x—b) 13a- + 5 . 
 
 6. 3a-9^^^1^^^ ^^^'-^^- (^"'~^")=-A- An, 
 a + 3 a + 3 a + S* 
 
 ^ . y' ar' — y' + y' a:' 
 
 ar— y a^— y ^— y 
 
 8 ,__ ^'-4a:*4-8 _ a:'-3x' + 4-(a;«-4a;' + 8) a;'~4 
 
 • ^+ (a;-2)' - (ar~2)' ~ (^^" " 
 
 ar + 2 
 a:— 2* 
 
 o . . I. . M «' + ^' a'-6»-(a' + 6') 26' , 
 
 9. a'+a6 + 6' -= ^ ^ — ~,Ans. 
 
 a—b a — b a — b 
 
 T y-r y 1- y -r j__^, ^__^, (l + y)(i-y) 
 
 -, Ans, 
 
 1-y 
 (69-70) 
 
ADDITION. 23 
 
 11 (, ly (^-l)'^ 4^-i)'-('^-i)- ^ (^-i)(^-i)' 
 
 ' ^ ^. X X X 
 
 (x-iy 
 
 12. a:» + 5xy+y* + - 
 
 , Ans, 
 
 X 
 
 21a;y ar* + 2xy-25^y4-/ + 21a:y 
 
 
 i , ^n*. 
 
 a;' — 5ary + y' x* — bxrj-\-y 
 
 ADDITION. 
 (IJIJO, page 75.) 
 , Zx 2x x 63a;4-30j-4-35.r 128a; , 
 
 ^- T+y+3= Wo =ToF'^"'- 
 
 a a-\-b ac + ah + h* , 
 6 c he 
 
 ^ a* a* + x* a' + a'ar + 3a'+ 3a;' , 
 
 3. —H = — r , Ans, 
 
 3 a + ar 3(a + ar) ' 
 
 g + 6 a — h a^ + 2ah + }}''-{- a" — 2nh ■]-})* la* + 2h* 
 
 6. The sum of the entire quantities is 6a. 
 
 q + 3 2a— 5 _ 4a + 12 + 10a— 25 _ 14a — 13 
 
 5 "^ 4 ~ 20 ~" 20 ' 
 
 „ ^ 14a-13 . 
 Hence, 6a h -^ , Ans, 
 
 x—2 2ar— 3_5ar' — 10a; + 6ar--9_5a;' — 4a;— 9 
 3 5x ""~ 15^ "" 15^ ' 
 
 „ ^ 6a;'-4a;-9 , 
 
 Hence, 9j; + — , Ans, 
 
 15.C 
 
 ^ a 2c c _a*— ac + 2ac + 2c'4-«c— c' 
 
 a+c a— c a+c 
 
 
 ( 70-75 ) 
 
24 FRACTIONS. 
 
 ^- — ^3 + K. >..»- + - 
 
 52:' 52V 10/ 
 
 2xy — 6y*4-6j;* + 6y* + a;y~-6jr* 2x^y^ ■\- x'^y* ly + x 
 
 9. "NVe find, by inspection, that tlie least common multiple of the 
 given denominators is {b—c){c—a){a — h). We must, therefore, 
 multiply the numerator and denominator of the first fraction by 
 (a — ft), of the second by (h—c)^ and of the third by {c—a)^ to re- 
 duce the fractions to their least common denominator. Thus, 
 
 a + 6 a'-6' 
 
 {b-c){c-a)~ (h-c){c—a){a — b) ' 
 
 6 + c h'-c" 
 
 {c-a){a — b)~{b—c){c—a){a—b) ' 
 
 c-\-a c^—a* 
 
 (a-b){b-c)~ {h-c)(c-a){a-by 
 Taking the sum of the numerators, we have 
 
 a^-b' + b'-c' + c'-a* 
 
 (b-c)(c-a)(a-b) -(6_f)(c-a)(a-6) 
 
 0, Ans, 
 
 10. By (109), we find that the least common multiple of the 
 given denominators is (a — i)(a — 1)(6 + 1). Hence the terms of the 
 first fraction must be multiplied by 6 + 1, of the second by — (a— 1), 
 and of the third by —(a— 6). Thus, 
 
 g»_6 _ a^b-b'-\-a'-^b , 
 (a_6)(a-l)~(a-6)(a-l)(6 + l) * 
 
 ft' + a — 06' — a'4-6' + a 
 
 (64-i)(6^=(a-6)(a-l)(6 + l)' 
 
 (l-a)(l + 6)^(a-6)(a-l)(6 + l)' 
 Taking the sum of the numerators, we have 
 a'b — b^-ha'—b-ab'-a' + b^+a—a-a^b 
 
 = 0, 
 
 (a-6)(a-l)(6+l) -(a_6)(a-l)(6 + l) 
 
 (75) 
 
ADDITION. 25 
 
 11. The least common denominator is {a—b){a—c)[b—c). 
 
 be b'c-bc' ^ 
 
 {a-b)(a-^~(a — b){a-c){b^cy 
 
 ac ac^ — aV 
 
 (6_c)(6-a)'^(a-/>)(a-^)^6^)' 
 
 ab a^b — ab* 
 
 {c—a){c—b) ~ (a — l)j{a-c)(b—cy 
 
 ^ . . a*b-ab* + b'c-bc' + ac^-a'c 
 
 Their sum is -^ iT-,-n r^, — i r = 1» ^^^^' 
 
 ao—ab' + b^c — bc'-\-ac—a^c 
 
 12. By (llO), we find the least common multiple of the given 
 denominators to be x* — Qx^ + 11-^—0. Dividing this by the denomi- 
 nators of the given fractions, the respective quotients are (a; — 3), 
 (ir— 2), and (r— 1). Multiplying the terms of the first fraction by 
 (ir — 3), of the second by (-^—2), and of the third by (x—l)^ we 
 have 
 
 x—3 a-'— 6.C + 9 
 
 x^ — 3x-i-2~x' — Qx* + Ux — ^ ' 
 
 T—2 a:' — 4^ + 4 
 
 a:^—4x-{-3~x*—6x*-i-llx-G ' 
 
 x—l a:' — 2a; + l 
 
 x^^5x+6~x' — Qx^-{-llx^Q' 
 
 Taking the sum of the numerators, we have 
 
 <r»_6.r4-9 + a?'~42r-f4 + ar* — 2a- + l 3a-'— 12a; + 14 
 
 , Ans. 
 
 x'—^x^ + llx—G x'-6x''-\-llx-6 
 
 13. The least common multiple of the given denominators is found 
 to be x' + 6x^ 4- 1 la; + 6. Hence, multiply the ternis of the first frac- 
 tion by (x* 4- 5ar + 6), of the second by (a: + 3), and of the third by 1. 
 
 z _ x^ + bx^-^-Gx 
 ar+l "^H-ear' + lLr+G ' 
 
 X* x" + 3x' 
 
 a:' + 3.r + 2 a:' + 6a:' + lla;+6' 
 a;' — 2ar' — 3a: 
 
 «' + 6a:' + llx-f 6 a:' + 6a;'^+ lla: + 6 
 (75) 
 
S6 FRACTIONS. 
 
 Taking the sum of the numerators, we have 
 
 X* -f 52-' -\-6x + x^-\- 3x^ + x* — 2x^—Sx_ Sx^ -\- Qx* + Sx 
 
 SUBTRACTION. 
 
 ( 130, page 76.) 
 
 Sx 2x_2lx—l4x _l2x 
 ^' y "T- 63 -"6F ' ^'''' 
 
 ^ 1x 2ar— 1 21ar— (43-— 2) lYa: + 2 ' 
 
 x—y x + y X —y x—y 
 
 4. The difference of the entire quantities is a, and the difference 
 of the fractions is 
 
 11a — 10 3a— 5_77a — YO — (45a — 75)_32a4-5 
 \h '1~~' 105 ~ 105 
 
 XT . 32a4-5 . 
 
 Hence, a H — , Ans, 
 
 a-\-b a—b_a* + 2ab-{-b'' — (a^ — 2ab + b^)_ 4ab 
 
 6. We first take the difference of the fractions. 
 
 x—y ^•\-y x^ — 2xy-\-y'^ {x^ ■\-2xy ■\-y^^ _ — 4.ry _ — 4y 
 ^■\-xy~x^—xy~'~x{J^^^^ x{x?-y'') ~ x{x^—y^~ x^-y*' 
 
 Hence, x-j^-^, Ans, 
 
SUBTRAC 
 
 V. The difference of the entire quantities 
 of the fractions is 
 
 X ( X — a\ X X — a cx-{-hx — ah 
 
 (x—a\ X 
 
 b \ c I b c be ' 
 
 Hence, ^X"^"" — ^^- <, Ans, 
 
 8. By ( 1 05 ), we find the greatest common divisor of the de- 
 nominators to be (2:r— 3). 
 
 (2ar' — ll.r+12)-^(2j:-3)r=:r-4; (2a;' + 5ar-12)^(2a;-3)=2: + 4. 
 
 Hence, the least common denominator of the fractions is 
 (2a;— 3)(r— 4)(.r + 4) ; and we have 
 
 x*-{-x— 5 _ x' 4-^-5 _ ir' + 5ar'— ar— 20 
 2j:''-llj; + r2~(2j:-3)(.c-4)~(2i:-3)(a;-4)(a: + 4')'* 
 
 a r' + ar-^l _ ar'+ar- l _ a;' - 3i:'^-5^-|-4_ 
 2x»+5^-12""(2jr-3)(a' + 4)"~(2i:-3)(a:-4)(ar + 4')* 
 
 Taking the diflference of the numerators, we have 
 
 x*+bx* — x-20 — (x*-.'^x'*- br-\-A) _ 83:'4-4a;-24 
 
 (2z-3)(x-4)(2: + 4) ~{2x-^)(x-^){x-\-A) 
 
 _ (23;— 3)(4.r + 8) _ 4a; + 8 
 ■"(2x-3)(a:'-16) ~ ^-16 ' • 
 
 9. We may factor the denominators of the given fractions by 
 inspection. Thus, 
 
 a' + 3aft-f 2i'=:(a+ h){a-{-2h)', 
 a' + 5a& + 66' = (a + 26)(a + 3&). 
 
 Hence {a-\-h)(a-{-2b)(a-{-^h) is the least common denominator of 
 the fractions ; and we have - 
 
 3a + 6 3a'+10a6+36' 
 
 wy 
 
 Hence, 
 
 a* + 3a6 + 26' (a + J)(aH-26)(a + 36) ' 
 cr + 76 o' 4-8^6 + 76' 
 
 a' + 5a6 + 66' (a + 6)(a + 26)(a + 36) 
 (76) 
 
i58 FRACTIONS. 
 
 3a'-f 10afe4■36'-(a' + 8(/^>4-T6') _ 2a* -\- 2ab - 4h* 
 
 (a-f 6)(a + 26)(a + 36) ~(a + 6)(a + 26)(a + 36) 
 
 _ 2(a + 26)(a-6) 
 ~" (a + 6)(a + 26)(a + 36) 
 
 2(a-fe) . 
 
 10. The denominators may be transformed as follows : 
 
 1ab{a-b)-2(a*-b') = {a-b)[1ab-2{a'-\-ab-\-b')] = 
 
 (a-6)(5a6-2a'-26'); 
 
 3ab{a + h)-2(a' + b') = (a + b)[3ab-2{a*-ab + b')]=: 
 
 {a + b)(5ab-2a*-2b'). 
 
 Hence the least common denominator is (a' — 6')(5«6 — 2a' — 26') ; 
 and the terms of the first fraction must be multiplied by (« + 6), and 
 the terms of the second fraction by (a — b). We shall have 
 
 4q — 36 4a* + a6-36' ^ 
 
 'ra6(a-6)-2(tf*T6^ ~ (a'-6')(5a6-2a'-26') ' 
 
 8a — b 8a'-9a64-6' 
 
 3a6(a4-6)-2(a'+6') ~ (a'-6')(5a6-2a»-26') ' 
 And the difference is 
 
 4a' + o6-36' — (8a'-9a6 + 6») iOa6-4a'-46' 2 
 
 (a'-6')(5a6-2a'-26') -(a«-6')(5a6-2a'-26')~"a'-6'* 
 
 Ans, 
 
 MULTIPLICATION. 
 
 (132, page 77.) 
 
 1. Multiply J- by -. 
 
 Canceling the common factor 6, we have - , Aits, 
 
 a-\-x 5a a-{-x 5« ^ a 
 
 ' "SF^ ^^^"^0^"^ 3(7+^) "^18' 
 (76-77) 
 
MULTIPLICATION. 29^ 
 
 2a 5z bx ' 
 
 4/ 15y-30 2yx2?/ 3(5y— 10) ^ . 
 5y--10 2y 5y-10 2y ^* 
 
 * a + b^ab-h* (M^6) ^6(a-6)~ 6 * 
 
 Ans, 
 
 1. elfz:^^ 6a ^^(« + ^)(^-^)^ 6^ -3(a + ;r),^n.. 
 a 2aa:— 2aj' a 2x(a— a;) ^ ^' 
 
 . ar aft + ar , y ob—y 
 
 aft + a; aft — y a^h'*-\-ahr. — nhxi — xu . 
 
 __ X -^=. j^-—- , .!»,. 
 
 3a;*— 5a; 7a _a-(3r— 5) 7a _ ffl(3a;— 5) 
 
 14 ^2a;»-3a;~ 7x2 ^ ar(2.r»-3)~2(2a;'-3) 
 
 Sax— 5a . 
 -r—,Ans. 
 
 4a:' -6 
 
 ,« X —y X a (x + y)(x—7/) x a 
 
 10. ^x X ^)^-LlJS — Jlx X =a,An8. 
 
 x x + y x—y X x + y x—y 
 
 ,, 4a«-166' b'l 4(a4-2i)(a — 26) 
 
 11. ri— X— - — — — — — =-^ — . — ^^— ^X 
 
 a — 26 8'*' + 32a6 4- 326' (a — 26) 
 
 56 56 _ o6 
 
 2 x4(a4-26)(a + 26)"~2(a + 26)~2a + 46 
 
 , Ans, 
 
 2a a + 6 1 2(a + 6) ' 
 
 ax a' ^^ a' — x* a*—b^ a' 
 
 13. a H — ■. Hence, r-x -x =: 
 
 a—x a — x a-\-b ax + x a-^x 
 
 (a+ar)(a-a:) ^ (a + 6)(a-6) ^ «' ^ «'(«-^) ^^^ 
 + 6 a"(a + a') a—x x ' 
 
 (78) 
 
SOfc FRACTIONS. 
 
 a*-z* ^ a-\-b a-b ^ {a'-^x'){a-{-x){a-x) a + b 
 
 ^*- a'^b' a'-hx' a-z (a + b)(a-b) a'+a;''' 
 
 ^ a—b . 
 
 =:a-^x. Ans, 
 
 a—x 
 
 15. ^'-^V/'+^^ ^' ^ (•^+^)(»^-^) ^ ^'+^V ^ ^^ - 
 
 6c ft+c ir— 6 be b-\-c x—b 
 
 16. _J^fczfL_x--^ti±^x^^' 
 a' + 2ac + c' a' — 2ac + c' ac'x 
 
 ^ ^(«-^) ^ ^(« + ^) x ^'"^'^^''~''^ =^ Arts. 
 (a + c)(a4-f) (a — r)(a — c) acw «a;' 
 
 ^ (a + 6— c)(rt — & + c) C4-&— rt 
 
 17. -5^ ^r ^X;^ ; .-r. — 
 
 a — b—c {c — b—a)(b—c—a) 
 
 {a-{-b—c)(a—b+c) c-\-b—a 
 
 ■ (_i)(c + 6-a) ^(-l)(a + 6-c) x (-l)(a-6+c) 
 
 (1)(1)(1) _1_ 1,^,, 
 
 (-1)(-1)(-1) -1 
 
 DIVISION. 
 (133, page 80.) 
 
 I5ab lOac _l5ab {a-\-x){a^x) _Sh{ a-^x) 
 a—x a^—x a—x \Oac 2c 
 
 2j:' — 7 . a' _ 2a;' — Y (x + a){x + a) _ 
 
 x-\-a ' x^ + 2ax + a^~ x + a a' ~ 
 
 . ^'-^* . ar + 6_ (:r' + ft')(.r + 6)(a:-6) ^-6_ 
 • ;e*-26a;+6'' • a:-6~ {^x-b)(x-b) x + b '^' 
 
 Ans, 
 
 (78-80) 
 
DIVISION. :31 
 
 Q :; zzz ^ — — ^ X =^ '■ 
 
 a^—x^ ' a—x (a—x){a^ + ax+x'') x a!^ + ax-\-x^* 
 
 Ans, 
 14.r— 3 IO2:— 4_14i;— 3 5x5 __70jr— 15 
 ^- 5~""^'^5~~~~5~"^ 10^-4"" lOx-4 » *• 
 
 9a;' — 3j; a:' ar(9jr— 3) 5 _9a:— 3 
 5 5 5 x^ X 
 
 Qx—1 x—l Qx — 1 3 18.C-21 , 
 
 I x-\-x* 2ax+2ax^_x + x* 1 _ "^ j 
 
 ^^- "3^"^ 7 -~J^'^2a{x+x^)-Q^^' ^''*- 
 
 a'— a:* a' + ar + a:' a' +a;')(a — a:)(a' + aa; + a;') 
 
 a^ — 2ax-^x^ ' a—x {a—x){a — x) 
 
 a °~ , a =a' + a;', ^w?. 
 
 5 5 5 yxy y 
 
 na—nx ma — mx n(a — x) a + b n 
 
 13. — r — i — =:-^^^ — r-^X— 7 -=~,Ans, 
 
 f a-\-b a-i-o a-\-o m{a — x) m 
 
 X a a x-\-y x—y a:'—?/ 
 
 14. a-^ X =- X — - X ^-= ^ , Ans, 
 
 x+y x—y I x a x 
 
 3(a;'-l) , /a; + l\/g-l\_ 3(a; + l)(-r-l) 2a(a + 6 _ 
 
 2(a + 6) • V 2a /U + 6/ 2(a + 6) ^ (ar + l)(a;-l)~" 
 
 3«, u4w*. 
 
 ■^ • 10a6 — 3a' — 36'"^V6^3a/W"~10a6— 3a' — 36'^6(3tt + 6)~' 
 (3tf + 6)(a + 3^») a(6-3a)_3a?>4-a' 
 (6-3a)(a^^ ^"6(3a + 6)~a6-36" '^** ' 
 
 ^ a' 1 a' 4- a:* a 1 1 a^—ax + x"^ . 
 
 17. — , + -= T-; -r— +-= , ; hence, 
 
 x'^ a ax^ ^ x^ X a ax^ 
 
 a^-\-x* a^—ax + x^ {a-{-x){a* — ax+ x* ax* a + x 
 
 ax* "^ ^' = {ax')(x) ^a'-aa;Hh^'~ x ' 
 
 (80) 
 
a FRACTIONS. 
 
 a — 1 6 — 1 c— 1 1 1 1 
 
 18. +-r-+ 1 = 1 + l-r + l 1=2- 
 
 a h c a b c 
 
 REDUCTION OF COMPLEX FORMS. 
 ( 135, page 81.) 
 
 2. Multiplying both numerator and denominator by 6c, we have 
 
 6 
 a + - 
 
 c _ahc-\-b^ 
 
 c ahc + c" 
 
 3. Multiplying both numerator and denominator by a'6V, wo 
 have 
 
 ^+^ 
 
 6c' a'c_a*6-f6*c 
 
 6'c ac' 
 
 4. Multiplying both numerator and denominator by mn, we have 
 
 x—\__x-\-\ 
 
 m n __nx— n— mar— m_ar(n— m) — (n + wi) 
 
 arH-1 a:— 1 nz+n-^mx—m ar(n + m) + (n— w) ' 
 
 Til n 
 
 6. Multiplying both numerator and denominator by a6, we have 
 
 ^±1-2 + ^^ 
 
 _b a _ a« + a — 2a6+6'—6 _(a— 6)* + (a — 6) _a— 6 + 1 
 
 g-l 6 + l ~a'— a-2a64-6' + 6~(a — 6)'-(a— 6y~a— 6-1 * 
 
 . 6 a Ans, 
 
 6. Multiplying both numerator and denominator by a6c, we have 
 
 ^+A+4 
 
 .3 . A« I ^a 
 
 ah ac be a^6'' + aV+6V 
 c 6 a 
 
 r:, » >4"«- 
 
 (80-82) 
 
REDUCTION OF COMPLEX FORMS. 33 
 
 7. Multiplying both numerator and denominator by (c4-rf)(c— c?), 
 or its equal c^—cT, we have 
 a+b a—b 
 
 c-\-d c—d ac-\-bc—ad—bd-\-ac—bc-\-ad — bd 2ac — 2bd 
 
 a-\-b a—b ac + bc-{-ad + bd-\-ac—bc—ad-{-bd 2ac-\-2bd 
 
 c—d c-^d ac—bd 
 
 — — - , Ans, 
 ac-\-bd 
 
 8. Multiplying both numerator and denominator by 
 
 (a* — 6') (a' + 6'), or its equal a* — b\ we have 
 
 a* + b' «*-y 
 
 a'-&' (^^IT' a* -f 2a'6' + b* - {a* - 2a'b' + b*) 
 
 a + b a-b ~a* + 2a'b + 2a'b' + 2ab'-^b*-{a'-2a'b+2a'b'—2ab'-\-b*) 
 a—b^ a+b 4a'6' _ ^^ a 
 
 9. The least common multiple of the denominators of the frac- 
 tional parts is {x* — 1 )(y' — 1 ) . Therefore, 
 
 + - 
 
 x—l^x-{-l _ xf — x + i/* — l-\-xi/*—x-'y*-^l 
 1 _l_""ar'y-y+ic' — l+xV— y-ar' + l 
 
 y — l"^yf 1 22ry'-2^ xy^—x_x(y^ — \ 
 
 2x*y-2y xhj-y y\x'-lf 
 
 ,0. i±i+i+i_£±l_l+I=i+l+i+l_i_l_i4 
 
 aoca a6ca 
 
 Hence the numerator =--f- -, 
 
 abed 
 
 Substituting this expression for the numerator in the complex 
 
 fraction, and multiplying by abcd{c-\-d){a-^b)^ we have 
 
 1 1__1 1 
 
 a b c~~d {c-\-d)(a-\-b){bcd + acd—ahd—abc) 
 
 cd ab ~ abcd{a + b)cd~abcd(c+d)ab 
 
 c^d a+b 
 
 (c + d){a-\-b)(bcd + acd—abd—abc) 
 
 abcd{bcd + acd—abd—abc) 
 
 _(c + d)(a + b) 
 
 aUd '^'''' 
 
 (82) 
 
34 
 
 SIMPLE EQUATIONS. 
 
 1. Given, 
 fcultiplying by 12, 
 
 2. Given, 
 multiplying by 42, 
 
 3. Given, 
 
 SIMPLE EQUATIONS. 
 
 ( 151, page 88.) 
 
 X 9x 3x ,^ 
 _4._--=10; 
 
 6j: + 8jr— 9x=:120, Ans, 
 3jr_2j: + 3_a:— 5 ^ 
 18j— 6jr— 9 = 2a;— 10, Ans, 
 
 -^ + -i 
 
 d 
 
 multiplying by (j;*— a'), axi-a* + cx—ca=dy Ans, 
 
 x—a 2x—Sa x + ac 
 
 4. Given, 
 multiplying by aV", 
 
 5. Given, , 
 
 * 8c 10a 4ac ' 
 
 multiplying by 40ac, ba^x—bahx—Ac^x + 4:acx=10hx— lOcar, ^n*. 
 
 c ac' a' * 
 
 aVj;— a*c— 2aa; + 3a'=:c'a; + ac*, .4n». 
 
 aa; — hx ex— ax hx — ex 
 
 6. Given, 
 
 5x Zx Z—x bx—2 
 
 12 10 24 20 
 
 multiplying by 240, 100a;— 45ar + 30 — 10^;— 60a; + 24=480, Ans, 
 
 1. Given, 
 multiplying by abcx, 
 
 1 a b c 
 
 abc box acx abx^ 
 a;=a' + 6' + c', Ans. 
 
 (88) 
 
REDUCTION. 35 
 
 REDUCTION OF SIMPLE EQUATIONS. 
 (156, page 92.) 
 
 1. Given, 
 
 1x-l6 = Sx-4t', 
 
 transposing and uniting, 
 whence, by division, 
 
 4:rri:12; 
 x=3, Ans, 
 
 2. Given, 
 
 3a;+9 = 5a; + l ; 
 
 transposing and uniting, 
 dividing by —2, 
 
 -2a:=:-8; 
 x=4^ Ans. 
 
 8. Given, 
 
 4x + 1=x-h2l — 3-{-x; 
 
 transposing and uniting, 
 
 2x=n ; 
 x=5^j Ans, 
 
 4. Given, 
 
 5x + l6=x-[-52\ 
 
 transposing and uniting, 
 
 4a:=36; 
 x=:9, Ans, 
 
 6. Given, 
 
 5ax—c = b — 3ax*^ 
 
 transposing and uniting. 
 
 8ax=b + c; 
 
 dividing by 8a, 
 
 b-\-c . 
 ^- 8a »^"*- 
 
 6. Given, 
 
 ax-\-b=:9x-{-c \ 
 
 transposing and factoring. 
 
 x{a-9)=c-b', 
 
 dividing by (a— 9), 
 
 ^-^_^,Ans, 
 
 7. Given, 
 
 H--^ 
 
 clearing of fractions, 
 
 3ar + 2:r= 10x12; 
 
 uniting, 
 dividing by 5, 
 
 5ar=10xl2; 
 x= 2x12; 
 ir=:24, Ans, 
 
 
 (92) 
 
da 
 
 SIMPLE EQUATIONS. 
 
 8. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 
 9. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 "whence, 
 
 10. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 
 11. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 
 12. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 dividing by (—18), 
 
 13. Given, 
 
 clearing of fractions, 
 transposing and uniting, 
 whence, by division, 
 
 14. Given, 
 
 clearing of fractions, 
 transposing and uniting. 
 
 3x X ^^ 
 
 Y=l + '*' 
 
 6x=x-}-96 ; 
 5x=96', 
 x=19i, Ans, 
 
 3a; + 5_15a:— 1 ^ 
 ~~2 ~ 8 ' 
 12x4-20=15x— 1; 
 _3j:=-21; 
 ar='7, Ans, 
 
 x+l Sx—5 9x ^ 
 3 "^ 5 "^10*' 
 10jr+10 + 18x — 30=:2'7a;; 
 
 x=20, Ans. 
 
 2x-\-l 'iX — l5_l1x-\-Z 3^ 
 '~2 "^ 6 ~ 8 '~2* 
 40jr + 20 4-562r— 120 = 85a:H-15 — 60 ; 
 lla:=55; 
 x = 5y Ans. 
 
 X X 5x 5x dx 
 
 -4--4- — =r 1 18 : 
 
 2^3^12 7^4 ' 
 
 42arH-282: + 352: = 60a: + 63^ — 18 x 84 ; 
 — 18x= — 18x84; 
 a?=84, Ans. 
 
 I7ar—12_6j:--16__10a;— 3_6a;— 7 ^ 
 3 4 6 2~* 
 
 68jr— 48 — 15j: + 48 — 20a: + 6 = 36ar— 42 ; 
 -3x=— 48; 
 a; =16, Ans, 
 
 21 
 
 3X-11 5x—5 9l — 1x 
 
 16 8 2 
 
 336+3a?— 11 = lOjr— 10 + 776 — 56a;; 
 49ar=441 ; 
 x=9, AnM. 
 (92) 
 
15. Given, 
 multiplying by 21, 
 
 transposing and uniting, 
 
 clearing of fractions, 
 
 whence, 
 
 and 
 
 16. Given, 
 multiplying by 36, 
 
 dropping 9j?, 
 
 clearing of fractions, 
 transposing and uniting, 
 whence. 
 
 17. Given, 
 
 , . 20jr , 36 
 dropping _+- 
 
 clearing of fractions, 
 transposing and uniting. 
 Whence, 
 
 1 8. Given, 
 
 clearing of fractions, 
 transposing, <fec., 
 dividing by (—3), 
 
 19. Given, 
 
 clearing of fractions, 
 transposing, <fec., 
 whence^ 
 
 REDUCTION. 
 
 Ix+IG 
 
 37 
 
 x+8 
 
 21 
 
 4^-11 3* 
 
 H . ,^ 21^ + 168 ^ 
 
 4X-11 
 
 16: 
 
 21a! + 168 
 
 4^—11 ' 
 6ix—ll6 = 2lx+lQS; 
 43x=3U; 
 x=8j Ans» 
 
 92:4-20_4a; — 12 x^ 
 36 bx-4: "^4' 
 
 ^ ^^ 144^-432 . ^ 
 0.r + 2O = — :: : \-9x; 
 
 20 = 
 
 5x — 4 
 144a;— 432 
 
 5x-4: ' 
 lOOar— 80 = 144^?— 432; 
 — 44ir= — 352; 
 x=8y Ans, 
 
 20ar 36 5a; + 20_4jr 86^ 
 25 25 9x-16~'"5" 25* 
 5a; + 20 _50_ 
 9ar-16~25"" ' 
 6x+20 = 18ar— 32; 
 — 13a:=— 52; 
 ar=4, Ans. 
 
 3x x—\ 
 
 = Qx- 
 
 20ar4-13 
 
 4 2 4 ' 
 
 3j;_2a; + 2 = 24a;— 20a;--13 ; 
 — 3x= — 15; 
 a; =5, Ans, 
 
 ar— 3 X ^^ a: + 19 
 
 8ar-.9 + 2a;=120— 3a;— 57 ; 
 8a;=72; 
 a;=9, Ana, 
 (92-93) 
 
38 
 
 SIMPLE EQUATIONS. 
 
 20. Given, 
 
 clearing of fractions, 
 transposing, <kc^ 
 whence, 
 
 21. Given, 
 
 clearing of fractions, 
 transposing, &c., 
 dividing by ( — 11), 
 
 22. Given, 
 
 dropping —29, 
 
 , . 6ar-12 ^ ^ 
 reducing 1 to 
 
 X — iS 
 
 the form of a fraction, 
 
 clearing of fractions, 
 dropping 5ar', transposing 
 and uniting. 
 
 23. Given, 
 
 removing the parenthesis, 
 
 clearing of fractions, 
 transposing and uniting. 
 
 24. Given, 
 
 removing the parenthesis, 
 
 clearing, of fractions, 
 transposing and uniting, 
 dividing by (—175), 
 
 x + 1 a;-f2 
 
 16 
 
 ar + 3 
 
 2 ' 3 4 ' 
 
 6j; + 6 + 4x + 8==192 — 3a:— 9 ; 
 13^=169 ; 
 a;=13, Ans. 
 
 x + 3 ,^ 12a; + 26 
 2._-_+i5=— ^— ; 
 
 3O2;— 5jr— 15 + 225 = 36a: + 'r8; 
 — llx= — 132; 
 a;=12, Ans, 
 
 6X-12 
 
 5j;4-5 
 
 29: 
 
 X + 2 X-r2 
 
 6ar + 5 6j;— 12 
 
 -30 
 
 a:+3L x—2 
 
 x + 2~~ x—2 * 
 
 5j;'_5j;_10 = 5j:' — 20 ; 
 — 5ar= — 10; 
 x=2f Ans. 
 1x-{-9 / 2ar— 1 
 
 X- 
 
 1 ; 
 
 ; + 9 / 2ar-l\ 
 
 1— r-^=^' 
 
 1x + 9 
 
 ■x + - 
 
 9 
 2x 
 
 ■=1 
 
 1 + 9x 
 
 63ar + 81 — 36j: + 8j;— 4 = 252; 
 35x=l75; 
 x=5f Ans. 
 
 63 + 81ar— 36-f 8 — 4a:=:252a;; 
 -1753;=— 35; 
 . a:=i, Ans, 
 (93) 
 
REDUCTION. 
 
 ^ 
 
 25. Given, 
 
 x-{-l x-\-2_x- 
 
 3 x-4 
 
 2 3 4 6 
 
 clearing of fractions, 6j;+6H-4ar-f 8 = 32:— 9 + 2a;— 8 + 36 ; 
 
 transposing and uniting, 5j?=5 ; and a;=l, Ans. 
 
 26. Given, 
 
 2 + 3+4 + 5 = "' 
 
 clearing of fractions, 
 
 30x + 20x + 15x+12x=11 xQtO; 
 
 uniting, 
 
 11x=11x60; 
 
 whence, by division. 
 
 a; = 60, Ans, 
 
 27. Given, 
 
 I+H=-' 
 
 clearing of fractions. 
 
 6a; + 4a; + 3a;=130x 12; 
 
 uniting, 
 
 132r=130xl2; 
 
 
 a:=10xl2; 
 
 
 ar=120, Ans. 
 
 28. Given, 
 
 i+i+^=-= 
 
 clearing of fractions. 
 
 6a; + 2a;4-a: = 90xl2; 
 
 
 9x = 90xl2; 
 
 
 a:=10xl2; 
 
 
 ic=120, Ans, 
 
 29. Given, 
 
 M+r-' 
 
 clearing of fractions. 
 
 21aJ4-14a:+6d;=82x42; 
 
 uniting. 
 
 4l2;=82x42; 
 
 
 a:=2x 42 = 82, Ans, 
 
 30. Given, 
 
 l+?+^+^+^=«««' 
 
 clearing of fractions. 
 
 84a; + 60ar + 35a; + 21ar + 202^ = 660 x 420 ; 
 
 uniting, 
 
 220a;=660x420; 
 
 dividing by 220, 
 
 a:=3x420; 
 
 whence. 
 
 ar=1260, Ans, 
 
 (93) 
 
40 SIMPLE EQUATIONS. 
 
 31. Given, a'x + 2ac— c'ar=a' + c' ; 
 
 transposing, a'x— c'a:=a' — 2ac + c' : 
 
 dividing by (a*—c^)^ and factoring, x=^ —c)[a—c) ^ 
 
 (a—c)(a + c) ' 
 
 or. 
 
 X——— , Ans, 
 a-\-c 
 
 32. Given, Ahx—2a=Zab — Qh''x 
 
 transposing, 66'a; + 46x= 3a6 + 2a ; 
 
 Zab + 2a ^ 
 W+Ab' 
 a 
 26 
 
 whence, 
 
 reducing, xz^~, Am, 
 
 33. Given, a{x—h)-\-h{x—c)-\-c{x — a)=0\ 
 
 removing parentheses, ax—ab + hx — hc-\-cx—ac=0\ 
 
 transposing, ax + bx + cx=ab-\-bc-\-ac*, 
 
 factoring, x(a + 6 + c) =a6 -f 6c + ac ; 
 
 , ab + bc-\-ac ^ 
 
 whence, ar= Ans. 
 
 a+b+c ' 
 
 34. Given, a'(ar— 1) 4-am(a:— 2)=m'; 
 removing parentheses, a*x—a*-\-amx—2am=m* ; 
 transposing and factoring, a[a-\-in)x=a* + 2am -\-m* ; 
 
 whence, x= , Ans. 
 
 a 
 
 So, Given, ax-{-cx + x=b-\ ; 
 
 c 
 
 clearing of fractions, acx + c*x-\-cx=bc-\-b—ax; 
 
 transposing, acx-\-ax-^c^x-{-cx=bc + b\ 
 
 factoring, ax{c + l)-\-cx{c-\-l) = b(c + l) ; 
 
 dividing by (c + 1 ), ax-{-cx=b] 
 
 whence, x= , Ans, 
 
 * a-[-c* 
 
 (93-94) 
 
REDUCTION. 41 
 
 ^. a-\-x c — x a 
 
 36. Given, __ + __=_; 
 
 cleari ng of fractions, ad-\-dx-\-bc— bx = ad ; 
 
 transposing, —bx + dx=--bc; 
 
 or by (150, III.,) b.e^dx=bc', 
 
 whence, x=j-—^,Ans, 
 
 X X X X 
 
 37. Given, - + - — — ---— — -=1. 
 
 a— 1 6—1 a+\ 6+1 
 
 . X X 2x X X 2x 
 
 Observing that -— — r-r=i — 7» ^^^ I — \ — TTT=T5 — 7» 
 
 ^ o— 1 a + 1 a' — 1 6—1 6 + 1 6' — 1 
 
 we have ^rri+fti^i^^* 
 
 clearing of fractions, 26'x-2ar+2a'x-2a:=(a»-l)(6'-l) ; 
 uniting and factoring, 2(6'— 2 + a').ci=(a'-l)(6' — 1) ; 
 
 (.«-l)(6» -i) 
 
 «— 1 X \ 2 
 
 38. Given, rH — —r= - + 
 
 c— 1 c + 1 c— 1 (c — 1)" 
 
 car— a; 2 
 
 multiplying by (c— 1), *~"^ + 7+Y-^ +^Z:i ' 
 
 car— ar 2 
 
 uniting, :r— 2H —- = ; 
 
 *" c + 1 c— 1 ' 
 
 clearing of fractions, c'a;— ar— 2c' + 2+c'a;— 2ca; + ar=2c + 2 ; 
 
 transposing and uniting, 2c'ar— 2ca; = 2c' + 2c ; 
 
 dividing by 2c, and factoring, (c— l)a:=:c+l ; 
 
 whence, x— -, Ans, 
 
 c — 1 
 
 XXX 
 
 39. Given, -+T^ — =a6 + ac + 6c; 
 
 a 6 c 
 
 clearing of fractions, a;(6c + ac-\- ab) — {ab-\-ac + 6c)a6c ; 
 
 whence, x=abc, Ans. 
 
 (94) 
 
SIMPLE EQUATIONS. 
 
 X — h — c T — a — c X — a — h 
 
 + -6— + — 
 
 3; 
 
 42 
 
 40. Given, 
 
 clearing of fractions, 
 
 hex — (6 ■{■€)})€ -Y acx— (a + c)ac ■i-abx-—(a. + h)ab=iSahc ; 
 Transposing, 
 
 bcx-^acx + abx=(b + c)bc + (a + c)ac + (a + i)a6 + Sabc. 
 
 Removing the parentheses, and arranging the second member of 
 the equation in three terms with reference to the coefficient of ar, 
 we have 
 
 (be + ac + ab)xz= (abc + tt'c+a'6) + (b^c + abc + ab^) + [be'' + ac* + abc\ 
 dividing by {be -\-ac-\- a6), a*= a + 6 + c, Ans. 
 
 41. Given, 
 transposing, 
 uniting, 
 dividing by .875, 
 
 42. Given, 
 transposing and uniting, 
 dividing by 2.844, 
 
 43. Given, ^ ^ . 
 
 ' 2.8 4 .7 
 
 clearing of fractions, 2.4a;— .12 + 3.22j;— 2.52 = 2.56a;— .192 ; 
 
 transposing and uniting 3.06a; =2.448 ; 
 
 x=,8, Ans. 
 
 1.25X— 6.125 + .25x=.625a:; 
 
 1.25a; + .25a;— .625a; = 6.125 ; 
 
 .875a: = 6.125; 
 
 a: =7, Ans 
 
 3.164a:— 4.266 = . 24a; + . 08a: ; 
 ra: = 4.266; 
 a;=1.5, Ans. 
 
 2.4a;— .12 4.6a:— 3.6 .64.r— .048 
 
 PROBLEMS 
 
 PRODUCING EQUATIONS WHICH CONTAIN ONE UNKNOWN QUANTITY. 
 
 (160, page 97.) 
 
 1. Let X represent the number ; then by the conditions, 
 
 (a;-6)ll = 121; 
 whence, a:— 6 = 11 ; 
 
 a;=l7, Ans. 
 (94-97) 
 
PROBLEMS. , '43 
 
 2. Let anytime past ; then 
 
 20— a:=time to come. 
 Now by the conditions of ) a;_20— a? ^ 
 
 the problem, we have i 3 2' 
 
 whence, 2j:=:60— 3j;; 
 
 5a: =60, and 2-= 12, Ans, 
 
 3. Let X represent the number ; then by the conditions, 
 
 XXX 
 
 ^ + 2-^3 + 4^^^^' 
 whence, 25a;=250 x 12 ; 
 
 a;:= 10 X 12 = 120, ^7W. 
 
 4. Let if = one part ; then 
 
 7 7 — a; = the oth er part ; 
 and by the conditions of) x 'J^'7— a;_ 
 
 the problem, we have j 7 3"' 
 
 8ar + 539 — 7^=315; 
 
 _4a:=_224; 
 whence, x—56y one part; 
 
 and 77— a:= 21, th» other part. 
 
 5. Let a; = the greater number; then 
 
 75— a;= the lesser number; 
 
 and by the conditions of ) , . x 
 
 the problem, we have j ^ '~3' 
 
 3^—225 +3a:r=a:; 
 
 5ar=225; 
 whence, a;=r45, the greater ; 
 
 and 75— a;=30, the less. 
 
 6. Let X represent the amount of money at first ; then by the 
 conditions, we have 
 
 ^ ^ «« 
 
 20a:— 5a;— 4jr = 66x20; 
 lla: = 66x20 ; 
 whence, x— 6 x 20 = 120, ^w*. 
 
 (97) 
 
44 
 
 SIMPLE EQUATIONS. 
 
 7. Let 30jr represent my money at first ; then 
 
 30.i: — = 20j:=: remainder after the firet payment; 
 
 3 
 
 20x J- = 152:=: remainder after the second payment; 
 
 15z 
 
 whence, loz =12a;=24 ; 
 
 5 
 
 x=2; 
 30ar=60, Ans. 
 
 8. Let X represent the number ; then by the conditions, 
 
 
 ^(^-5) = 40; 
 
 
 2ar-10=:120; 
 
 
 ar=65, Ans, 
 
 9. Let 
 
 arr^price of the horse ; then 
 
 
 200— ar=:price of the chaise ; 
 
 and by the conditions, 
 
 1=^(200-.) ; 
 
 
 3jr=400-2ar; 
 
 
 a:=80, the price of the horse. 
 
 
 200— ar=120, the price of the chaise. 
 
 10. Let 
 
 a?=the greater part ; then 
 
 
 48— ar= the lesser part; 
 
 and by conditions, 
 
 6^ 4 -^' 
 
 
 4a; + 288 — 6a;=216; 
 
 whence, 
 
 ar= 36, the greater; 
 
 and 
 
 48— a;=12, the lesser. 
 
 11. Let X represent the value of the estate; then by the con- 
 ditions, . * ; 
 
 (97-98) 
 
PROBLEMS. 45 
 
 X 
 
 --f 200= what tlie first received, 
 4 
 
 /»• 
 
 - + 340= what the second received, 
 5 
 
 /p 
 
 - + 300 = what the third received, 
 6 
 
 /p 
 
 - + 400 = what the fourth received, 
 8 
 
 whence, T^f + ^ + ^ + 124Q=ar ; 
 
 4 o o o 
 
 30ar-f 24«+20ar+15a;+1240 x 120 = 120x ; 
 
 31j;=1240x120 ; 
 
 ar=40 X 120 = 4800, ^n«. 
 
 12. Let X represent the number ; then by the conditions, 
 
 whence, lOi: — 910 = 3 j; 
 
 7^=910, and «=130, Ans, 
 
 13. Let ar= A s stock ; then 
 
 3j= B's stock ; 
 12ar= C's stock ; 
 5^= D's stock. 
 By the conditions, 21ar= V3500, and ar=$3500, Ans, 
 
 15. Let *lx— A's share; then 9^;= B's share; and by the con- 
 ditions, 
 
 7^ + 9^= $2000; 
 whence, '7a:=$875, A's share ; 9x=$1125, B's share. 
 
 16. Let Zx= the rye ; then ^x= the oats, and 5x— the peas. 
 By the conditions, 
 
 3i: + 4a; + 5a:=72 ; 
 whence, ar=6 ; 3jr=18; ^xz=z2A^ and 5ar=30; 
 
 therefore, Rye, 18 bushels; oats, 24 bushels; peas, 30 bushels, -4w5, 
 
 (98) 
 
SIMPLE EQUATIONS. 
 
 17. Let 6j*= the quantity of wine drawn from one cask; then 
 1x=z the quantity of wine drawn from the other. 
 By the conditions, 
 
 whence, 
 
 6x 
 '7x—16=—=3x 
 2 
 
 x=4, 7.r = 28, and 6a;=24. 
 
 18. Let 
 then, 
 whence, 
 
 5x= less part, and a=204 ; 
 a—5x= greater part ; 
 a— Yjr = 20j— f(a — 5.r) ; 
 204j;=10a = iox204; 
 a:=10; 
 5.r=50, the lesser; 
 204 — 5^=154, the greater. 
 
 19. Let 
 
 Now by the conditions, 
 
 whence, 
 and, 
 
 Sx= the price of the horse ; then 
 a—8x= the price of the chaise. 
 
 2a — 16x—Sx=2ix—^{a — Sx) ; 
 148 — 133ar=:1682:— 5a-+-40jr, 
 341ar=19a=19x341 ; 
 ar=19; 
 8ar=152, horse, 
 341 — 8j:=189, chaise. 
 
 21. Put I560 = a ; then 9560 = a + 2000. 
 
 Let x= each private's share ; then 
 
 2lx= what all the privates received ; 
 2*Jx-{-a= the prize. 
 By conditions of the problem, we have 
 
 27a: + a=:25j: + a + 2000; 
 2a;=2000; a:=1000; 
 
 27a; + 7560: 
 
 34560, Ans. 
 
 22. As he annually increases his unexpended capital by one third 
 of itself, four thirds of this unexpended part will represent his capi- 
 tal at the end of each year. 
 
 (99-100) 
 
PROBLEMS. 
 
 47. 
 
 Let 
 
 therefore, 
 
 flr= his original stock; put 1000=ff; then 
 
 4 . . 4x — 4a_ stock at the end of 
 3^ 3 ~ the first year ; 
 
 4/4x—4a \ 16j?— 28.c_ stockat the endof 
 3\ 3 /~ 9 the second year J 
 
 4/16ar— 28a \ 64:c— 148a_ stock at the end of 
 3\ 9 /~ 27 ~ the third year; 
 
 27 
 64a; — 148a 
 
 2x 
 
 27 
 
 64ar— 148a = 54a;; 
 10a:=148a=148xl000; 
 a;=: 14800, Ans, 
 
 23. Puta=99. Let 
 Now by the conditions, 
 
 whence, 
 
 a'zsthe time past ; then 
 a— a? = the time to come. 
 
 2a: 4. , 
 
 10a;=12a — 12a;; 
 22a;=12a = 12x99 ; 
 a;=6x9=:54; 
 99— a;=: 45, Ans. 
 
 24. Let 
 
 6a;=the amount of gunpowder; then, 
 4a; + 10= the nitre; 
 ar— 4|=the sulphur; 
 4a;+10 
 
 By the conditions. 
 
 whence, 
 
 2=the charcoal. 
 
 4a;— 10 
 
 4a; + 10+a;-44 + 
 
 4a;— 10 
 
 2 = 6x; 
 
 1 •^- 2' 
 
 8a; + 20 — 14a;=— 49; 
 
 6a;=69, ^w*. 
 
 (100) 
 
48 
 
 SIMPLE EQUATIONS. 
 
 25. Put a= 183. Let 
 
 By the conditions, 
 
 whence, 
 
 a:=what the first received ; then, 
 
 a— x^what the second received. 
 
 40jr=21a — 21J-; 
 
 21 xl83; 
 
 a;=:21 x3 = 
 183— ar^ 120 ) 
 
 26. Let 
 
 By the conditions, 
 whence. 
 
 a:=the greater part; then, 
 68— :r=the lesser part. 
 84— ar=3[40-(68-a-)]; 
 84— ar=120 — 204 + 3J:; 
 4z=168, 
 
 68 — a;=26 ) 
 
 27. Let 
 
 l/2jr 3j-\ 
 
 3(t+t) 
 
 2z= distance from A to B ; then 
 32r= distance from C to D ; 
 
 :— — distance from B to C. 
 
 By the conditions of the problem, 
 5ar+y=34; 
 
 1Vj:=34x3; 
 
 Whence, 
 
 x=6. 
 
 2x=12, distance from A to B ; 
 3x=:18, distance from C to D; 
 ^x=z 4, distance from B to C. 
 
 28. Let 3ar=the number of sheep at first; then 
 
 2a;— 6=the number after the first plunder, 
 2x— 6 — (ar— 3) — 10 = 2; 
 
 a:=:lo; 3x=^45, Ans, 
 
 Whence, 
 
 29. Observe that for every vessel broken, he lost 12 cents,- 
 3 cents fee, and 9 cents forfeiture. 
 
 (100) 
 
PROBLEMS. 
 
 49 
 
 Let 
 
 30. Let 
 
 By the conditions, 
 
 x= the number he broke ; then 
 300 — 12j?=:240; 
 
 x=5j Ans. 
 
 x= hia indebtedness to A ; then 
 2x= his indebtedness to B ; 
 6x= his indebtedness to C. 
 9j:=270 ; and x = 30. 
 
 31. Let ()Xz= A's share, and 9x= B's share ; then 
 
 20a:=C's share, and 433ar=:D's share. 
 By the conditions, l8^z=3lo ; whence, 
 
 Gx=24, 9x=S6, 20jr=80, 433a:=:l75. 
 
 32. Let 5 j:=: his money at first; then 
 
 4x4- 4 = what he had after first losing and winning; 
 3ar + 34-3 = what he had after second losing and winning. 
 As he loses } of this, he must have f of it left ; therefore 
 
 i(3a:-f6) = 20; 
 15a; 4-30 = 120; 
 whence, 15a:=90, and 5a:=30, Ans. 
 
 33. Let 3.c=:his income ; then 
 
 2j;=his family expenses. 
 The remainder of his income is a;, | of which he spends in improve- 
 ments, leaving | of a; to lay by ; whence, 
 
 X 
 
 - = 70 ; and 3ar=630, Ans. 
 
 34. Let 
 
 By the conditions, 
 whence, 
 
 35. Let 
 
 By the conditions, 
 
 orrzithe lesser part ; then 
 60— a:=:the greater part. 
 x{eO-x) = dx''; 
 60~a:=3a:; 
 
 4ar=60, and ar^lS, A7is. 
 
 a?=the value of the saddle ; then 
 8jr=the value of the horse. 
 9a:=90 ; whence a;=10, Ans. 
 
 (100) 
 
50 
 
 36. Let 
 By the conditions, 
 
 SIMPLE EQUATIONS. 
 
 ar=:what one receives ; then 
 10^= what the other receives. 
 Il2r=462 ; whence a*=42, Ans. 
 
 37. Let armrent of the estate last year ; then by the conditions, 
 
 8x 
 ^ + 100 = ''''' 
 
 108x= 189000, and x=ll50, Ans, 
 
 ar=the greater ; then 
 840-x=the less. 
 
 3^— 2520 + 3^=a? ; 
 
 5irr=2520, and x=504, Ans, 
 
 whence, 
 38. Let 
 
 By the conditions. 
 
 whence, 
 
 39. Let «=hi8 income ; then by the conditions, 
 .-g + 10o)=| + 35; 
 
 10a?— 2a:— 1000 = 5ar + 350 
 
 whence. 
 
 3a:=1350, and a; =450, Ah 
 
 40. Let 
 
 a;=A's part; then 
 
 
 100+ir=B's part. 
 370+a;=C's part. 
 
 By the conditions, 
 whence, 
 
 3a:+470 = 1520; 
 
 a;=350, A's part. 
 
 "•Let )(^.:,^. 
 
 By the conditions, 
 whence. 
 
 7a; = the income ; then 
 3"= A's annual debt; 
 
 — =what B saves annually. 
 2(y) = 2a: + 32; 
 
 a;=40; 
 
 
 7a; =2 80, Ans, 
 
 (101-10^) 
 
PROBLEMS. 
 
 51 
 
 42. A^s rate of travel is | miles per hour ; 
 
 B's rate of travel is f miles per Lour; 
 
 66 
 
 — -=the distance between A and B, when B sets out. 
 5 
 
 Let «=the number of hours ; then by the conditions, 
 
 6x_*ix 56 
 
 25a:=2U + 168; 
 
 whence, ' 0^=42, number of hours; 
 
 *bx 
 -—=^0^ number of miles. 
 
 43. Let ar= the number of days ; then both working together 
 will do ~ of the work in one day. But A does | of the work in 
 
 X J • 
 
 one day, and B j*j ; therefore, 
 
 1 J^_l. 
 8'*"l2~ar' 
 
 X X 
 
 whence, 
 
 8 12 
 3ar + 2a;=24, and a;=4f, Ans, 
 
 44. Let 
 
 x=. the distance ; then 
 -= hours he rides ; 
 
 By the conditions, 
 whence. 
 
 -=: hours he walks. 
 4 
 
 X X ^ 
 
 8j:=48, and a;=16, Ans, 
 
 45. Let 3a: = the time in which C can dig the trench ; 
 
 then 2ar= the time in which B can dig the trench ; 
 
 and x=. the time in which A can dig the trench. 
 
 (102) 
 
52 SIMPLE EQUATIONS. 
 
 Now in one day A can dier - of it, B -— of it, and C — of it ; 
 
 but all together can dig - of it in one day. Hence, 
 >^^ , ^ 1111 
 
 ^ 7 J ar=ll. As time, 
 
 .-40=^ /S '^^ 2x=22, B'stirae, 
 
 ^j.'^<?..-.'^^>t-^ g^^33 c'stime. 
 
 / 46. Let x= the distance between A and B. Since C and B 
 A travel at the respective rates of 5 and 4 miles an hour, when they 
 
 5 5x 
 
 have met, C has traveled -- of the whole distance, or — miles. In 
 y y 
 
 3 3 bx 
 
 the meantime, A, traveling - as fast as C, has traveled - of — , or 
 
 5 5 9 
 
 X 5x X 2jj 
 
 - miles. Hence,— — — , the distance between A and C when 
 
 3 9 3 9 
 
 C turns back. 
 
 5 5 
 
 Now C in traveling back to meet A, goes - of this distance, or - 
 
 8 8 
 
 of — = — miles. But the whole distance traveled by C is 60 
 9 36 "^ 
 
 miles; hence, 5x 5x ^^ 
 
 ¥+3-6 = '*°' 
 
 x=l2y Ans. 
 
 47. Let x= the valueof one sheep; then 
 
 l2x-\-92x — 164x= the value of the flock. 
 By the conditions, 
 
 92*— |35 = 72:r+$35, 
 
 a-=$3i; whence, 164a;=$574,^n*. 
 
 48. Let x= the rate of the current per hour. As the current 
 will retard the boat by its whole velocity in going up the river, and 
 accelerate it the same in going down, 12— x= actual rate of row- 
 ing up stream, and 12 +a;= actual rate of rowing down stream. By 
 
 the conditions, 
 
 7(12-ar) = 5(12+ar); 
 
 84 — 7.r=:G0 4-5x; 
 
 12i: = 24; 
 
 whence, ar= 2, Ans. 
 
 (102) 
 
TWO UNKNOWN QUANTITIES. 5& 
 
 SIMPLE EQUATIONS 
 
 CONTAINING TWO UNKNOWN QUANTITIES. 
 
 (irO, page 109.) 
 
 1. Given i ^^ + ^^= ^^' (^) 
 
 ^' •il2;r + 7^=100; (2) 
 
 multiplying equation (1) by 3, and equation (2) by 2, we have 
 
 24.r + 15?/ = 204; (3) 
 
 24ar+14y = 200; (4) 
 
 subtracting (4) from (3), y=4; 
 
 substituting this value of y in (1), 8x + 20=68 ; 
 
 whence, 
 
 8j-r=48, and ar=6, Ans. 
 
 2. Given, 
 
 (5.r + 2y=y9, (1) 
 \lx-6y= 9; (2) 
 
 multiplying (1) by 3, 
 adding (2) and (3), 
 whence, 
 
 15a; + 6y = 57; (3) 
 22.r=66; 
 
 ar=3, and y=2, Ans. 
 
 3. Given, 
 
 3x+1y=l9, (!) 
 ir + 4y=38; (2) 
 
 multiplying (2) by 3, 
 subtracting (1) from (3), 
 whence, 
 
 3;r + 12y = 114; (3) 
 6y=35; 
 y=1y and a;=10, Ans. 
 
 4. Given, 
 
 ( 5x-3y=SQ, (1) 
 ^2a: + 9y = 96; (2) 
 
 multiplying (1) by 3, 
 adding (2) and (3), 
 whence. 
 
 15a;-9y=108; (3) 
 I72:zr:204; 
 a;=12, and y=8, Ans. 
 
 5. Given, 
 
 f x-\-My=5i, (1) 
 J3a:-25y = 10; (2) 
 
 multiplying (1) by 3, 
 subtracting (2) from (3), 
 whence. 
 
 3ar + 51y = 162; (3) 
 76y=152; 
 
 y=:2, and a;=20, Ans. 
 
 
 (109) 
 
54 
 
 BIMFLE EQUATIONS. 
 
 6. Given, 
 
 6:r-4y= 40, (1) 
 a:-6y=~97; (2) 
 
 multiplying (2) by 5, 
 subtracting (3) from (1), 
 whence. 
 
 6a;-25y=-485; (3) 
 21y= 525; 
 
 y= 25,anda;=28,^w«. 
 
 v. Given, 
 
 multiplying (1) by 3, 
 multiplying (2) by 4, 
 subtracting (4) from (3), 
 
 f8ar4-15y= 9, (1) 
 
 J6a:-12y=-l; (2) 
 
 24a; + 45y=: 27; (3) 
 
 24ar— 48y=— 4; (4) 
 
 93y= 31;y=^i=i 
 
 substituting the value of y in (1), 8a;4-5= 9 ; x 
 
 :t1 
 
 8. Given, 
 
 j Yx + 7y=30, 
 ]3x + 4y=l7; 
 
 multiplying (1) by 3, 21;r + 21y=90; 
 
 multiplying (2) by Y, 21a;4-28y = 119; 
 
 subtracting (3) from (4), 1i/ = 29 
 
 substituting the value of 1y in (1), 7^+29 = 30 
 
 
 9. Given, 
 
 multiplying (1) by 2, 
 adding (2) and (3), 
 whence, 
 
 10. Given, 
 
 multiplying (2) by 2, 
 adding (1) and (3), 
 whence, 
 
 11. Given, 
 
 multiplying (1) by 3, 
 multiplying (2) by 5, 
 subtracting (4) from (3), 
 whence, 
 
 j 8a; + 3y=:25, 
 |5j;-6y = 55; 
 162: + 6y = 50; 
 21a;=105, 
 ar=5, and y: 
 
 Ans, 
 
 (1) 
 
 (2) 
 (3) 
 W 
 
 Ans. 
 
 (1) 
 
 (2) 
 (3) 
 
 —5, Ans, 
 
 ( I5x-8y= 9, (1) 
 
 ( 10ir + 4y=-43. (2) 
 
 20jr + 8y= — 86; (3) 
 
 35x= — 11; 
 
 ar= — 2 j, and y = — 5|, ^7W. 
 
 j 9a;-5y= 950, (1) 
 
 ( 2a;-3y=— 450; (2) 
 
 27a:— 15y= 2850; (3) 
 
 10a:-15y=-2250; (4) 
 I7.r= 5100; 
 
 ar=300, and y = 350, Ans, 
 (109-UO) 
 
TWO UNKNOWN QUANTITIES. 55 
 
 :20, (1) 
 
 12. Given, 
 
 -—1- 
 2 4" 
 
 ^+1=10; (2) 
 
 multiplying (2) by 2, f+f^^*' ' (^) 
 
 adding (1) and (3), -^=40 ; 
 
 3 
 
 whence, 2.r=120; x=QO. 
 
 substituting in (1), 30— 1=20 ; y=40. 
 
 13. Given, 
 
 multiplying (1) by 2, 
 multiplying (2) by 3, 
 
 subtracting (4) from (3), 
 
 clearing (5) of fractions, 
 whence, 
 
 substituting this value of y in (3), a:+ 10 = 16 ; 
 
 ar=6. 
 
 2 + 3- ®' 
 
 (1) 
 
 |-F-> 
 
 (2) 
 
 "1= "; 
 
 (3) 
 
 .-!=-., 
 
 (4) 
 
 14= »: 
 
 (5) 
 
 10y + 9y=19xl5; 
 
 
 y=15; 
 
 
 14. Given, 
 
 3^-|=3|, (1) 
 
 4y-|=.7; (2) 
 
 multiplying (1) by 2, Qx—yz=i Y; (3) 
 
 multiplying (2) by 5, 20a;--y=35; (4) 
 
 subtracting (3) from 4, 14x=28; 
 
 whence, ar=2, and y=5, ^ns. 
 
 (UO) 
 
56 
 
 SIMPLE EQUATIONS. 
 
 15. Given, 
 
 adding (1) and (2), 
 
 clearing (3) of fractions, 
 or, 
 
 clearing (1) of fractions, 
 subtracting (6) from (7), 
 whence, 
 
 + 8y=194, 
 
 ^-1-8j;=131; 
 8 
 
 + 8(j;-fy) = 325; 
 
 8 
 
 x-\-y + e4(x + 7j) — d25 X 8 
 
 65(.c + y)=:325 x8 
 
 x + 7j^= 5x8 
 
 a;+64y = 194x8 
 
 63j/=189 X 8 
 
 y=24, and x 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 
 (5) 
 (6) 
 (n 
 
 16, A-v. 
 
 16. Given, 
 
 X 
 
 3 + 
 
 =21, 
 
 + 3j- = 29; 
 
 (1) 
 
 (2) 
 
 adding (l) and (2), i(-r + y) + 3(.r + y) = 50 ; (3) 
 
 whence, 10(a; + y) = 50 x 3 ; (4) 
 
 or, a:4-y= 5 x3; (5) 
 
 from (1), a: + 9y=:21 x 3 ; (6) 
 
 subtracting (5) from (6), 8y = 1 6 x 3 ; 
 
 whence, y=6, and a: = 9, Ans, 
 
 17. Given, 
 
 adding (1) and (2), 
 
 whence, 
 
 or, 
 
 from (1), 
 
 subtracting (5) from (6), 
 
 whence. 
 
 + 7y=99, 
 
 4-7a;=51; 
 
 4(^+y) + 7(a;4-y) = 150; 
 
 50(a; + y) = 150x7; 
 
 x-\-y= 3x7; 
 
 a;+49y= 99x7; 
 
 48y= 96 X 7 ; 
 
 y=14, and x-. 
 
 (110-111) 
 
 (1) 
 
 (2) 
 
 (3) 
 
 (4) 
 (5) 
 (6) 
 
 :7, Ans, 
 
18. Given, 
 
 TWO UNKNOWN QUANTITIES. 67 
 
 4 4 
 
 ---=1. (1) 
 
 ---=H; (2) 
 
 2 1 
 subtracting (2) from (1), "=o> ^°^ y=^ 
 
 y -^ 
 
 4 . 
 substituting value of y in (1), -=2, and a;=2 
 
 u4n5. 
 
 19. Given, 
 
 dividing (1) by 147, i-^=2T' (^> 
 
 ,.,.,>,» 17 17 68 
 
 multiplying (3) by 17, V""7=2T' ^^^ 
 
 . , . . , ^ 73 41 68 219 , , 
 
 subtracting (4) from (2), y=-g — ^=-^' w) 
 
 13 1 
 
 dividing (5) by 73, -=— =^, and y=7 
 
 17 1^ 
 substituting value of y in (3), -=—=:-, and ar=3 
 
 20. Given, 
 
 
 Reducing the first members of equations (1) and (2) to mixed 
 quantities, we have 
 
 ar +— =a;+— -; 3) 
 
 4 a? + y 
 
 27 54 ,^. 
 
 (111) 
 
d8 SIMPLE EQUATIONS. 
 
 dropping x in (3), and ) 14^ 
 
 dividing by 1 7, 3 4~«Ty' ^^ 
 
 dropping y in (4), and ) 12 
 
 dividing by 27, f h^x—y * ^^^ 
 
 clearing (5) of fractions, a?+y=16 ; (7) 
 
 clearing (6) of fractions, «— y=10; (8) 
 
 adding (7) and (8), 2ar=26, and ar=13 
 
 subtracting (8) from (7), 2y= 6, and y=: 3 
 
 >■ Ans, 
 
 21. Given, 
 
 2y— ^ ^^ 59 — 2a; 
 
 -^^ =20 
 
 23— a: 2 
 
 ar-^>— =20 —, (1) 
 
 ^+Ss=--^^-. (^) 
 
 multiplying (1) by 2, 2a;— ^—^= 40-59 + 2aj, (3) 
 
 or. 
 
 4v— 2a; 
 clearing (4) of fractions, 437 — 19a;=4y— 2a;, (5) 
 
 or, 
 
 I7a; + 4y = 437; (6) 
 
 multiplying (2) by 3, 3y + ?^^=90-73 + 3y, (7) 
 
 or. 
 
 a:-18 
 3y-9 
 
 = 1V; (8) 
 
 a;— 18 
 
 clearing (8) of fractions, 3y— 9=l7a;— 306 ; (9) 
 
 transposing, 17a;— 3y=297 ; (10) 
 
 subtracting (10) from (6), 7y=140, and y=20 j . 
 
 whence, from (10), 17a; =3 5 7, and a;=21 j 
 
 |6a;'-24y' + 130 
 2.-4y + 3 -^^ + ^y+^> (1) 
 
 9a;y-110 , 151 -16a; 
 -^]^3^+-43^^r-=3-r; (2) 
 
 clearing (1) of ) ^ , ^ , 
 
 frac^ons, \ ^^ "24^' + 130 = 6.'-24y' + lla:+14y+3 ; (3) 
 
 whence, lla;+14y=127 ; (4) 
 
 (111) 
 
TWO UNKNOWN QUANTITIES. 59 
 
 (15l — l6x)(Su — i) 
 
 whence, 110 — 12^;=^=^ ^^-^^ ^; (6) 
 
 4y— 1 
 
 or, 440y — 48a;y— 110 + 12a' = 453y— 48.ry— 604 + 64.r; (7) 
 
 or, 52.r + 13^ = 498; (8) 
 
 or, 4a: + y = 38; (9) 
 
 adclmg.(4) and (9), 15.r + 15y = 165 ; (10) 
 
 ^4-y-ll; (11) 
 
 subtracting (11) from (9), 3a;=27, and x=9 \ 
 
 substituting value of x in (^,i// ) y = 2 ) 
 
 a; + 3y_7.c— 21_3a; — 15_8jr — 9y 
 
 , 3 6 " 4 12 
 
 23. Given, / 
 
 ^ 2a; + y 9.r— 7 3y + 9 4.r + 5y 
 
 (1) 
 
 2 8 4 16 ' ^^^ 
 
 dealing (1) of fractions, 4j--f 12y— 14i:4-42 = 9;r— 45 — 8a; + 9y , (3) 
 
 transposing and uniting, 1U- — 3^ = 87 ; (4) 
 clearing (2) of fractions, 16a:+8y— 182:+14=:12y + 36 — 4;r— 5// ; (5) 
 
 whence, 22: + y = 22; (6) 
 
 multiplying (6) by 3, 6x + 3i/=GQ\ (7) 
 adding (4) and (7), l7a;ii::153, and x=:9 ) 
 substituting value of a: in (6), 18+y= 22, and 
 
 x=z9 I 
 y = 4 J 
 
 24. Given, 
 
 ax +hy =d^ 
 ' a'x + b'y=d'', 
 
 (1) 
 (2) 
 
 multiplying (1) by b\ 
 
 b'ax + b'bi/=b'd; 
 
 (3) 
 
 (2) by 6, 
 
 ba'x + b'by = bd'\ 
 
 (4) 
 
 subtracting (4) from (3), 
 
 {b'a-ba')x=b'd-bd' 
 
 (5) 
 
 whence, 
 
 b'd-bd' 
 '"'-'b'a-ba' ' 
 
 
 multiplying (1) by a', 
 
 a' ax -\- a' by— a' d ; 
 
 (6) 
 
 (2) by a, 
 
 a'ax-\-ab'y=ad' ; 
 
 P) 
 
 subtracting (7) from (6), 
 
 {a'b--ab')y = a'd-ad' ', 
 
 (8) 
 
 whence, 
 
 a'd-ad' 
 y-a'b-ab'' 
 (111-112) 
 
 
60 
 
 SIMPLE EQUATIONS. 
 
 25. Given, 
 
 multiplying (2) by 6, 
 
 subtracting (3) from (1), 
 
 clearing (4) of fractions, 
 whence, from (1), 
 
 a 
 
 ah ab ' 
 
 a a 
 
 (1) 
 
 (2) 
 (3) 
 
 |_|=a6-i'=J(a-6); (4) 
 
 y(a—h) =ab^(a—b)^ and y=ab^ 
 X * \ Ans, 
 
 --\-ab=2ab^ 
 
 and x=a''b 
 
 26. Given, 
 
 ax-\-cy: 
 
 ex 4- ay -. 
 
 multiplying (1) by a, a*x-{-acyz 
 
 a*+c* 
 
 ac 
 
 multiplying (2) by c, c*x-}-aci/= 
 
 ac' 
 
 a' + c' 
 
 (1) 
 (2) 
 
 (3) 
 
 subtracting (4) from (3), (a'-c')a:r^^^*-^-ti'=:'^'('^'7^') ; 
 
 whence, 
 or, 
 
 o'(a*-c') _ g' 
 
 
 substituting value of a; in (2), - + ay= ; 
 
 c ac 
 
 whence, 
 
 a'+c* a c* c 
 ay— =_=:_, 
 
 ac c ac a 
 
 on 
 
 c 
 
 (112) 
 
TWO OR MORE UNKNOWN QUANTITIES. 61' 
 
 SIMPLE EQUATIONS 
 
 CONTAINING MORE THAN TWO UNKNOWN QUANTITIES. 
 
 (172, page 116.) 
 
 r 2ar + 4y— 3^ = 22, (1) 
 
 1. Given, < 4x—2i/ + 5z=l8, (2) 
 
 ( 6x + 1i/- z=6S. (3) 
 
 As the coeflScients of x in (2) and (3) are multiples of the coeiRcient 
 of X in (1), combine (1) with (2), and then with (3), to eliminate x. 
 
 Multiplying (1) by 2, 
 
 4x + S7/—6z=4i, 
 
 (4) 
 
 bringing down (2), 
 
 4x-^2y-\-5z=l8; 
 
 
 by subtraction, 
 
 10y-llz=26; 
 
 (5) 
 
 multiplying (1) by 3, 
 
 6x + 12y— 92 = 66; 
 
 (6) 
 
 bringing down (3), 
 
 6x + 1y-z=e3; 
 
 
 by subtraction, 
 
 5y-8z= 3; 
 
 (7) 
 
 multiplying (7) by 2, 
 
 107/-162= 6; 
 
 (8) 
 
 subtracting (8) from (5), 
 
 5z=20, and z = 
 
 =4; 
 
 substituting value of z in (7), 
 
 5y=35, and y= 
 
 ='7; 
 
 " values of z and y 
 
 'in(l), xz 
 
 =3. 
 
 
 ( 3ar + 9y + 82=41, 
 
 (1) 
 
 2. Given, 
 
 < 5x + 4r/—2z=20, 
 
 (2) 
 
 
 ( lla: + 7y-62=37; 
 
 (3) 
 
 multiplying (2) by 4, 
 
 20x + 16y—8z= 80; 
 
 W 
 
 bringing down (1), 
 
 3ar + 9^4-82= 41; 
 
 
 by addition. 
 
 23ar + 25y=121 ; 
 
 (5) 
 
 multiplying (2) by 3, 
 
 16a;4-12y— 62= 60; 
 
 (6) 
 
 bringing down (3), 
 
 Ux + 1y-6z= 37; 
 
 
 by subtraction, 
 
 4ar + 5y= 23; 
 
 0) 
 
 multiplying (7) by 5, 
 
 20a; + 25y=115; 
 
 (8) 
 
 subtracting (8) from (5), 
 
 dx=z 6, and xz 
 
 = 2; 
 
 whence, by (8), 
 
 2by=1b^ and y- 
 
 =3; 
 
 and by (1), 
 
 82= 8, and z- 
 (116) 
 
 = 1. 
 
62 SIMPLE EQUATIONS. 
 
 ja: + y + z=32, (1) 
 
 3. Given, i x + y—z=25, (2) 
 
 ix^y-z= 9; (3) 
 
 adding (1) and (3), 2ar=40, and a:=20 ; 
 
 subtracting (3) from (2), 2y=16, and y= 8; 
 
 ** (2) " (1), 22= 6, and z= 3. 
 
 rar4-y + 2=26, (1) 
 
 4. Given, "j ^— y =4, (2) 
 
 ( x-z = 6 ; (3) 
 
 adding the three equations, 
 
 
 8ar=36, and ar= 
 
 12; 
 
 substituting value of x in (2), 
 
 
 y= 
 
 8; 
 
 U U U 
 
 (3), 
 
 
 z= 
 
 6. 
 
 
 
 ^■ 
 
 -y-z= 6, 
 
 (1) 
 
 5, Given, 
 
 
 i3y- 
 
 -x-«=12, 
 
 (2) 
 
 
 
 (y^- 
 
 -y— jr=24. 
 
 (3) 
 
 Assume 
 
 
 ar-fy + 2=«; 
 
 
 equation (1) becomes 
 
 
 
 2j:=6 + *, 
 
 w 
 
 - (2) « 
 
 
 
 4y=12+», 
 
 (5) 
 
 " (3) « 
 
 
 
 8z=24 + «. 
 
 (6) 
 
 multiplying (4) by 4, 
 
 8x=24 + 4«; 
 
 (^) 
 
 (5) by 2, 
 
 
 
 8y=24 + 2«; 
 
 (8) 
 
 bringing down (6), 
 
 
 
 8y=24+«; 
 
 
 by addition, 
 
 
 
 8«=72 + '7«; 
 
 
 whence. 
 
 
 
 «=72; 
 
 
 substituting value of s in 
 
 W, 
 
 
 a:=39; 
 
 
 (( u u 
 
 (5), 
 
 
 y=21 ; 
 
 
 tt U (( 
 
 (6), 
 
 
 2=12. 
 
 (1) 
 
 6. Given, 
 
 
 
 \3y=M+a: + 2, 
 j 42=tt+u; + y, 
 ( u=a:-14; 
 
 (2) 
 (3) 
 (4) 
 
 Assuming x + y-\-z-{-u=s^ and adding ar to both sides of (1), y to 
 both sides of (2), and z to both sides of (3), 
 
 (116) 
 
TWO OR MORE UNKNOWN QUANTITIES. 63 
 
 equation (1) becomes 3x=s 
 
 « (2) " 4y=s 
 
 « (3) " 62=5 
 
 7. Given, 
 
 (5) 
 
 (6) 
 (7) 
 
 60a;=20«; 
 
 (9) 
 
 60y=15s; 
 
 (10) 
 
 60z = 12«; 
 
 (11) 
 
 60w = 20s— 840; 
 
 (12) 
 
 " (4) « « = 3-U; (8) 
 
 multiplying (5) by 20, 
 
 " (6) by 15, 
 
 (1) by 12, 
 
 " (8) by 60, 
 
 by addition, 60s = 67s— 840; 
 whence, *=120; 
 
 substituting value of s in (5), a:=40; 
 
 '' (6), y = 30; 
 
 " " " (7), z=24; 
 
 " " " (8), tt = 20. 
 
 ANOTHER METHOD. 
 
 Subtracting (2) from (1), 2x—St/=y—Xy or 3x=4y; (5) 
 
 (2) " (3), 42-3y=y-0, or5«=4y; (6) 
 
 adding (3) and (4), 4z = 2x+y—U\ (1) 
 
 multiplying (7) by 5, 20z=10x + 5y — 10\ (8) 
 
 (6) by 4, 20x=16y; (9) 
 
 by subtraction 10x-lly=70; (10) 
 
 multiplying (10) by 3, 30a:— 33^ = 210; (11) 
 
 (5) by 10, 30ar-40y=:0; (12) 
 
 subtracting (12) from (11), Yy=210, and y=SO ; 
 
 from (5), (6), and (4), ar=40, 2=24, u=26. 
 
 u + 3x-y- z=7, (1) 
 
 2tt — 2a;-f-y + 32=8, (2) 
 
 3u— ar + y — 4z = 8, (3) 
 
 4u+ x^y-2z=1\ (4) 
 
 adding (1) and (2), 3m + a; + 2^=15 ; (5) 
 
 " (1) " (3), 4M + 2a: — 52 = 15; (6) 
 
 " (2) " (4), 6m- x-{- z = 15; (7) 
 
 " (5) " (7), 9m + 32=30; (8) 
 
 u (3) « (4), 7m-62 = 15; (9) 
 
 (116) 
 
u 
 
 SIMPLE EQUATIONS. 
 
 adding (8) and (9), 
 " (8) " (10), 
 
 substituting value of m in (8), 
 
 " values of tt and z in (5), 
 
 ♦* " of w, z and x in (2), 
 
 16m-3z=46; (10) 
 
 26tt='75, andw=3 ; 
 
 z=l; 
 a:=4; 
 
 y=7. 
 
 ( 5x- y + 7z=61, 
 < 4x + dy + 3z = 8, 
 ( Sx— y—5z = S; 
 
 9x—3y— 152=9 ; 
 lSx—l2z = l1 ', 
 2ar + 12z = 58; 
 
 15a; = 75, and a;= 
 substituting value of x in (6), 122=48, and z= 
 
 " values of x and 2 in (1), y= 
 
 8. Given, 
 
 multiplying (3) by 3, 
 adding (4) and (2), 
 subtracting (3) from (1), 
 adding (5) and (6), 
 
 5; 
 
 •8. 
 
 9. Given, 
 
 Adding the equations, 
 
 ■whence, 
 
 subtracting (Y) from (1), 
 (7) « (2) 
 (7) " (3), 
 (7) " (4), 
 (7) " (5), 
 
 u + v+x+z + 2y=50, 
 u + v+y+z+2x=49, 
 u+x+y+z+2v=46, 
 
 6» + 6»+6x + 6y + 62=240; 
 u+v + x+y+z=:iO; 
 2=12; 
 y=10; 
 x= 8; 
 e= 6; 
 u— 4. 
 
 (1) 
 (2) 
 (3) 
 
 (*) 
 (5) 
 (6) 
 
 (1) 
 (2) 
 (3) 
 (4) 
 (5) 
 
 (6) 
 
 ANOTHER METHOD. 
 
 Assuming u + v-\-x-\-i/ + z=s, and subtracting this from each given 
 equation, 
 
 equation (1) becomes 2=52— s; (6) 
 
 , " (2) " y=50-«; (7) 
 
 « (3) " ar=48— »; (8) 
 
 « (4) " v=46~5; (9) 
 
 ** (6) " w=44-s; (10) 
 
 (116-117) 
 
TWO OR MORE UNKN^If' QUANTITIES. ^^'^^ 65 
 
 by addition, s—240 — 5s', (11) 
 
 whence, 5~ 40; ' (12) 
 
 substituting value of s in (6), (7), (8), (9) and (10), 
 
 2=12, y=10, x=8, v = 6, u = 4, Ans. 
 
 2x + i/—2z =40, (1) 
 
 4i/—x-{-3z =35, (2) 
 
 10. Given, ^ 3u + t =13, (3) 
 
 ij + u + t =15, (4) 
 
 3^— y + 3<— M = 49, (5) 
 
 Eliminate t and u first according to practical suggestion (170, 2. 
 
 Adding (4) and (o), 3x + 4t= 64; (6) 
 
 multiplying (3) by 4, 12m + 4<=52; (7) 
 subtracting (7) from (6), 3^;— 12w= 12, 
 
 or, x—4u= 4 ; (8) 
 subtracting (3) from (4), y — 2u= 2, 
 
 or, 2y—4u= 4; (9) 
 
 subtracting (9) from (8), a:— 2y=0, orir=2y; (10) 
 
 from (1) 6jr + 3y— 6^=120; (11) 
 
 from (2), 8y-2.r + 63= 70; (12) 
 
 by addition, 4.r + lly=190; (13) 
 from (10) and (13), 8y + lly=190, or y=10; 
 by substitution in (10), a;=20 ; 
 " (1), z= 6; 
 
 " " " (9), u- 4; 
 
 " ** " (3), tz= 1. 
 
 ( ^+ y- 2=1, (1) 
 
 11. Given, < 8ar + 3y — 6^=1, (2) 
 
 ( 32-4a:- y=l. (3) 
 
 Adding (1) and (3), 2z—Zx—2 ; (4) 
 
 multiplying (1) by 3, 3a: + 3y— 3«=3 ; (5) 
 
 subtracting (2) from (5), 3z— 6a;=2 ; (6) 
 
 (4) " (6), «-2a;=0,orz=2^; (7) 
 substituting value of 2 in (4,) 4a;— 3a;=2, ir=2 ; 
 whence, 5?=4 ; 
 substituting values ofx and 2 in (1), y=3. 
 
 (117) 
 
66 
 
 SIMPLE EQUATIONS. 
 
 12. Given, 
 
 2w + 2x + 2w+ 2 = -3, (1) 
 
 Su + 3x+Sz-h2j/=: 3, (2) 
 
 4u + 4i/ + 4z + Sx= — 2, (3) 
 
 5j;+5y + 52 + 4M= 2. (4) 
 
 Assuming u + x + i/ + z=s^ and adding z to both sides of (1), y to 
 (2), x to (3), and u to (4), 
 
 equation (1) becomes 2*=— 3+^, (5) 
 
 (2) « 35= 3+y, (6) 
 
 (3) " 4«=-2+.r, (7) 
 « (4) " 55= 2 + w; (8) 
 
 by addition, 145=5. (9) 
 
 But (9) can be true only when s=0; hence by substituting this 
 
 value of 5 in (5), (6), (7), and (8), we have 
 
 ^=3, y = -3, 
 x=2y u = — 2. 
 
 13. Given, 
 
 Clearing (1) of fractions, 
 
 « (2) 
 
 « (3) " 
 
 multiplying (4) by 4, 
 subtracting (5) from (7), 
 multiplying (4) by TO, 
 " (6) " 3, 
 
 by subtraction, 
 multiplying (8) by 4, 
 whence, 
 
 2 3^4 
 X y z 
 
 X y , z 
 
 + - + -: 
 
 4 5 6 
 
 :62, 
 
 :47, 
 :38. 
 
 6jr + 4y + 3z: 
 
 20x-\-\by+\2z 
 
 15^+12y + 10z 
 
 24ar + 16y+120 
 
 A:X + y 
 
 60x + 40y4-302 
 
 45j; + 36y + 302 
 
 15a; + 4y 
 
 16a; + 4y 
 
 ar=24, y 
 
 (117) 
 
 (1) 
 (2) 
 (3) 
 
 = 744; (4) 
 
 = 2820; (5) 
 
 = 2280; (6) 
 
 = 2976; (7) 
 
 = 156; (8) 
 
 = 7440; (9) 
 
 = 6840; (10) 
 
 = 600; (11) 
 = 624; 
 = 60, and 2= 120, ^715. 
 
TWO OR MORE 
 
 UNKNOWN QUANTITIES. 
 / 1 1 
 
 
 
 U*r'- 
 
 (1) 
 
 14. Given, 
 
 Ill 
 
 (2) 
 
 
 \ - + -=3; 
 \y z 
 
 (3) 
 
 adding the three equations, 
 
 H^"^ ■ 
 
 (4) 
 
 dividing by 2, 
 
 y,f- 
 
 (5) 
 
 subtracting (1) from (5), 
 
 1=2, or 2=4 ; 
 
 
 " (2) " (5), 
 
 l=l,ory=l; 
 
 
 " (3) " (5), 
 
 -=1, or x=l. 
 
 
 
 lx-\-a=y + z, 
 
 (1) 
 
 15. Given, 
 
 K+a=-ix+2z, 
 
 (2) 
 
 
 {z + a=2x + 3y\ 
 
 (3) 
 
 by transposition, 
 
 X — y — z^= — a ; 
 
 (4) 
 
 
 ~1x+y — iz=—a; 
 
 (5) 
 
 
 —3x—3y + z=—a; 
 
 (6) 
 
 adding (4) and (6), 
 
 —2x—4y=—2a; 
 
 0) 
 
 or, 
 
 -x-2y=-a; 
 
 (8) 
 
 multiplying (4) by 2, 
 
 2x—2y—2z=—2a; 
 
 (9) 
 
 Bubtnicting (5) from (9), 
 
 ix—Sy=—a; 
 
 (10) 
 
 multiplying (8) by 4, 
 
 — ix—8y=—4a ; 
 
 (11) 
 
 adding (10) and (11), 
 
 lly= 5a; 
 
 
 whence. 
 
 y=A«. «=t't«. 2=Vt« 
 
 , Ans. 
 
 16. Assume x-{-y-\-z=s] then the equations become, 
 
 si-z = 2(b + c), (1) 
 
 s + y=2{a + c\ (2) 
 
 s-\-x=2(a-\-b). (3) 
 
 (117) 
 
68 SIMPLE EQUATIONS. 
 
 Add i ng and uniting, 4* = 4 (a 4- 6 + c) ; 
 
 sz=a-\-b-\-c ; 
 by substitution in (3), x=za-\-b—c ) 
 
 " (2), y=a + c-& [jns. 
 
 (1), z=b-\-c-a 
 
 ( U-3i/=a, (1) 
 
 lY. Given, < 5t/—llx=a, (2) 
 
 ( 9y—5z=a; (3) 
 
 multiplying (1) by 5, 35.r— 15y=:5a; (4) 
 
 (2) by 3, — 33jr+15y = 3a; (5) 
 
 adding (4) and (5), ' 2x=Sa] 
 
 whence, ar=4a, y=9a, and 2 = 16a. 
 
 18. Given, 
 
 ^ + y_^_y (I) 
 
 a^b-b a' ^^^ 
 
 Clearing (1) of fractions, bx-\-ay=:ax—bf/] 
 
 or, (a-6)j;-(a + % = 0; (3) 
 
 multiplying (2) by (a+b), (a + b)x + {a + b)7/=-—^ ; (4) 
 
 11. /«x , /.N « ^a'ft ' . 2a6 
 adding (3) and (4), 2ax= -^ and x= r ; 
 
 substituting value of ar in (3), 2a6 — (a + Z>)y = 0, and y=: 7. 
 
 a -}- 
 
 19. Given, 
 
 Clearing (3) of fractions, 
 and transposing, 
 
 ax + by + cz=ab-\-ac-{-bc, (l) 
 
 a'x + bhj + c^z=3abc, (2) 
 
 l-^+.v=f.+ y=2+!r«=o. (3) 
 
 be uc ac ab ^ 
 
 I ax + 2b)/ + cz=2ac-}-ab + bc ; (4) 
 
 ac 
 
 subtracting (1) from (4), by^=ac^ and y=^-r 
 
 substituting value of y in (1), ax-\-cz=iab-\-bc\ (5) 
 
 « " " "(2), a'x4-cV-2a6c; (6) 
 
 (117-118) 
 
TWO OR MORE UNKNOWN QUANTITIES. 69 
 
 multiplying (5) by r, acx+c^z=ahc-i-bc'' ; (7) 
 
 be 
 subtracting (7) from (6), a{a—c)x=bc(a—c)y and x= — ] 
 
 substituting value of x in (5), bc + cz=abi- bc^ and z = — . 
 
 c 
 
 { cx + y + az~2a, (1) 
 
 20 Given, J c*x + y + a^z = 2ac, (2) 
 
 ( acx—y -f ac2=a' + c'. (3) 
 
 Adding (1) and (3), car + «c.r + (/2 + ac2 = 2a + rt' + c* ; (4) 
 
 or, c(l +a)jr + o.(l+c)2=2a + a' + c'; (5) 
 
 adding (2) and (3), c{c-\-a)x-\-a{c-\-a)z=c'^ + 2ac-{-a^^ 
 
 or, dividing by (c + a), cx + az:=c + a\ (6) 
 
 multiplying (6) by (1+c), c{\-\-c)x-{-a(\-\-c)z=c + a + c' + ac\ (7) 
 
 subtracting (7) from (5), c{a—c)x=.a'^ —ac ■\-a—c=::{a'-c){a + 1) ; (8) 
 whence, x=a-{-\ ; 
 
 substituting in (6), i 
 
 " (1), y=^a-c 
 
 a ' 
 
 r a'x + ay +a2 =«, (1) 
 
 21. Given, I ax-^a''y + az =a\ (2) 
 
 ( ax -{-ay +a'«=a'. (3) 
 
 Dividing each equation by a, aar + y + r = 1 ; (4) 
 
 x-\-ay-{-z=a] (5) 
 
 ir + y + a2 = a'; (6) 
 subtracting (5) from (4), (a — l)x + {l—a)y = l—a^ 
 
 ora-— y= — 1; (7) 
 (6) - (2), (a-l).r4-(a'-l)y==0; 
 
 orx+{a-hl)y=0] (8) 
 
 ^ " (7) « (8), (a + 2)y=l; or y: ^ 
 
 substituting in (7), 
 
 a + 2 
 
 + 2 a + 2 
 
 whence, .= (^, 
 
 ' a + 2 
 
 (U8) 
 
70 
 
 SIMPLE EQUATIONS. 
 
 PROBLEMS 
 
 PRODUCING EQUATIONS CONTAINING TWO OR MORE UNKNOWN 
 QUANTITIES. 
 
 (174, page 119.) 
 
 1. Let a:= the first number, and 3/= the second; then by the 
 conditions of the problem, we have 
 
 2jr + 3y=105, (1) 
 
 3i: + 2y= 95; (2) 
 
 whence, by elimination, ar=15, and y=25, Ans. 
 
 By adding the equations, and taking | of their sum from the first 
 equation, we have the value of y ; taking the same from (2), we 
 have X, 
 
 2. Let x=z the first, y= the second, 2= the third. 
 
 By the conditions, 
 
 clearing (1) of fractions, 
 subtracting (3) from (4), 
 clearing (2) of fractions, 
 subtracting (3) from (5), 
 whence, 
 
 :, + r:__ = 120, (1) 
 
 y+'-i^= 90, (2) 
 
 « + ar + y = 190; (3) 
 
 22; + y + «=240; (4) 
 
 xz=z 50; 
 5i/+x-hz = 450; (5) 
 
 4yr=260; 
 y=^65, and 2=75. 
 
 3. Let x=z A's share, y= B's, and z= C*s. 
 
 By the conditions, 
 
 x-^(y + z) = l20=a, (1) 
 
 y-^(^ + ^) = 120=a, (2) 
 
 z-^{i/ + x) = l20=a; (3) 
 (119) 
 
TWO OR MORE UNKNOWN QUANTITIES. 71 
 
 clearing of fractions, 1x—4:y—4z=^a, (4) 
 
 Sx-{-8i/ — 3z = 8a, (5) 
 
 ^2x—2y-[-9z=9a', (6) 
 
 multiplying (4) by (2), l4x — 8]/-8z—Ua; (1) 
 
 adding (7) and (5), Ux—llz = 22a,orx—z=2a'^ (8) 
 
 multiplying (6) by 2, — 4j~4y + 182 = 18a ; (9) 
 
 subtracting (4) from (9), ^llx-^22z=zlla^OT —x + 2z=a] (10) 
 adding (10) and (8), 2=:3a=360, 
 whence, a;=:600, and y=480. 
 
 4. Let x= A's daily wages, y= B's, and z=: C's. 
 
 I 6(x + y)=40, 
 
 x + y=6l; 
 
 (1) 
 
 By the conditions, < 9 (a; + «) = 54, 
 
 x+z=6; 
 
 (2) 
 
 ( 15(y + 2) = 80, 
 
 y+2=5i-. 
 
 (3) 
 
 One half the sum of the equations is 
 
 x+y+z=9; 
 
 (*) 
 
 subtracting (3) from (4), 
 
 ^=3|; 
 
 
 (2) " (4), 
 
 y=3; 
 
 
 (1) " (4), 
 
 z=2i. 
 
 
 5. Let X, y, z and u represent their ages respectively. 
 
 ix + y + z=18, (1) 
 
 a; + y-fu=16, (2) 
 
 x + z -\-u = 14^ (3) 
 
 y + «4-w = 12. (4) 
 
 One third the sum of the equations is, a; + y + « + M = 20; (5) 
 subtracting (4) from (5), ar= 8 ; 
 
 " . (3) " " y= 6; 
 
 « (2) " " z= 4; 
 
 it (1) « « u= 2. 
 
 6. Let x=: A's shillings, y= B's, and 0= C's ; then by the con- 
 ditions of the problem, after the first game each will have as follows : 
 
 x—y—z— A's shilHngs, 
 2y= B's " 
 2z=C's " 
 (119) 
 
72 SIMPLE EQUATIONS. 
 
 After the secofnd game, 
 
 2x— 2y — 22= A's shillings, 
 2i/ — {x—y—z) — 2z = 3t/—x—z=B's " 
 
 4z= C's 
 After the third game, 
 
 
 4ar~4y— 42=16, 
 
 (1) 
 
 
 6y-2i:-2z = 16, 
 
 (2) 
 
 
 72_ar-y = l6; 
 
 (3) 
 
 Adding the equations, 
 
 x + i/ + z = 4S] 
 
 (4) 
 
 " (3) and (4), 
 
 8z=64, and z= 8 ; 
 
 
 " iof(l)to(4), 
 
 2a:=52, and x=26 ; 
 
 
 substituting in (4), 
 
 y=u. 
 
 
 T. The same equations as in Ex. 6th, page 116. 
 
 8. Let ar= the value of the better horse, and y= the value of 
 the poorer. ^ 
 
 -fMo =:/r;rv?-7-<-)/7^,_/^-^,^^-U+15 = -(y + 10), (1) 
 
 By the conditions, } 
 
 ^^:t:;:^u («+io=i5(y+i5); (2) 
 
 Jy^^ i,i-^~\i = ^-^ 15 
 
 adding 5 to both sides of (2), a: + 15 = --(y + 15) + 5 ; (3) 
 
 lo 
 
 co.npan„g^(l) and (3), J 4^^^,„)^15(^^^,)^, . ^^^ 
 
 clearing of fractions, 52y + 520=45y + 675 + 195 ; (5) 
 
 whence, 7y=350, and y=50, poorer horse ; 
 
 and by substitution in (1), a; =65, better horse, 
 
 9. Let ar= the price of a dozen of sherry, 
 
 y= " " " " brandy. Put a=r78. 
 
 By the conditions, j f ^o^^f^ o ^1 
 
 ^ ' ( 7a: + 2y = 9a + 9; (2) 
 
 multiplying (1) by 2, 4:c+2y = 6a ; (3) 
 
 subtracting (3) from (2), 3jr=3a4-9, 
 
 or, . ar=a + 3 = 81, sherry ; 
 
 whence, from (1), y=a— 6 = 72, brandy, 
 
 jz^ - > -/, (119-120) 
 
 J V ^- ^ 4 J 
 
TWO OR MORE UNKNOWN QUANTITIES. 7Sl 
 
 10. Let x= the time in which A can do the work, 
 -.__ « ti *' B " " " 
 
 then -= the part of the work A can do in one day, 
 
 u u u J5 « « u 
 
 y 
 
 In four days, both working together would do \ of the work. 
 
 Therefore. 
 JL ^ ~l ^~L 1 36 3 12 1 
 
 whence, / r" , y = 48, and ;r=24, ^«*. 
 
 11. Let a;= numerator, and y= denominator. 
 
 2i: 2 
 
 "^-f^ /4 4 1 
 
 By conditions. 
 
 y + 7-3' 
 
 x^2 3 
 
 2y ~5 
 
 (1) 
 
 ; (2) 
 
 clearing of fractions, and transposing, %x—2y=z 
 
 bx—Qyz=i- 
 multiplying (3) by 3, 18^— 6y = 
 subtracting (4) from (5), 13j;= 
 whence, {c= 
 Hence the fraction is 
 
 14, (3) 
 -10; (4) 
 42; (5) 
 62, 
 
 4, y=b. 
 1, Ans, 
 
 12. Let x=A'8 money, and 
 
 y=B'8, 
 
 
 1 
 
 By the conditions, \ 
 
 ^ + |^=240=a, 
 ^x 
 
 (1) 
 
 { 
 
 y +— =240=:a; 
 
 (2) 
 
 clearing of fractions, 
 
 subtracting (3) from (4), 
 substituting in (3), 
 
 3x + 2y = 3a, 
 3a; + 4y=:4a ; 
 2y=a, y= 
 3i: + a = 3a, x= 
 (120) 
 
 (3) 
 (4) 
 
 = 120} 
 
 = 160, 
 
74 SIMPLE EQUATIONS. 
 
 13. Let rr= the number of persons, and y=. what each paid ; 
 then xy= the amount of the bill. 
 
 By conditions, 
 
 
 C(^ + 4)(y-l) = a-y; 
 
 (1) 
 
 
 
 (2) 
 
 expanding (1), 
 
 
 XyJ^\y—X — \=iXy, 
 
 (3) 
 
 " (2), 
 
 
 xy — Zy^x—Z—xy\ 
 
 (*) 
 
 dropping xy 
 
 
 4y— a;— 4 = 0, 
 
 (5) 
 
 
 
 -3y + ar-3r=:0; 
 
 (6) 
 
 adding (5) and 
 
 (6), 
 
 y-7 = 0; 
 
 
 whence, 
 
 
 y=i7, a: = 24. Arts, 
 
 14. Let x~ the digit in the place of tens, and y= the digit in 
 the place of units ; then 10i: + y will represent the number. 
 
 ■KT 1 . ,. . ( 10x + y = 4x4-4y, (1) 
 
 Now by t.ie conditions, ^ _ ^ , ^ /^( 
 
 ^ ' ( 10a: + y + 27=a:+10y;(2) 
 
 transposing and reducing (1), 2x—y\ (3) 
 
 (2), y-^=3; (4) 
 
 adding (3) and (4), ar==3, whence, y—^. 
 
 Hence, 36, Ans, 
 
 15. Let X represent the number of hundreds, y the number of 
 tens, and z the number of units ; then lOO^ + lOy + z will represent 
 the number. 
 
 r x + y+2-11, (1) 
 
 By the conditions, \ z=2x, (2) 
 
 ( 100xi-l0y + z-\-29l=x + l0y + 100z; (3) 
 
 transposing and re- ) _ q . {a\ 
 
 ducing (3), 3 
 
 substituting value ) 9 _ _o . 
 
 of z in (4), J 
 
 whence, x=Sj z = Gj and y=2. 
 
 Hence, 326, Ans. 
 
 16. Let Xj y, and z represent the parts. 
 
 (x-\-y-\-z=90, (1) 
 
 By the conditions, < 2a; + 40 = 3y + 20, (2) 
 
 ( 40-|-lO=:2a; + 4O; (3) 
 
 (120-121) 
 
TWO OR MORE UNKNOWN QUANTITIES. 75 
 
 from(2> y= — ^ — ; 
 
 2ar + 30 
 " (3) ^=—4—' 
 
 , . . . ,,v 2ar+20 2i? + 30 ^ 
 by substitution m (1), x-\ f- — =90, 
 
 ar=35, first part, 
 y=30, second " 
 2=25, third ** 
 
 ANOTHER METHOD. 
 
 By the conditions, twice the first part plus 40, three times the 
 second part plus 20, and four times the third part plus 10, are all 
 equal to the same number, which we may represent by x ; then 
 
 a:— 40 
 
 — - — z=z the first part, 
 
 ^—20 ,, , „ 
 
 = " secofi'l *' 
 
 3 
 
 and ^j^^ „ ^j^.^^ ,, 
 
 4 
 
 ^ ^ a:— 40 x—20 x—\0 ^^ .. 
 
 Therefore, -y- + — ^ + -^ = 90, (1) 
 
 whenccy a;=110. 
 
 — - — = 35, = 30, and — - — = 25, Ans. 
 
 17. Let x= the part at 5 per cent, and y the part at 4 per cent. 
 Put $100,000=0, 4640=6. 
 
 I ar + y=a, (1) 
 
 By the conditions, -j 5a; ^V _j^, /n\ 
 
 ( ioo'*"ioo~ ' ^ ' 
 
 from (1), y=za-~x\ 
 
 5x 4ff 4aj 
 
 " (2), by substitution, i5o"^"Tocr=*' 
 
 whence, a;=1006 — 4a=$64000 ) ^^^ 
 
 and y= a— x=$36000 J 
 
 (121) 
 
i6 
 
 SIMPLE EQUATIONS. 
 
 18. To avoid the labor of using large numbers in the operation, 
 put a = 5000; then 2a = 10000, 3a =16000, 
 
 10 
 
 lOr/ 
 1500, aud ^-- = 800. 
 
 Let x=: the principal of the first person, and y= the rate. 
 
 ar+2a= 
 and x-\-3a= 
 
 By conditions, 
 
 clearing of fractions ) 
 and dropping ary, j 
 multiplying (3) by 2, 
 subtracting (4) from (5), 
 substituting in (3), 
 
 Hence, 
 
 " second " y+l= " 
 
 " third " y + 2= " 
 
 ^y ry4-2ay4-a: + 2a 16a 
 
 iob~ 100 Too* 
 
 ary a-y + 3a t/ + 2.r + 6a 3« 
 
 iob~ Too ~~"" 10 
 
 = 2</y4- ar + 2a — 16a 
 
 = 3ay4-2a; + 6a— 30a; 
 = 4ay-f 2jr + 4a — 32a ; 
 at/ — 4a^ and y=4 ; 
 2:= 6a =$30000. 
 
 (1) 
 
 (3) 
 (4) 
 (5) 
 
 Principals, $30,000, $40,000, $45,000 ; 
 
 Ratt 
 
 4, 
 
 5, 
 
 6, per cent* 
 
 19. Let T, y and ar represent their respective ages. 
 
 By the conditions, 
 
 adding (1) and (3), 
 substituting in (2), 
 (3), 
 subtracting twice (5) from (4), 
 whence, 
 
 5y + 2z— ar=147, 
 x-\-y-\-z= 96; 
 
 2a:= 96, and x 
 6y + 22=195 
 y + Z= 48 
 3y= 99 
 y= 33, 2;=15. 
 
 48; 
 
 0) 
 
 (2) 
 
 (3) 
 
 (4) 
 (5) 
 
 20. Let x= what A is worth, y= what B is worth, and z= what 
 C is wortb ; also let s— what they all are worth. Put 100=a. 
 
 
 
 
 ix-\-3y + dz=4la, Y ^<>^ 
 
 (1) 
 
 By the conditions. 
 
 
 
 K y+4a; + 4z=58a, y^<^ 
 ( 2 + 5a; + 5y = 63a; ^S^^ 
 
 (2) 
 
 (3) 
 
 adding 2x to (1), 
 
 
 
 ds= 47a + 22; : 
 
 (4) 
 
 " 3y to (2), 
 
 
 
 48= 58aH-3y; 
 
 (S) 
 
 " 42 to (3), 
 
 
 
 5s= 63a + 45?; 
 (121-122) 
 
 (6) 
 
 4y ^/2 y ^/Z c^ - / 
 
 /' r 
 
 e^-t? 
 
 (jj ZOyL -^2J>^ ^ 4 SL ^ -2- y 
 
 -2 ^ 
 
 
 ?^ 
 
 1^0 
 
 
TWO OR MORE UNKNOWN QUANTITIES. 77 
 
 multiplying (4) by 6, 
 
 
 18s=282a + 12.r; (1) 
 
 (5) by 4, 
 
 
 16s = 232a + r2y; (8^ 
 
 (6) by 3, 
 
 
 i5s=zl89a-\-12z', (9) 
 
 by addition, 
 
 
 49s = 703rt + 12s; 
 
 whence. 
 
 
 s= 19a; 
 
 substituting value of s 
 
 in (4), 
 
 2xz= 10a, x=5a = 500 ; 
 
 U U t( 
 
 (5), 
 
 3y= 18a, y=:6a = 600; 
 
 tt tt u 
 
 (6), 
 
 4z= 32a, 2 = 8a = 800. 
 
 21. Let x=z the cost of a pound of tea, t/— the cost of a pound 
 
 of coffee; then oOx= the whole cost of the tea, and 30?/= the whole 
 
 cost of the coftee. Ilis gain in selling the tea is y'^ of its cost, or 
 
 5x ; his gain in selling the coffee is \ of its cost, or 6y. 
 
 ^> -^ ^Oy-J.4^jy^ ( 5i:+ 6y= 2.90, (1) 
 
 Bf the conditions, 1 .- T „« «k . J /«( 
 
 ^^j -i^y- U5a;+36.y = 27.40; (2) 
 
 subtracting 6 times (1) from (2), 25x= 10.00 ; 
 
 whence, ar=$.40, and y = $.15, Ans. 
 
 22. Let a*, y, ar, w and v respectively represent their money. 
 
 Then by the conditions, 
 
 substituting value of v in (4), 
 ** " ti in (3), 
 
 ** " z in (2), 
 
 z+|=30, 
 
 
 (1) 
 
 M = 30. 
 
 
 (2) 
 
 |-i = -. 
 
 
 (3) 
 
 Su V 
 
 
 {*) 
 
 5v „^ ^ 
 
 -—=30, whence v= 
 6 
 
 = 36; 
 
 (5) 
 
 ?^=24, " « = 
 
 =32; 
 
 
 1=2. .. .= 
 
 = 33; 
 
 
 1=19, " ,-. 
 
 =38; 
 
 . 
 
 19=30, " a:= 
 
 = 11. 
 
 
 122) 
 
78 SIMPLE EQUATIONS. 
 
 23. Let x^ y and z represent respectively their money. Put 
 a=1000. 
 
 
 / ar + |=2000= 
 
 =2«, 
 
 (1) 
 
 J5y the conditions. 
 
 / y + ^ = 2000 = 
 ) 
 
 = 2a, 
 
 (2) 
 
 
 ( 2r 4-^ = 2000 = 
 
 = 2a; 
 
 (3) 
 
 
 i 2x+y=. 4a, 
 
 
 (4) 
 
 clearing of fractions, 
 
 ^ 3y + 2= 6ff, 
 
 
 (5) 
 
 . 
 
 (42+ar= 8a; 
 
 
 (6) 
 
 multiplying (5) by 4, 
 
 12y + 42=24a; 
 
 
 (7) 
 
 subtracting (6) from (7), 
 
 \2y — x=\Qa\ 
 
 
 (8) 
 
 " (8) from 12 times (4), 25a:=32a; 
 
 32a 
 whence, ar=— — ; 
 
 Zo 
 
 or, a:=1280; 
 
 by substitution, y=1440; 
 
 and z=1680. 
 
 24. Let ar= the hourly rate of the first courier, and y= the hourly 
 rate of the second courier. Now the distance divided by the rate 
 will be the time ; hence, 
 Z _ -/ - hi - J^ 147 147 
 
 ^ .L/. £p ^ y 
 
 These equations are the same as those in Ex. 19, page 111, and may 
 be solved in a similar manner. 
 
 y ^ 3 
 
 -7 
 
 25. Let x=. the greater, and y= the less. 
 By conditions. 
 
 X y 
 
 2+1=13, (1) 
 
 (122) ^ ^2^^-/3 
 
 J 7 
 
TWO OR MORE UNKNOWN QUANTITIES. 79 
 
 3^ 
 
 multiplying (1) by 3, 
 
 
 y+y=39; 
 
 (3) 
 
 (2) by 2, 
 
 
 2-e 
 
 (*) 
 
 by addition, 
 
 
 13' o« 
 6 =««' 
 
 
 ■whence, 
 
 
 ir=:18, and y = 12. 
 
 , Ans. 
 
 26. Let X— the first, 
 
 y=1 
 
 the second, and 2= third. 
 
 
 
 ( 
 
 ^ + My+«) = 51=flr, 
 
 (1) 
 
 By conditions, 
 
 
 y + i(a?+2) = 51=a. 
 
 (2) 
 
 
 ( 
 
 z^\{x^y) = h\=a', 
 
 (3) 
 
 
 ( 
 
 a^ + (^+y + e) =2a. 
 
 (4) 
 
 clearing of fractions, 
 
 
 2y+ (a; + y + 2r)=3a. 
 
 (5) 
 
 
 ( 
 
 3z + (a: + y + 2) =4a. 
 
 (6) 
 
 Let (ar + y4-«)=s, anc 
 
 I multiply (4) by 6, (5) by 3, and 
 
 (6) by : 
 
 equation (4) becomes 
 
 
 6ar + 6s = 12a; 
 
 (7) 
 
 " (5) " 
 
 
 6y + 35= 9a; 
 
 (8) 
 
 (6) " 
 
 
 6z+2s= 8a; 
 
 (9) 
 
 by addition, 
 
 
 65+ll«=29a, or«=:87. 
 
 
 substituting value of s in 
 
 'W. 
 
 ir=2a — «, or ar=15; 
 
 
 (( U (( 
 
 (5). 
 
 2y=3a— 5, or ^=33 ; 
 
 
 ti H M 
 
 (6), 
 
 3r=4a — 5, or 2 = 39. 
 
 
 2*7. Let a;= A's, y= B's, and z= C's sheep. 
 
 rar + 8-4=y + 2-8, (1) 
 
 3y the conditions, •< i(y + 8)=a; + 2 — 8, (2) 
 
 ( M^ + 8)=^+y-8; (3) 
 
 clearing of fractions and uniting, a; — y - « = — 1 2, (4) 
 
 — 2a: + y — 20=— 24, (5) 
 
 — 3a;— 3y + 2-— — 32; (6) 
 adding (4) and (6), _2a;-4y=-44; (7) 
 or a; + 2y = 22; (8) 
 subtracting twice (4) from (6), — 4a; + 3y=0; (9) 
 adding four times (8) to (9), lly=:88 ; 
 
 vhence, y— 8, a:= 6, and 2=10. 
 
 (122-123) 
 
80 SIMPLE EQUATIONS. 
 
 28. Let - represent the fractions. 
 
 By the conditions, 
 
 clearing of fractions, <fec., 
 
 whence, by subtraction, 
 consequently the fraction is 
 
 29. Let - represent the fraction. 
 
 /x + l_ 
 
 ) ^ 
 
 1 
 
 "3' 
 
 
 
 (1) 
 
 I X 
 
 1 
 ~4^ 
 
 
 
 (2) 
 
 3z—y= 
 4x^y= 
 
 X- 
 
 = -3; 
 
 = 1; 
 
 = 4, 
 
 and y- 
 
 = 15; 
 
 (3) 
 (4) 
 
 Ans, 
 
 ar4-2 5 
 
 By tho conditions. 
 
 y 7 
 
 X 1 
 
 (1) 
 (2) 
 
 y+2 3' 
 
 clearing of fractions, <kc., Yj:— 5y=: — 14; (3) 
 
 Zx-y= 2 ; (4) 
 
 subtracting (3) from five times (4), 8i:= 24, whence ar=3 ; (5) 
 substituting in (4), y=^^\ 
 hence the fraction is, ^, Am. 
 
 30. Let .r, y, z, and u represent the parts of an acre which the 
 men respectively can mow in one hour. 
 
 iar + 3y+ 22+ 2w = l, (1) 
 
 3a: + 2y+ 42 + 11m = 2, (2) 
 
 5a; + 4y + 12z+ 5m = 3, (3) 
 
 9-r + 7y+ 6z+ 8if = 4, (4) 
 
 subtracting twice (I) from (2), x— 4y-\- 1u=z 0; (6) 
 
 " sixtimes(l)from(3), — a;— 14y— 1u = —3; (6) 
 
 adding (5) and (6), 18y=3, whence y=l ; 
 
 subtracting (4) from three ) 2m--1 • ^7^ 
 
 times (1), \ -6x + 2y-2u- 1, (7) 
 
 substituting the value of ) ar— 1 + 7?^= 0; (8) 
 
 y in (5) and (7), f -6ar + i-27^ = -l ; (9) 
 
 (123) 
 
TWO OR MORE UNKNOWN QUANTITIES. 81 
 
 transposing, and multiply- ) , ^ 
 
 ing(8)by6, 3 ^ \ ) 
 
 adding (10) and (9), 40u = ^^ w=tV» 
 
 by substitution, ^=h ^^^ ^— ti* 
 
 Hence, A can mow an acre in 5 hours; B in 6 hours; C in 12 
 hours ; and D in 15 hours. 
 
 (1) 
 
 (2) 
 
 31. Let x= A's money, and i/=Ws. 
 
 By the conditions, ] """f "!^/ "^ ^} 
 
 ^ * (a: + 5 = 3(y-5); 
 
 V 5 
 by subtraction, 10 = 3y— 15— -— -; (3) 
 
 Z 2 
 
 whence, 20==5y— 35, and .y=:ll ; 
 
 by substitution, a:=:13. 
 
 32. Let x= the number of bushels of wheat flour, 
 and y= " " " " barley " 
 Then l0x + 4i/= the cost of the whole, 
 
 and ll^ + lly= the value of the sale at 11 shillings per bushel. 
 
 Now, Tn — Inh ' *"^ ^^ *^^® conditions, 
 
 fii(10x + 4y) = ll.r + lly, (1) 
 
 or, 6'750ar+2300y=:4400^ + 4400y, (2) 
 
 or, 1350.c = 2100y, 
 
 or, 9j: = 14y. 
 
 Converting the equation into a proportion, by placing the factors 
 of one member for the extremes, and the factors of the other for 
 the means, we have 
 
 ir : y : : 14 : 9 
 or, wheat : barley : : 14 : 9, Ans, 
 
 33. Let X represent the number of tens, and y the number of 
 units; then the number expressed in units, must be 10^ + y. 
 
 By the conditions. 
 
 (123-124) 
 
82 SIMPLE EQUATIONS. 
 
 Reducing (1), we have y=l» 
 
 (2), " " x=4. 
 
 The number, therefore, is 41, Ans. 
 
 34. Let x=z the seniors, y= the juniors, z= the sophomores, and 
 u= the freshmen. Place 119=a. 
 
 Now by the conditions of the problem, we have 
 
 *+!=-. (1) 
 
 y+l^a, 
 
 (2) 
 
 ^ + ^=«, (3) 
 
 u+l=a; 
 
 (*) 
 
 subtracting (2) from twice (1), 
 
 2«-|=«; 
 
 (6) 
 
 adding (3) to three times (5), 
 
 6j!+|=4o; 
 
 («) 
 
 subtracting (4) from four times (6), 
 
 24j;-|=15a; 
 5 
 
 
 or, 
 
 119z=75o; 
 
 
 whence, a;=75, y=88, z= 
 
 =03, and u=104. 
 
 
 35. Let Xy y, «, and u represent the numbers respectively. 
 
 / 3^ + y=:a=359, (1) 
 
 By the conditions, ) ^^ "*" "^ =''' (^) 
 
 J 52 + «— a, (8) 
 
 \6u+x—a] (4) 
 
 subtracting (2) from four times (1), l2x—z = Sa ; (6) 
 
 adding (3) to five times (5), 60x+u — l6a; (6) 
 
 subtracting (4) from six times (6), 359.e= 95a ; 
 whence, 
 
 a:=95, y=:V4, 2=63, and w=44, 
 
 (124) 
 
GENERAL PROBLEMS. 83 
 
 GENERAL SOLUTION OF PROBLEMS. 
 (179, page 126.) 
 1. Let xz=. the greater part, and y— the less, 
 conditions, \ , 
 
 (1) 
 (2) 
 
 transposing (2), a:— y=6—a; (3) 
 
 n + 6 — a 
 adding (I) and (3), 2a;=« + 6— cr, ar=- ; 
 
 subtracting (3) from (1), 2y=n + «— o, y— ■ 
 
 2 
 84 4-58-16 
 2. Substituting the numerals, 
 
 2 =^^' 
 
 84 + 16-58 „, 
 
 3. Let a-, y, and « represent 
 
 the numbers. 
 
 
 /ar + y+2=5, (1) 
 
 By the conditions, 
 
 ^ :r=:y-«, (2) 
 
 ( y-^-&, (3) 
 
 subtracting (3) from (2), 
 
 a;-2y + «=6-a; (4) 
 
 " (4) « (1), 
 
 . s-Va — h 
 3y_s + a 6, ory=: — ^ ; 
 
 pubstituting in (2), 
 
 s-\-a — b s—2a—h 
 
 X- g a, or ^_ ^ 
 
 «{3) 
 
 54-0 — & , s + a4-26 
 2- ^ +6, org- . 
 
 4. Let ar= my indebtedness to A ; 
 
 then nxzz, « " B ; 
 
 and m.r= « *< C. 
 
 Now by the conditions, x-\-nx-^mxr=^a\ 
 
 whence, ^—r-. — ; — ^ Ans, 
 
 ' 1 + w + m ' 
 
 (128) 
 
84 SIMPLE EQUATIONS. 
 
 6. Substituting the numerals, x=- — - — -=$131, Ans, 
 
 1 +24-3 
 
 6. Let xz= the number of days he was idle, and 
 
 a—x= " " " worked; 
 
 then (a—x)b=: the money due for his labor, and 
 
 cx= " " he forfeited. 
 Now by the conditions, we have 
 
 (a—z)b—cx=d ; 
 
 whence, x= ~ , Ans. 
 
 o-\-c 
 
 ANOTHER METHOD. 
 
 6. Let x= the number of days he was idle. He would have re- 
 ceived ab cents, had he worked all the time. Every day he was 
 idle he lost his wages, 6, and forfeited c cents ; his whole loss, there- 
 fore, for the idle days, was (6 + c)x. ITierefore, 
 
 ab—{b-i-e)xz=d; 
 whence, ^~~h ' '^^^' 
 
 v. Let ar= the value of the saddle, and nx= the value of th« 
 horse ; then 
 
 (» + l)x=a; 
 
 whence, x= -, and nx= -, Ans. 
 
 n+1 »+l 
 
 8. Let x= the rent last year; then by the conditions, 
 
 nx 
 ^ + 100='' = 
 
 100a 
 whence, '=100+;i' ^"'• 
 
 (126) 
 
GENERAL PROBLEMS. 85^ 
 
 9. Let X represent his income ; then by the conditions, 
 
 X X 
 
 Ux=2](a-^b), 
 whence, x= — ^— — -, Aru, 
 
 10. Let X represent his income ; then by conditions, 
 
 X X 
 
 — + a-\-~-hb=x; 
 m n 
 
 whence, nx-[-nix-\-nin{a-\-h=mnx)^ 
 
 x(mn—m—n)=mn(a-{-b)^ 
 
 mn — m — n 
 
 11. Let ar= the number of days required when all work together. 
 Since A can do the work in a days, in one day he will do a part of 
 
 the whole work expressed by -, and in x days he will do a part of 
 the work expressed by -. Reasoning similarly, in x days B will do 
 
 X 
 
 a part of the work expressed by y, and C a part of the work ex- 
 pressed by -. Therefore, 
 
 XXX 
 
 -+t4— =1; 
 
 a b c 
 
 whence, x{ab + ac + be) = a5c, 
 
 abc . 
 
 x=— — Ans. 
 
 ab-{-ac + bc 
 
 6x8x12 5^6 
 
 12. Substituting the numerals, x=-— — -— — —-=——=2*^, Arts. 
 
 ^ * 48 + 72 + 96 216 '' 
 
 13. Let X represent the number ; then by the conditions, 
 
 ax—c^bx-{-d ; 
 
 whence, (a—b)x=c + dj 
 
 c-hd . 
 
 x= r» ^'**» 
 
 a—b 
 
 (126-127) 
 
86 ' SIMPLE EQUATIONS. 
 
 14. Let x= the number of bushels of oats, 
 y= " " " . peas. 
 
 Then by the conditions, i , ^ )Jk 
 
 ^ \ax-hby=cd\ (2) 
 
 subtracting b times (l)from (2), (a — b)x=c{d—b) ; (3) 
 
 c(d-b) 
 
 whence, x=z-^ r-^,oats; 
 
 a — 
 
 subtracting (2) from a times (l), (a—h)t/=c(a — d) ; 
 whence, y = -^ ^ , peas 
 
 15. Let x= the number of nuts. The party of a boys lost bm 
 nuts by the snatching, and the party of b boys lost am nuts. The 
 party of a boys had x—bni nuts U) ha divided, and each would have 
 
 . The party of b boys had x — ani nuts to be divided, and 
 
 each would have — i — . Therefore, 
 
 
 
 X — bm X — atn 
 T'""~b~' 
 whence, (6 — a)xz=.m{b'* — a'). 
 
 mib'-a") , ,, , 
 
 x=-^. '=m{a-\-b\ Ans. 
 
 b—a ^ ' 
 
 16. Let a;, y, sr, and u respectively represent the numbers. 
 
 ax-^yz=zm,, (1) 
 
 ■r. , ,. . J by-\-z=.m^ (2) 
 
 By the conditions, / \ [ 
 
 ^ cz ■\-uz=m, (3) 
 
 du-{-xz=zm. (4) 
 
 Subtracting (2) from b times (1), abx—z=im(b — 1) ; (5) 
 
 adding (3) to c times (5), abcx-i-u=m(bc~c + l) ; (6) 
 
 subtracting (4) from c? times (6), abcdx—x=m(bcd—cd + d—l); (1) 
 
 , m(bcd — cd + d—1) 
 
 whence, x——^ -— ^; 
 
 abed— I 
 
 (127) 
 
GENERAL PROBLEMS. 8?- 
 
 substituting in (4), du=m ^^ will ' 
 
 mi abed — bed -\-cd — d) 
 
 ''"^ ^cirri • 
 
 _ m(abc-hc + c—\) _ 
 
 ... , - . / X m(abc — bc-\-c — \) 
 substituting value oiu in (3), cz=m ^ j~- — , 
 
 miabcd—abc ^-bc—c) 
 or, cz= —^ ^ • 
 
 abed — 1 
 
 m(abd — ab + b — 1) 
 dmdingbye, ^= ^^— ^ ; 
 
 substituting in (2), 
 
 m(acd—ad-\-a — 1) 
 abed — 1 
 
 17. Let a;= the number of pupils ; 
 then ax — c= the whole attendance in one school, 
 and bx—d=: " " " the other school. 
 
 na= the number of days' attendance for which A pays, 
 mb= " " " " B " 
 
 Now either patron will pay such a part of the whole money raised 
 in his school, as his number of days' attendance is part of the whole 
 attendance. Therefore, by the conditions, 
 
 mb na 
 
 bx—d ax—c ' 
 whence, abmx — cbm=^anbx — adn\ 
 
 or, x{ab) (w — n) = bcm — adn ; 
 
 bcm — adn 
 
 ab(m — w) 
 
 Ans, 
 
 18. Let the several parts be represented by a?, ax, a'ar, and a'x 
 respectively. Then by addition, 
 
 a*x-{-a*x + ax-^xz=ni ; 
 
 whence, x = -r =- — — - , Ans, 
 
 (127-128) 
 
0) 
 
 (2) 
 
 88^. SIMPLE EQUATIONS. 
 
 19. Let X and y represent the numbers. 
 By the conditions, i ^~^| 
 
 by addition, 2j:=:5 + rf, and x— ; 
 
 2 
 
 by subtraction, 2y=«— rf, and y= . 
 
 20. Let a-, y and ar represent the numbers. 
 
 rar + yzra, (1) 
 
 By the conditions, < ar + «=6, (2) 
 
 (y+2=c; (3) 
 
 subtracting (3) from the sum of (1) and (2), 1x-=ia 4- 6— c, 
 
 a4-6— c 
 
 (2) " « (l)and(3), 2y=a + c-6, 
 
 y= 
 
 2 
 
 " (1) " " (2) and (3), 2z=6 + c-a, 
 
 6-f-c— a 
 
 21, Let «= the number of tens ; 
 and y= " " units; 
 
 then 10x-f-y= the number of units in the given number. 
 
 By the conditions, ] J^^+2^ ^t'/"''* ^}1 
 
 ^ ( 10x + y-l-c = 10y + ar; (2) 
 
 UlO-a)i: + (l-a)y = 0, (3) 
 
 ( 9a;-9y=-c; (4) 
 
 multiplying (3) by 9, 9(10— a)a;4- 9(1 —a)y=0 ; (5) 
 
 " (4)by(l-a), 9(l-a)ar-9(l-a)y=ac-c; (6) 
 
 by addition, 9(11 — 2a)ar=c(a— 1), 
 
 whence, '= 9(n-2a) ' *"°°' 
 
 substituting in (4), -^j — — ^+c=9y, 
 
 c(lO-a) 
 
 2^=9(Trr^)'""^''- 
 
 (128) 
 
NEGATIVE RESULTS. 89 
 
 INTERPRETATION OF NEGATIVE RESULTS. 
 ( 18^5 page 133.) 
 
 1, Let X represent the number; then by the conditions, 
 
 X X 
 
 -rir ''■' 
 
 whence, x=144y Ans, 
 
 In an arithmetical sense, the fourth part of a number can never 
 exceed its third part; and the absurdity of this supposition is indi- 
 cated by the negative result. Hence the question must bo modified 
 so as to read, 
 
 What number is that whose third part exceeds its fourth part by 
 12? 
 
 2. Let X equal the number of years ; then by the conditions, 
 
 30 + jr= 3(15 + x); 
 ar= — VJ, Ans. 
 As the man was 30 years old, and his wife 15 years old, at the 
 time of marriage, she is already one half as old as he ; he can, 
 therefore, never become three times as old as she. Ilencc, the 
 problem must be modified as follows : 
 
 A man was 30 ijeai's old when he married, and his wife 15. flow 
 many years be/ore their marriaye was his aye three times the aye of 
 his wife ? 
 
 )x -\- u — s 
 \ whence, 
 x—y—d, 
 
 If we make «=120 and rf=160, we have 
 
 120 + 160 ,^^ 
 xz= = 140, greater; 
 
 120-160 ^^ , 
 y— = — 20, less. 
 
 s-\-d 
 
 s-d 
 "~2~ 
 
 Algebraically considered, —20 added to 140 is 120, the sum ; and 
 —20 subtracted from 140 is 160, the difference, 
 
 (133) 
 
90 SIMPLE EQUATIONS. 
 
 4. Let x= B's money ; then 3^:= A's money. Now by the con- 
 ditions of the problem, we have 
 
 32: + 400 = 2(j; + 150); 
 whence, ar=: — 100, 
 
 3ar= — 300. 
 
 The negative results show indebtedness^ instead of capital, at the 
 commencement of business. Hence, the question should read as 
 follows : 
 
 A owes three times as much as B. A gains by trading $400, and 
 B $150, when A has twice as much money as B. What was the 
 indebtedness of each at first ? 
 
 5. Let a:= the father's wages, and y= the son's. 
 
 (2) 
 
 T> .1 :,• . S Vjr + 3y=22, (1) 
 
 By the conditions, i ^ , „ ^ ' 
 
 subtracting (l) from three times (2), 8j:= 32 ; 
 
 or, «= 4; 
 
 whence, y= — 2. 
 
 The negative value of y shows that the boy was charged for board 
 each day two shillings more than his wages. 
 
 6. Let ar, y and z represent the wages of the man, his wife, and 
 son, respectively. 
 
 I 102:+ 8y+ 62=10.30, (1) 
 
 By the conditions, ^ 12jr + 10y+ 42=J3.20, (2) 
 
 ( 15j; + 10y + 12^=13. 85; (3) 
 
 subtracting (3) from twice (1), 5x-\-Qy= 6.75 ; (4) 
 
 " (3) from three times (2), 21ar + 20y = 25.75 ; (5) 
 
 multiplying (4) by 10, 50a: + 60y = 67.50 ; (6) 
 
 (5) by 3, 632; + 60y = 77.25 ; (7) 
 
 subtracting (6) from (7), 13a;=: 9.75 ; 
 
 whence, a;=.75, y= .50, and z=— .20. 
 
 The negative value of z shows that the boy was charged for board 
 $.20 each day above his daily wages. 
 
 (133-134) 
 
NEGATIVE RESULTS. 91 
 
 V. Let ar, y and z represent the wages of each respectively. 
 
 i 10.r + 4y + 3z=rll.50, (1) 
 
 By the conditions, < 9a: + 8y + 6z=:12.00, (2) 
 
 ( 7a: + 6y + 42=: 9.00; (3) 
 
 subtracting (2) from twice (1), ll.r = 11.00 ; 
 
 whence, a:=:1.00, y=0, and 2= .50. 
 
 The value of y shows that the wife's board and wages were equal. 
 
 8. Let x= the numerator, and y= the denominator. 
 
 ar+l 3 
 By the conditions, 
 
 From (1) 
 " (2) 
 
 \ 
 
 y 
 
 '5' 
 
 
 y 
 
 5 
 
 (y+i~ 
 
 ''1' 
 
 bx 
 
 -^y= 
 
 :-5; 
 
 1x- 
 
 -5y = 
 
 : 5; 
 
 25x- 
 
 .I5y= 
 
 :-25, 
 
 2\x- 
 
 ■I5y = 
 
 : 15, 
 
 
 4x= 
 
 :-40, 
 
 (1) 
 
 (2) 
 
 ar= — 10, y=— 15; hence, — —~yAns, 
 
 — 15 
 
 The modified example will be as follows : 
 
 What fraction it that which becomes | when 1 is subtracted from 
 
 its numerator^ and ^ when 1 is subtracted from its denominator^ 
 
 The equations will now be, 
 
 ar— 1_3 X _5 
 
 whence, ar=10, y=15 ; hence {^y Ans, 
 
 9. Let Xy y, z and u represent respectively the net capital or 
 insolvency of each. 
 
 Ix + y + z + u = 5l80, (1) 
 
 x + y + z =7950, (2) 
 
 y + 2 4-^ = 2220, (3) 
 
 X +z + u = l320, (4) 
 
 (134) 
 
92; SIMPLE EQUATIONS. 
 
 Subtractiog (3) from (1), x= 3560, A's net capital. 
 
 " (4) " (1), y= — 1540, B's net insolvency, 
 
 « (2) " (1), M = — 2170, D's net insolvency, 
 
 by substitution, z= 5930, C's net capital. 
 
 Tlie positive values of z and z show net capital for A and C ; the 
 negative values of y and u show net insolvencj/ for B and D. 
 
 10. Let xz= the number of hours Jiftcr six o'clock, when A passed 
 B. Now, at six o'clock, B's distance from Boston was n -}- 46 miles. 
 Hence, at the moment of passing we have 
 
 m — ajr=A's distance from Boston; 
 n + ib-bxz=Ws " " " . 
 
 Therefore, in — ax=n + 4b — bx\ 
 
 m — n — 46 
 
 a-b 
 
 hours, yins. 
 
 11. Substituting the given vahus for 7n, n, a, and 6, we hav 
 
 36 — 28—12 —4 
 
 x= = = — 2. 
 
 5 — 3 2 
 
 That is, A passed B — 2 hours after six, or 2 hours be/ore six, 
 which is 4 o'clock. 
 
 12. Let ar=r the greater, and x—a=: the less; then by the con- 
 ditions, 
 
 J x-{-5{x—'a) = b', 
 
 64-5a . 
 whence, x= — - — , the greater ; 
 
 o 
 
 6 — 3a , , 
 
 and X — a = , the less. 
 
 8 
 
 If a=:24 and 6=48, then 
 
 48 + 120 48— -K©- ^ 
 
 21, greater; x^a= =— Syless. 
 
 8 '° ' 8 
 
 Arithmetically speaking, there is no number —9. But considering 
 the quantities, 21 and — H^ algebraically^ they fulfil the conditions 
 of the problem. 3 
 
 (134-136) 
 
DISCUSSIONS. 93 
 
 (193, page 142.) 
 
 1. Let x^ y, z, and u represent the parts of the contents of the 
 cistern which will flow through the pipes respectively in one hour. 
 
 
 
 I5x + l5y + 15z + 15u = l, 
 
 (1) 
 
 
 
 5x+ 8y+ 1z+ 3« = J, 
 
 (2) 
 
 Bv the conditions, 
 
 
 1 ^ * 
 
 
 J ' 
 
 
 3x+ 4y+ 30+ u^\, 
 
 ^^) 
 
 
 
 4.r+ 2y+ 32-f 2^ = 1. 
 
 (^) 
 
 Subtracting J of (1) from 
 
 (2) 
 
 2a: + 5y + 45f=:-i-'^, 
 
 (5) 
 
 " •i'Tof(l) " 
 
 (3), 
 
 2j: + 3y + 23 = y\, 
 
 (6) 
 
 " ftof(l) " 
 
 (4), 
 
 2-1^ + z=i^\ 
 
 (^) 
 
 multiplying (5) by 3, 
 
 
 C.r + 45y + 122— y^; 
 
 (8) 
 
 (6) by 5, 
 
 
 10ar + 15y+10;2r:=ff ; 
 
 (0) 
 
 subtracting (8) from (9), 
 
 
 \x -2z=H; 
 
 (10) 
 
 multiplying (7) by 2, 
 
 
 4x +20-n; 
 
 (11) 
 
 adding (10) and (11), 
 
 
 8^= ^f, and:r= /^=r -,\ 
 
 
 subtracting (10) and (11) 
 
 
 4z=-if, and z=—^^^-^- 
 
 
 substituting in (5) 
 
 
 5y= f^andy::^ ^V= tj 
 
 
 (1) 
 
 ) 
 
 ^5w=-?f, and t^ = -/^=--V. 
 
 
 Therefore, the contents of the cistern will flow through the first 
 pipe in 12 hours, through the second in 15 hours, through the third 
 in 20 hours, and through the fourth in 30 liours. The positive 
 values of x and y indicate receiving pipes ; the negative values of z 
 and u indicate discharging pipes. 
 
 2. Let m represent what A will have in 2 days ; then m4-2 will 
 be what B will have in 4 days. 
 
 Let ar= the number of days hence, when A and B have the same 
 money; then A will have m + 5(.r— 2) dollars, and B will have 
 m + 2 + 3 (z~ 4) dollars. Hence, 
 
 w + 5(jr— 2) = m + 2 + 3(:r — 4) ; 
 (5— 3)ar=0; 
 
 
 5 — 3" 
 
 That is, they now have the same sum. 
 
 (142-143) 
 
 0. 
 
94 INEQUALITIES. 
 
 3. Let X represent the period ; then by the conditions, 
 3a;-10=:3(4j-4-8); 
 -whence, 12a;— 40 = 12x4-24 ; 
 
 (12-12)a:=64; 
 
 64 64 
 
 =— - = oo. 
 
 12-12 
 
 The value of x is an expression for infinity, according to (188, 1.). 
 The period of the comet, therefore, is a number of years greater 
 than any assignable number. 
 
 4. Let X represent the monthly wages ; then by the conditions, 
 2(9j: — 450) = 3(6.c — 300) ; 
 whence, 1 8a; — 900 = 1 8 j- — 900 ; 
 
 (18-18)j:=900-900, 
 
 _900-900_0 
 ^~ 18—18 ~0' 
 The value of ar is a symbol of indetermination, according to 
 (I885 4.) The monthly wages of each may therefore be any 
 number of dollars. If they receive more than $50 a month, they 
 •will each lay up the same sum. If they receive less than $50, they 
 •will become equally indebted. 
 
 INEQUALITIES. 
 (301, Page 149.) 
 
 1. 6jr> 5^ + 14; 
 
 clearing of fractions, lOx > 3a--(-28 ; 
 
 dropping 3a-, 7a; > 28 ; whence, a; > 4, Ans, 
 
 2a; 2a; 2a; 
 
 2x 2a; 
 
 dropping y, ""T ^ ""^ ' 
 
 2x X 
 
 changing signs by ( 199, III.), y < 2, or - < 1 ; 
 
 clearing of fractions, ^ < 3, Ans, 
 
 (143-149) 
 
INEQUALITIES. 95 
 
 3. ^ + 7<-^+.n> 
 
 5x 5 ^11 ,1x 
 8 4^6 12 
 
 clearing of fractions, 75ar+150 < 220 + ^Oar; 
 
 whence, 6;c < 70 ; 
 
 X < 14, Ans, 
 
 Sx x — l 20a; + 13 
 
 whence, 3x—2x-^2 < 24a — 20j;— 13 ; 
 
 or, —3x < — 15 ; 
 
 changing all the signs by (109, HI.)) 3a; > 15 ; 
 
 ar > 5, Ans. 
 
 6. 
 
 transposing. 
 
 dividing by (a—c). 
 
 6. 
 
 multiplying by aft, 
 transposing, 
 dividing by (a + 6), 
 
 7. (a—x)(m—x) — a(m—c)<^x* ; 
 
 1 < , aV 
 removing parentheses, am—ax—mx-{-x^—am-{-ac<^x^ ; 
 
 ax—b > 
 
 cx + c?; 
 
 ax— ex > 64-c?; 
 
 ^> 
 
 6+fl? . 
 
 's Ans, 
 
 a—c 
 
 ^< 
 
 ^-b 
 
 aar— a' < 
 
 ab—bx ; 
 
 aa;-|-6a; < 
 
 a'i-ab] 
 
 ^< 
 
 0, ^ns. 
 
 
 ; . , aV 
 
 transposing, 
 
 
 a'c 
 
 -ax—mx < —ac ; 
 
 m 
 
 changing all the signs. 
 
 
 a'c 
 
 aa;+ma; > acH ; 
 
 m 
 
 multiplying both sides by w, 
 
 
 m(a 4- m)a; > ac(a + w) ; 
 
 dividing by m(a + m). 
 
 (149) 
 
 a;> — , Ans. 
 m 
 
^6 INEQUALITIES. 
 
 (dOd, page 150.) 
 
 1. Given, i2. + 4y>30, (1) 
 
 (3x + 2y = 31; (2) 
 
 dividing (1) by 2, a- + 2y > 15 ; (3) 
 
 subtracting (3) from (2), ( 1 »9, IT.), 2.r < 16 ; 
 
 whence, a: < 8. 
 
 If we substitute 8 for ar in (2), the first member will be greater 
 than the second member ; thus, 
 
 24 + 2y > 31; 
 transposing, 
 whence, 
 
 2. Given, 
 
 multiplying (1) by 2, 
 
 subtracting (3) ffom (2),(199, II.), 
 
 whence. 
 
 Substituting 2 for y in (2) will make the first member less than the 
 second ; hence, 
 
 8x + 4 < 46 ; 
 transposing, 8j: <[ 42 ; 
 
 whence, x < b\. 
 
 3. Given, 
 
 multiplying (2) by 2, 
 adding (1) and (3), 
 whence, 
 substituting in (2), 
 
 whence, 
 
 2y> V; 
 
 
 y>H' 
 
 
 4x-3y < 15, 
 
 (1) 
 
 8jr + 2y = 46; 
 
 (2) 
 
 Sx-Qy < 30; 
 
 (3) 
 
 8y > 16 ; 
 
 
 y> 2, 
 
 
 4. Given, \ 
 
 From equation (2), 
 
 (150) 
 
 7ar~10y < 59, 
 
 (1) 
 
 4x+ 5y = 68; 
 
 (2) 
 
 8x-fl0y =136; 
 
 (3) 
 
 Ibx < 195; 
 
 
 x< 13; 
 
 
 52 + 5y> 68; 
 
 
 5y> 16; 
 
 
 y> 31. 
 
 
 + 3y> 121, 
 
 (1) 
 
 + 4y= 168; 
 
 (2) 
 
 1x 
 
 
 y=42--; 
 
 (3) 
 
POWERS OF POLYNOMIALS. 
 
 97 
 
 substituting this value of y in (1), we have 
 6^+126 — 
 
 >121; 
 
 multiplying by 4, 20x + 504— 21a: > 484 ; 
 
 transposing and uniting, — -^ > —20^ 
 
 whence by ( 1 99, III.), ^ < 20. 
 
 Substituting 20 for x in (2), we have 
 
 1404-4y > 168; 
 transposing, 4y > 28 ; 
 
 y> '?. 
 
 W 
 
 a;— 4 y — 10 
 
 5. Given, 
 
 8 
 
 6 
 
 >1. 
 
 3ar— 24 x—y 
 
 (1) 
 
 (2) 
 
 multiplying (1) by 24, 
 
 
 Sj;_12-4y + 40 > 24; 
 
 (3) 
 
 or. 
 
 
 3^-4y> -4; 
 
 (4) 
 
 multiplying (2) by 4, 
 
 
 3j;_24+2ar-2y = 52; 
 
 (5) 
 
 or, 
 
 
 5a;-2y = '76; 
 
 (6) 
 
 multiplying (6) by 2, 
 
 
 10jr-4y = 152; 
 
 V) 
 
 subtracting (4) from (7), 
 
 
 Va;< 156; 
 
 
 whence. 
 
 
 X < 22^, 
 
 
 substituting in (6), 
 
 
 lll|-2y> 76, 
 
 
 transposing, 
 
 
 -2y > -35f 
 
 
 or, 
 
 
 2y < 35^ 
 
 y < iH- 
 
 
 
 INVOLUTION. 
 
 
 (316, 
 
 page 157.) 
 
 
 1. By simple multiplication according to the rule, we shall have 
 
 (2a;'4-3y)'=4a;*+ 6jr'y + 6a:'y + 9y* 
 = 4i;' + 1 2x'y + 9y*, Ans. 
 
 (150-157) 
 
98 INVOLTUION. 
 
 2. Multiplying the factor bx—y* by itself, we get 
 (5a;' -y')' = 25j:' - lO^-y' +y*. 
 Multiplying this result by 5x—i/*, 
 
 (5a;— y')'= 125ar»— 50a:y + 5x/ — 25a;y + 10a;/— y* 
 = 125a;'— 75a;V + 15a-y*— y', ^n«. 
 
 8. {l-\-2x — 3xy = l-^4x-6x'+ 4j;'— 12a;' + 9a;* 
 
 = 1 + 4a;— 2ar' — 1 2x' + 9a;*, Ans, 
 
 4. (3a + 26 + c)' = 9a' + 1 2a6 + 6ac + 46' + 46c + c\ 
 
 Multiplying this result by 3a + 26 + c, we have 
 (3a + 26-fc)' 
 
 = 27a' + 36a'6 + 18tt'c-l- 12ff6' + 12a6c+ 3ac' 
 
 + 18a'6 + (86') + 24a6' + 12a6c+(86'c) + 26c' 
 
 + 9a'c + (46V) + 1 2«6c + 6ac' + 46c' + c' 
 
 = 2'7a' + 64a'6 + 2Ya'c-|-36a6' + 36a6c+ 86* +9ac' + 126'c 
 + 66c* + c', Ans. 
 
 In Examples 6th and 6th we ^lave the simple binomial ; see (70). 
 
 7. (a'c-' + a-'cy z=a*c-* + 2a'c° + a-*c* 
 
 =a*c-* + 2-ha-*c\ Ans. 
 
 8. (a' 4- 1 +a-')' =a* + 2a' + 2a» + 1 + 2a-' + a-* 
 
 = a* 4- 2a' + 3 4- 2a-' + a-*. 
 Multiplying again by a' + 1+ a~', 
 
 (a' + 1+ a-')' = a' + 2a* 4- 3a' 4- 2a'' 4- a-* 
 
 4- a*4-2a'4-3 4-2a-»4- cr* 
 
 4- a' 4- 2a" 4- 3a-' 4- 2a-* 4- a-' 
 
 (a' 4- 1 4- a--y=a'' 4- 3a* 4- 6a' 4- V 4- Qar* + 3a-* 4-a~*, Ans. 
 For Example 9th, see (71). 
 
 (217, page 159.) 
 
 All examples in this article are readily solved by a strict applica- 
 tion of the rule, which is an important one. In the first three 
 examples the symmetry of the answers should be noticed. 
 
 (158-159) 
 
SQUARE ROOT OF POLYNOMIALS. 99^ 
 
 6. (l—a + a'— a')«=l--2a-f 2a'— 2a' 
 
 4- a' — 2a' 4- 2a* 
 
 + a* — 2a* + a' 
 
 = 1 — 2a + 3a' — 4a' -f 3a* — 2a' + a\ Arts, 
 
 1. (3aa; + 2a' — 4a;' — 5)' 
 
 :9aV + 12a'a:— 24aa;' — 30a2; + 4a*— 16a'j;'— 20a' + 16j:* + 40a;' + 25 
 ;12a'x— 24ax'— 30aar + 4a* — Ya'jr' — 20a' + 16a:* 4-40a:' + 25, Ans. 
 
 8. (1— 2a:— y' + ary— a;')' 
 : 1 — 4a: — 2y' + 2a:y — 2a;' 
 
 + 4a;' 4- 4a-y' — 4a:'y + 4a:' + y* — 2a'y ' + 2a:'y' 
 
 + a:'y' — 2a:'y + a;* 
 
 : 1 — 4a: — 2y' + 2a:y -f 2a;' + 4a;y' — 4a:'y + 4a;' + y* — 2a;y' + 3a:'y' 
 
 — -2a;'y + a;*, Ans, 
 
 EVOLUTION. 
 (aaO, page 166.) 
 
 1. a' + 2a6 + 2ac + 6' + 26c + c\a + 6 + c, Ans, 
 
 a' 
 
 2d+b 2a64-2ac + 6' 
 2a6 +6' 
 
 2a+264-c 2ac +26c+c' 
 
 2ac +26c4-c' 
 
 2. a* — 6a'64-4a' + 96' — 126 + 4(a' — 36 + 2, -4w*. 
 
 2a' — 36 — 6a'6+ 96' 
 
 -6a'6+ 96' 
 
 2a«— 664-2 4a' -1264-4 
 
 4a' -1264-4. 
 
 (159-166) 
 
IPP EVOLUTION. 
 
 3. «' + 4j:* + 2x« — 2a:' + 6x*-.2a:+l(df' + 2a^— ar+l, Ans. 
 
 4x'-\-4x* 
 
 2x*-h4x*—x —2x* — 2x*-\-5z* 
 
 — 2x*-4x'-\- X* 
 
 2ar' + 4jr' — 2jr+l 2x'-\-4x*-~2x-\-l 
 
 2x' + 4a;' — 2ar + l. 
 
 4. The two first terms of the root will be l—rt, by inspection; 
 hence, 
 
 . I — 2a + 3a* — 4a* + 3a* — 2a* + a\\—a + a*'-a\ Ans, 
 1 — 2a+ a* 
 
 2 — 2a + a' 2a' — 4a' + 3a* 
 
 2a' — 2a'+ a* 
 
 2— 2a + 2a'— a' —2a" 4- 2a* — 2a' -fa' 
 
 — 2a' + 2a* — 2a' -fa* 
 
 5. The square root of the first term is 2a'6, and — 12a'6'-?- 
 4a'6=— 3a6; hence we proceed as follows : 
 
 |2a'6— 3a64-2a6', Ans, 
 
 4a*b* — 1 2a'6' -f 8a'6' + 9a'6' - 1 2a'6' -f 4a'6* 
 4a*6'-12a'6'4- 9a'6' 
 
 4a'6' — 6ab + 2a6' 8a'6' -^1 2a'6' + 4a'6* 
 
 8a'6' ■ — 12a'6' + 4a'6* 
 
 6. |3x' — 5a?*y— 4a:y' + 6y', Ans, 
 
 9ar'— 30yy4- a:*y' + 76x'y' — 44a;'y* — 48ary' + 36/ 
 . 9x'--30x'y + 25x*y' 
 
 6j?"— 10ar"y-4ry' -242:*y' + 76x'y'-44ar'y* 
 
 -24x*y' + 40.r'y»4-16a:Y " 
 
 6a;' — lOar'y— 8jry" + 6y' 362r'y' — 60Jry — 48a:y»4- S6y* 
 
 S6xy-Q0xy-48xy*-{-d6y* 
 (166) 
 
SQUARE ROOT OF POLYNOMIALS. 101 
 
 |a' — 36c4-2c</— c?', Ans, 
 7. a*-6o'6c+4aW-2aV + 96V-126c»tf-|-66cc?'+4cV'-4cc^"+rf* 
 a*-6a'6c +9/>V 
 
 2a*-^6bc+2cd 4aW-2aV —Ubc'd 
 
 Ad'cd —Ubc'd +4cV 
 
 2a*-6bc+4cd-d* -2aV +66crf' ^4cd'+.d' 
 
 -2aV +66ccf' _4crf'+c?* 
 
 . „ 3a«6« aJ^ 6' , , ab b' . 
 
 a*—a*b-{- — - 
 4 
 
 2 4 16 
 
 2ar* — 6a; + x-' 2x' — 62?-* + a;"* 
 
 2x' — 6^-*+a;~* 
 
 10. a"*-' — 10a6-' + 27 — 10a-'6+a-'6'(a6-' — 5+a-'6, ^n«. 
 
 a«6-»_10a6-» + 25 
 
 2a6-* — 10 4-a''6 + 2 — 10a-'6+a-'6' 
 
 2-10a-'6 + a-«6« 
 In this example we have the last multiplication, 
 2ab-' xa-'6=2a°6»=2. 
 
 11. a*** 4- 6a*"»c'' + 11 a""c'" + 6a V + c^^^a^^ + 3a'"c'* + c*", ^n*. 
 
 a*** 
 
 20'-" + Sa^'cT 6a'"*c'* + 1 la""c^ 
 
 6o""c"+ 9a»V 
 
 2a2V + 6aV+c*'' 
 (168) 
 
102 EVOLUTION. 
 
 SQUARE ROOT OF NUMBERS. 
 (934, page 169.) 
 
 1. "72,25(85, Ans. 2. 10,82,41(329, Ans, 
 
 a>- 64 9 
 
 2a+6-165 825 2<i+h-.62 182 
 
 2a6+6>- 825 124 
 
 2a+26 + c-649 5841 
 
 5841 
 
 3. 65,12,49(807, ^ns. 4. 9*7,41,69(987, ^n*. 
 
 _64 81 
 
 1607 11249 188 1641 
 
 11249 1504 
 
 1967 13769 
 13769 
 
 5. 5,09,85,64(2258, ^»*. 6. 66,34.10,25(81.45, ^»W. 
 4 64 
 
 161 234 
 161 
 
 42 
 
 109 
 
 
 84 
 
 445 
 
 2585 
 
 
 2225 
 
 4508 
 
 36064 
 
 
 36064 
 
 7. 
 
 18,12,88,60,8 
 
 
 16 
 
 82 
 
 212 
 
 
 164 
 
 845 
 
 4888 
 
 
 4225 
 
 8507 
 
 66360 
 
 
 59549 
 
 8514^ 
 
 J 681184 
 
 
 681184 
 
 1624 7310 
 6496 
 
 16285 81425 
 81425 
 
 8. .33,98,89(.583, ^7W. 
 25 
 
 108 898 
 864 
 
 1163 3489 
 3489 
 
 (169-170) 
 
SQUARE ROOT OF NUMBERS. 103 
 
 9. .00,52,41,76(.0724, Ans. 
 49 
 142 341 
 
 10. 
 
 41 
 
 4,77(21.8403+, Ans. 
 4 
 
 77 
 
 284 
 
 
 
 41 
 
 1444 5776 
 
 
 428 
 
 3600 
 
 5776 
 
 
 
 3424 
 
 
 
 4364 
 
 17600 
 17464 
 
 
 
 436803 1360000 
 
 
 
 
 1310409 
 
 i 
 
 11. 
 
 63 
 
 11.09(3.33016+, Ans, 
 
 9 
 
 209 
 
 
 189 
 
 
 
 663 
 
 2000 
 1989 
 
 
 ■> 
 
 66601 
 
 110000 
 66601 
 
 
 
 666026 
 
 4339900 
 3996156 
 
 
 
 12. The square root of a fraction is the square root of the numer' 
 ator divided by the square root of the denominator. Hence, 
 
 13,69(37 1,18,81(109 ^ ^^^ 
 
 9 1 109' "*' 
 
 67 469 
 469 
 
 13. 1,02,03,02,01(10101 1 . 
 
 ioioi'^'*"- 
 
 
 209 1881 
 1881 
 
 
 1,02,03,02,01(10101 
 
 1 
 
 201 
 
 0203 
 201 
 
 20201 
 
 20201 
 20201 
 
 (170) 
 
104 
 
 
 EVOLUTION. 
 
 14. 
 
 
 245 49 TT ^ . 
 720 = 144- Hence, -,^n,. 
 
 16. 
 
 43 
 
 = 5.57,14,28,57 + (2.3604 + , Arts. 
 4 
 
 157 
 129 
 
 
 466 
 
 2814 
 2796 
 
 
 27204 
 
 182857 
 168816 
 
 CONTRACTED METHOD. 
 
 («355 page 171.) 
 
 1. 56.00,00,00,0(7.4833147 + 
 49 
 
 144 
 
 700 
 
 
 576 
 
 1488 
 
 12400 
 
 
 11904 
 
 14963 
 
 49600 
 
 
 44889 
 
 14966 
 
 4711 
 
 
 4490 
 
 1497 
 
 221 
 
 
 150 
 
 150 
 
 71 
 
 
 60 
 
 15 
 
 11 
 
 
 11 
 
 2. 
 67 
 
 14.00,00,00 
 9 
 
 500 
 469 
 
 (3, 
 
 7416574 
 
 744 
 
 3100 
 2976 
 
 
 
 7481 
 
 12400 
 7481 
 
 
 • 
 
 7482 
 
 4919 
 4489 
 
 
 
 748 
 
 430 
 
 
 
 75 
 
 374 
 
 ~56 
 
 53 
 
 
 
 t 
 
 3 
 3 
 
 
 
 ( 170-171 ) 
 
SQUARE ROOT OF NUMBERS. 
 
 105 
 
 844 
 
 848 
 
 85 
 
 6. 
 
 142 
 
 1444 
 
 14483 
 
 14486 
 
 1449 
 
 145 
 
 14 
 
 18.00,00(4.2426 + 
 
 4. 
 
 19.00,00,00(4.3588984- 
 
 16 
 
 
 
 16 
 
 200 
 
 
 83 
 
 300 
 
 164 
 
 
 865 
 
 249 
 
 [ 3600 
 
 5100 
 
 33Y6 
 
 
 870( 
 
 4325 
 
 J 224 
 
 i 77500 
 
 169 
 
 
 
 69664 
 
 55 
 
 
 87l€ 
 
 ; 7836 
 
 61 
 
 
 
 6973 
 
 
 
 872 
 
 863 
 
 785 
 
 
 
 87 
 
 78 
 12 
 
 52.46,30,00 
 
 r 
 
 1(7.2431346 4- 6. 
 
 7.00,00,00,00(2.64575131 4- 
 
 49 
 
 
 
 4 
 
 346 
 
 
 46 
 
 300 
 
 284 
 
 
 524 
 
 276 
 
 6230 
 
 2400 
 
 5776 
 
 
 5285 
 
 2096 
 
 45400 
 
 30400 
 
 43449 
 
 
 52907 
 
 26425 
 
 1951 
 
 397500 
 
 1449 
 
 
 62914 
 
 370349 
 
 602 
 
 27151 
 
 435 
 
 
 
 26457 
 
 67 
 
 
 5291 
 
 694 
 
 58 
 
 
 
 529 
 
 "9 
 
 
 529 
 
 165 
 
 8 
 
 
 
 159 
 
 
 
 53 
 
 6 
 5 
 
 (172) 
 
106 EVOLUTION. 
 
 1. 6*=(5y=(125)^ 
 
 1,25.00,00,0(11.18034 + 
 I 
 
 21 25 
 
 21 
 
 221 400 
 
 211 
 
 2228 18900 
 
 18824 
 
 2236 76 
 
 67 
 
 22 9 
 
 9 
 
 (239, page 175.) 
 
 27a' + 108'i' + 144a + 64(3a+4, Arts. 
 
 |27a' 108a' + 144a + 64 
 
 9a + 4 36a + 16|27a'+36a + 16 108a' + 144a + 64 
 
 In this example we have, 
 
 Trial divisor, 27a' 
 
 First factor of correction, 9a + 4 
 
 Correction of trial divisor, 36a +16 
 
 Complete divisor, 27a' + 36a + 16 
 
 according to formula (a), (JSJ37.) 
 
 (172-175) 
 
CUBE ROOT OF POLYNOMIALS. 
 
 107 
 
 
 CO 
 1 
 
 1 
 
 
 1 
 
 CO 
 
 OS 
 
 1 
 
 CO 
 
 OS 
 
 
 + 
 
 + 
 
 "^ 
 
 s 
 
 -^ 
 
 
 + 
 
 1 
 
 + 
 
 2 
 
 1-H 
 
 
 + 
 
 1 
 
 1 
 
 5 
 
 
 
 CO 
 
 CO w 
 
 
 CO 
 
 
 I-H 
 
 e« 
 
 + 
 
 H 
 
 H 
 
 (N 
 
 M* 
 
 r-( 
 
 <M 
 
 + 
 
 1 
 
 m 
 
 
 H 
 
 « 
 
 <N 
 
 (M 
 
 
 
 + 
 
 + 
 
 « 
 
 •♦ 
 
 H 
 
 H 
 
 CO 
 
 CO 
 
 •I + 
 
 O H 
 
 w T 
 
 c« + 
 
 a 
 s 
 
 bo 
 
 s + 
 
 ^? 
 
 a CO 
 
 a 
 o 
 
 H 
 CO 
 CO 
 
 + 
 
 CO 
 CO 
 
 00 00 
 
 CO ** 
 
 «o <?^ 
 
 1 I 
 
 CO "^ 
 
 CO "5 
 
 
 r-l 
 
 tH 
 
 + 
 
 j 
 
 + 
 1 
 
 "5 
 
 CO 
 
 CO 
 CO 
 
 + 
 
 + 
 
 n 
 
 CO 
 
 CO 
 
 1 
 
 CO 
 
 CO 
 
 1 
 
 1 
 
 r-t 
 
 1 
 
 I— « 
 
 t- CO 
 
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 + + 
 
 CO CO 
 
 CO CO 
 
 
 (176) 
 
108 
 
 I 
 
 00 
 
 + 
 
 \ 
 
 o 
 
 + 
 J 
 
 3a 
 
 + 
 
 EVOLUTION. 
 
 + 
 
 
 
 
 • 
 
 
 
 1 
 
 ^0 
 1 
 
 
 
 1 
 
 1 
 
 
 
 ►O 
 
 -o 
 
 
 
 « 
 
 « 
 
 
 
 a 
 
 05 
 
 
 
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 n 
 
 
 
 <s 
 
 c 
 
 
 
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 ■^ 
 
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 w 
 
 CM 
 
 C^ 
 
 
 
 + 
 
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 1 
 
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 O 
 
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 e 
 
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 04 
 
 cs 
 
 
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 rO 
 
 
 
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 5S 
 
 
 
 a 
 
 
 
 ! 
 
 CT 
 
 
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 Ci 
 
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 c^ 
 
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 m 
 
 n 
 
 M 
 
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 8 
 
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 r-l 
 
 
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 ^ .« 
 
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 « 
 
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 CO CO 
 
 CO 
 
 CO 
 
 
 
 •• 
 
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 JC> 
 
 
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 1 
 
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 1 
 
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 c 
 
 
 
 
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 + 
 
 
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 8 
 
 
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 I I 
 
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 00 
 
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 1 
 
 1 
 
 
 
 
 
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 ^ 
 
 
 
 t- 
 
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 1 
 
 « 
 
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 J 
 
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 e 
 
 e 
 
 e 
 
 
 
 t- 
 
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 cs 
 
 cs 
 
 10 
 
 cs 
 
 CO 
 
 CO 
 
 cs 
 
 T—i 
 
 1-^ 
 
 r-i 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 
 
 
 
 ^ 
 
 :s 
 
 ^ 
 
 CO 
 
 lO 
 
 00 
 
 00 
 
 1 
 
 1 
 
 1 
 
 1 
 
 ^1 
 
 1 
 
 1 
 
 •a 
 
 1 
 
 
 
 "e 
 
 e 
 
 *« 
 
 05 
 
 >o 
 
 -rt^ 
 
 ■^ 
 
 lO 
 
 CO 
 
 cs 
 
 cs 
 
 T— < 
 
 
 
 
 4- 
 
 + 
 
 + 
 
 + 
 
 
 
 
 c 
 
 CO 
 
 <§ 
 
 S 
 
 cE 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 "e 
 
 "e 
 
 
 
 10 
 
 
 
 
 
 i-H 
 
 T-H 
 
 
 _u 
 
 
 (4 
 
 
 .8 
 
 
 CS 
 
 
 + 
 
 
 
 
 
 
 
 
 
 1 
 
 ^ 
 
 1 
 
 
 
 '« 
 
 cs 
 
 »- 
 
 t—t 
 
 CS 
 
 + 
 
 + 
 
 *> 
 
 •a 
 
 e 
 
 e 
 
 <N 
 
 cs 
 
 1 
 
 1 
 
 *« 
 
 « 
 
 CO 
 
 CO 
 
 o 
 
 CO 
 
 cs cs 
 
 + + 
 
 cs cs 
 
 
 5 
 
 + 
 
 
 
 1 
 
 cs*^ 
 
 1 
 
 
 e 
 
 10 
 
 
 »o 
 
 r-l 
 
 
 + 
 
 + 
 
 SI 
 
 c 
 
 c* 
 
 e 
 
 e 
 
 e 
 
 cs 
 
 1 
 
 1 
 
 CD 
 
 1 
 
 1 
 
 •• 
 
 1 
 
 1 
 
 8 
 
 e 
 
 "e 
 
 CO 
 
 CO 
 
 CO 
 
 (175) 
 
CUBE ROOT OF POLYNOMIALS. 
 
 109: 
 
 -f §■ 
 
 
 
 
 
 
 
 § 
 
 > 
 
 rO 
 
 
 
 > 
 
 5a 
 
 
 i "> 
 
 
 
 
 
 ^ 
 
 s^ 
 
 
 
 s 
 
 ^ 
 
 
 + 
 
 
 
 
 
 ^"^ 
 
 4- 
 
 
 
 + 
 
 + 
 
 
 1 
 
 
 
 
 
 CO 
 
 
 
 
 
 > 
 
 
 :b. 
 
 
 
 
 
 e 
 1 
 
 
 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 1-i 
 1 
 
 1 
 
 ^ 
 
 1 
 
 
 
 1 
 
 1 
 
 
 1 
 
 
 
 
 1 
 
 1 
 
 — 55 
 
 
 
 Sa 
 
 rO 
 
 
 + 
 
 
 
 
 + 
 
 1 
 
 •< 
 
 
 
 
 4- 
 
 1 
 
 
 's 
 
 
 
 
 5 
 
 s 
 
 > 
 
 
 
 "« 
 
 > 
 
 
 1 
 
 
 
 '8 
 
 8 
 
 1 
 
 1 
 
 + 
 
 
 
 
 V 
 
 
 + 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 2a 
 
 M 
 
 3a 
 
 
 
 
 '^ 
 
 
 
 M 
 
 '8 
 
 CO 
 
 M 
 
 & 
 
 1 
 
 1 
 
 1 
 
 
 
 
 1 
 
 
 
 1— ( 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 Sa* 
 
 
 
 ^ 
 
 
 
 a 
 
 
 I 
 
 ^ 
 
 1 
 
 's 
 
 
 
 
 
 1-H 
 
 
 
 
 
 
 
 CO 
 
 CO 
 
 
 CO 
 
 CO 
 
 
 + 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 1 
 
 1 
 
 
 1 
 
 1 
 
 
 § 
 
 1 
 
 •a 
 
 '8 
 
 •^ 
 
 'M 
 
 ^ 
 
 1 
 
 1 
 
 ^ 
 
 
 
 
 I— 
 
 
 1 
 
 
 
 1 
 
 1 
 
 ;§ 
 
 c§ 
 
 eg 
 
 
 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 3l 
 
 1 
 
 1 
 
 
 + 
 
 + 
 
 ^ 
 
 ;g 
 
 
 + 
 
 + 
 
 + 
 
 
 
 r-4 
 
 s 
 
 s 
 
 
 ;§ 
 
 c§ 
 
 
 1 
 
 1 
 
 2l 
 1 
 
 
 
 + 
 
 CO 
 
 + 
 
 CO 
 
 + 
 
 ^ 
 
 CO 
 
 CO 
 
 
 •i» "k 
 
 
 
 M 
 
 1 
 
 ^ 
 
 
 <N 
 
 
 O c 
 
 
 
 
 "« 
 
 8 
 
 CO 
 
 S 
 
 ft 
 1 
 
 1 
 
 1 
 
 
 1- 
 
 
 
 
 + 
 
 + 
 
 + 
 
 1 
 
 1 
 
 1 
 
 
 2 % 
 
 
 
 
 "^ 
 
 "^ 
 
 1 
 
 eg ^ 
 
 
 c< 1 •< 
 
 '^i 
 
 
 
 
 
 
 1 
 
 
 
 rO 
 
 rO 
 
 rG rO 
 
 
 
 
 1 
 
 1 
 
 1 
 
 
 
 
 
 *1» o 
 
 
 
 "h 
 
 'M 
 
 's 
 
 I 
 
 S 
 
 
 
 + 
 
 + 
 
 ^ ^ 
 
 + 1 
 
 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 
 
 rO 
 
 f 
 
 rO 
 
 
 
 1 
 
 % 
 
 1 
 
 1 
 
 1 
 
 1 
 
 
 
 S 
 
 l" 
 
 1 
 
 
 '^ 
 
 % 
 
 M 
 
 5§ 
 
 'M 
 
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 1 
 
 1 
 
 1 
 
 
 
 + 
 
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 f— ( 
 
 1 
 
 
 1— t 
 
 1— t 
 
 1—4 
 
 I-" 
 
 
 
 
 4. 
 
 
 
 1 
 
 1 
 
 "8 
 
 
 
 
 
 
 
 
 
 ^ 
 
 CO 
 
 + 
 1 
 
 1-1 
 
 1 
 
 8 
 
 
 
 1 
 
 =1 
 
 1 
 
 
 
 
 8 
 
 CO 
 
 
 
 
 CO 
 
 
 
 
 + 
 
 + 
 
 
 
 
 + ' 
 
 
 
 
 e< 
 
 e< 
 
 
 
 
 rO 
 
 CO 
 
 
 "8 
 
 1 
 
 1 
 
 1 
 
 w 
 
 
 i 
 
 1 
 
 
 
 1 
 
 35 
 
 
 1 
 
 
 1 
 
 •8 
 CO 
 
 
 
 
 1 
 
 
 1 
 
 
 (176) 
 
no 
 
 EVOLUTION. 
 
 + 
 I 
 
 
 « % 
 
 + + 
 
 + + 
 
 H 
 
 H. 
 
 H 
 
 ^ 
 
 O 
 
 -* 
 
 CO 
 
 CO 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 1 
 
 "h 
 
 ^i 
 
 H 
 
 'h 
 
 •«l 
 
 w 
 
 rn-* 
 
 «»* 
 
 o+« 
 
 o*# 
 
 
 
 H 
 
 H 
 
 
 
 + 
 
 + 
 
 
 
 *« 
 
 « 
 
 
 
 ©< 
 
 (N 
 
 
 
 1 
 
 1 
 
 
 + 
 1 
 
 •• 
 
 
 "h 
 
 "h 
 
 i? 
 
 
 CO 
 
 00 
 
 «»(* 
 
 
 r-H 
 
 -<*< 
 
 -1 
 
 
 + 
 
 + 
 
 + 
 
 
 -^ 
 
 "h 
 
 % 
 
 
 <N 
 
 T*« 
 
 "* 
 
 
 1 
 
 1 
 
 1 
 
 « 
 
 1 
 
 1 
 
 1 
 
 H 
 
 N 
 
 H 
 
 H 
 
 CO 
 
 CO 
 
 CO 
 
 CO 
 
 
 CO 
 
 
 ■t 
 
 
 
 
 + 
 
 
 + 
 
 
 ^ 
 
 
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 1 
 
 
 (M 
 
 
 1 
 
 
 i-t 
 
 
 •V* 
 
 
 1 
 
 
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 « N 
 
 
 
 
 
 
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 1— t 
 1 
 
 
 
 S 
 
 
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 1 
 
 
 
 
 
 fe ^ 
 
 1 
 
 
 
 + 
 
 
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 "•j^ 
 
 ^ 
 
 '^ '^ 
 
 r-t 
 
 
 
 C^ 
 
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 CO CO 
 
 O 
 
 
 
 05 
 
 CO 
 
 Oi Oi 
 
 + 
 
 
 
 rH 
 
 + 
 
 1 
 
 + 4- 
 
 i 
 
 
 
 5 
 
 1 
 
 CO CO 
 
 f-4 
 
 
 
 OS 
 
 a 
 
 a S 
 
 + 
 
 
 
 1 
 
 + 
 
 1 1 
 
 
 
 
 « 
 
 
 
 cS 
 
 CO 
 
 S 
 
 cs 
 
 '^ 
 
 :^ 5 
 
 J 
 
 T 
 
 + 
 
 1 
 
 J 
 
 CI ^ 
 1 1 
 
 
 e 
 
 
 »- 
 
 
 
 u 
 
 H 
 
 >i 
 
 H 
 
 *« 
 
 
 ■^ 
 
 HH 
 
 c^ 
 
 CI 
 
 (N 
 
 
 T 
 
 CO 
 
 1 
 
 + 
 
 rH 
 
 1 
 
 1— t 
 1 
 
 
 S 
 
 s 
 
 S 
 
 
 
 
 + 
 
 + 
 
 + 
 
 
 
 S 
 
 « « 
 
 
 
 
 f-H 
 
 + 
 
 + + 
 
 i 1 
 
 
 
 « 
 
 'n 
 
 :i 
 
 '^ '^ 
 
 
 
 irH 
 
 C<l 
 
 f— 1 1— ( 
 
 
 
 rt< 
 
 + 
 
 1 
 
 1 1 
 
 
 
 + 
 
 -H 
 
 14 
 
 "« '« 
 
 
 
 eg 
 
 C^ 
 
 C<l 
 
 d CI 
 
 
 
 r— ( 
 
 f-H 
 
 1— t r-t 
 
 
 
 + 
 
 + 
 
 + 
 
 + + 
 
 
 •« 
 
 « 
 
 « 
 
 •« 
 
 % •« 
 
 
 CO 
 
 CO . 
 
 CO 
 
 CO 
 
 CO CO 
 
 
 
 ^ 
 
 
 's 
 
 s 
 
 
 
 + 
 
 
 r^ 
 
 + 
 
 
 
 <a 
 
 
 + 
 
 f^ 
 
 
 
 S 
 
 
 CI 
 
 1 
 
 1 
 
 + 
 1 
 
 (176) 
 
 
 4^ 
 
 1 
 
 i 
 
 1 
 
 
 1 
 
 + 
 
 1 
 
 2l 
 
CUBE ROOT OF NUMBERS. 
 
 IM 
 
 In the preceding solutions, we have made a formal applicatio» of 
 the rule in each case. This will enable us to extract the cube root 
 of any algebraic expression by direct process ; but when the root is 
 a binomial, we can generally find it by simple inspection. Thus, in 
 the first example the cube root of the first term is 3a, and that of 
 the last term is 4 ; from this we infer, since the expression has but 
 four terms, that 3a + 4 is the cube root. This is easily verified. 
 
 Again, in example 4, the cube root of the first term is a" ; and 
 dividing the second term by 3a* we have +3ab for the second term 
 of the root. We have also for the cube root of the last term, —6' ; 
 and dividing the preceding term by 36*, we have + 3a6 for the term 
 of the root preceding —6'. Hence we should infer that the cube 
 root isa' + 3a6— 6*, which- might be verified by involution. 
 
 CUBE ROOT OF NUMBERS. 
 
 (a43J5 page 1V9.) 
 a* + 3a'b + Sab' + b' = U8,8l1 (53, Ans. 
 
 3a + b dab + b^ 
 
 3a' 
 
 
 153 459 
 
 7500 
 
 23877 
 
 3a' + 3a6 + 6' = 
 
 7959 
 
 23877 
 
 2. 
 
 571,787 {8S,Ans, 
 512 
 
 119200 59787 
 243 729119929 59787 
 
 Note.— This example may be solved by inspection, and all others 
 in "which the root has but two places of figures. 
 
 3. 
 
 183 
 
 549 
 
 1895 9475 
 
 10800 
 11349 
 
 256,047,875 (635, Ans, 
 216 
 
 40047 
 34047 
 
 1190700 6000875 
 
 1200175 6000875 
 
 (179) 
 
112 
 
 
 
 EVOLUTION. 
 
 4. 
 
 
 354,894,912 (708, Ans. 
 343 
 
 
 
 1470000 
 
 11894912 
 
 
 2108 16864|l486864 
 
 11894912 
 
 5. 
 
 
 11,852.352 (22.8, ^«*. 
 8 
 
 
 |1200 
 
 3845 
 
 
 62 124 
 
 1324 
 
 2648 
 
 
 
 145200 
 
 1204352 
 
 
 668 5344 
 
 150544 
 
 1204352 
 
 144,125,083,907 (5243, Ans. 
 125 
 
 152 
 
 304 
 
 7500 19125 
 7804 15608 
 
 
 1564 
 
 6256 
 
 811200 3517083 
 817456 3269824 
 
 
 15723 
 
 47169 
 
 82372800 247259907 
 82419969 247259907 
 
 128,100,283,921 
 125 
 
 (5041, Ans, 
 
 1504 
 
 6016 
 
 750000 3100283 
 756016 3024064 
 
 
 15121 
 
 15121 
 
 76204800 76219921 
 76219921 76219921 
 
 
 8. 
 
 127 
 
 1412 
 
 14166 
 
 105,555,569,176 (4726, Atu, 
 64 
 
 4800 
 8895689 
 
 41555 
 39823 
 
 662700 
 2824665524 
 
 1732569 
 1331048 
 
 |66835200 
 84996166920196 
 (179) 
 
 401521176 
 401521176 
 
CUBE ROOT I^F NUMBERS. 
 
 113 
 
 9. 
 
 731,189,187,729 (9009, ^n5. 
 729 
 
 10. 
 
 243000000 
 
 2189187729 
 
 27009 243081 243243081 
 
 2189187729 
 
 
 1,762.790,912 (12.08, Ans, 
 1 
 
 
 300 
 
 762 
 
 32 64 
 
 364 
 
 728 
 
 
 4320000 
 
 34790912 
 
 3608 28864 
 
 4348864 
 
 34790912 
 
 11. 
 
 1,061,520,150,601 (10201, Ans. 
 1 
 
 302 
 
 60^ 
 
 30000 
 [ 30604 
 
 61520 
 61208 
 
 30601 
 12. 
 
 3060] 
 
 312120000 
 L 312150601 
 
 312150601 
 312150601 
 
 33,212,361.641,984 (321.44, Ans, 
 
 27 
 
 92 
 
 184 
 
 2700 
 
 2884 
 
 6212 
 5768 
 
 961 
 
 961 
 
 307200 
 308161 
 
 444361 . :: 
 308161 
 
 9634 
 
 38536 
 
 30912300 
 30950836 
 
 136200641 
 123803344 
 
 96424 
 
 385696 
 
 3098938800 
 3099324496 
 
 12397297984 
 12397297984 
 
 (179) 
 
114 
 
 EVOLUTION. 
 
 13. 
 
 1,371,737,997,260,631 (111111, Ans. 
 
 
 
 1 
 
 
 
 
 300 
 
 371 
 
 31 
 
 31 
 
 331 
 
 331 
 
 
 
 36300 
 
 40737 
 
 331 
 
 331 
 
 36631 
 
 36631 
 
 
 
 3696300 
 
 4106997 
 
 3331 
 
 3331 
 
 3699631 
 
 3699631 
 
 
 
 370296300 
 
 407366260 
 
 33331 
 
 33331 
 
 370329631 
 
 370329631 
 
 
 
 37036296300 
 
 37036629631 
 
 333331 
 
 333331 
 
 37036629631 
 
 37036629631 
 
 14. 
 
 0.171,467 (0.55655+ Jin*. 
 125 
 
 15. 
 
 155 
 
 775 
 
 7500 
 
 46467 
 
 A1 0*7 ti 
 
 
 
 1655 
 
 8275 
 
 907500 
 915775 
 
 5092000 
 4578875 
 
 16655 
 
 83275 
 
 92407500 
 92490775 
 
 513125000 
 462453875 
 
 166655 
 
 833275 
 
 9257407500 50671125000 
 9258240775 46291203875 
 
 0.004,235,801,032 (0.1618, Ans, 
 
 1 
 
 36 
 
 216 
 
 300 
 516 
 
 3235 
 3096 
 
 481 
 
 481 
 
 76800 
 77281 
 7776300 
 7815004 
 
 139801 
 77281 
 
 4838 
 
 38704 
 
 62520032 
 62520032 
 
 (179) 
 
CUBE ROOT OF NUMBERS. 
 
 115 
 
 CONTRACTED METHOD. 
 (9^3, page 181.) 
 
 1^442249 + , ^w». 
 3.000000 
 
 1 
 
 
 
 300 
 
 2.000 
 
 34 
 
 136 
 
 436 
 
 1744 
 
 
 
 58800 
 
 256000 
 
 424 
 
 1696 
 
 60496 
 
 241984 
 
 
 
 62208 
 
 14016 
 
 432 
 
 86 
 
 62294 
 
 12459 
 
 
 
 6238 
 
 1657 
 
 4 
 
 ] 
 
 8239 
 
 1248 
 
 
 
 624 
 
 309 
 250 
 
 
 
 62 
 
 69 
 66 
 
 1 1.912931+ , Ans. 
 7.000,000( 
 1 
 
 
 
 300 
 
 6000 
 
 39 
 
 351 
 
 651 
 
 5859 
 
 
 
 108300 
 
 141000 
 
 671 
 
 571 
 
 108871 
 
 108871 
 
 
 
 109443 
 
 32129. 
 
 673 
 
 115 
 
 109558 
 
 21812 
 
 
 
 10967 
 
 10317 
 
 6 
 
 5 
 
 10972 
 
 9875 
 
 
 
 
 
 
 1098 
 
 442 
 329 
 
 
 110 
 
 113 
 
 
 
 
 110 
 
 (181) 
 
116 
 
 EVOLUTION. 
 
 16.38321261+, Ant, 
 156.000,000,000 
 125 
 
 
 
 7500 
 
 31000 
 
 153 
 
 459|7959 
 
 23877 
 
 
 
 842700 
 
 7123000 
 
 1598 
 
 12784 
 
 855484 
 
 6843872 
 
 
 
 86833200 
 
 279128000 
 
 16143 
 
 48429 
 
 86881629 
 
 260644887 
 
 
 
 86930067 
 
 18483113 
 
 16149 
 
 3230 
 
 86933297 
 
 17386659 
 
 
 
 8693653 
 
 1096454 
 
 161 
 
 16 
 
 8693669 
 
 869367 
 
 
 869369 
 
 227087 
 
 
 
 
 173874 
 
 
 86937 
 
 53213 
 
 
 
 
 52162 
 
 
 8694 
 
 1051 
 
 
 
 
 869 
 
 |32.643859 +, Ant. 
 34,786.000 
 27 
 
 92 
 
 148 
 
 2700 
 2884 
 
 7786 
 5768 
 
 966 
 
 5796 
 
 307200 
 312996 
 
 2018000 
 1877976 
 
 978 
 
 391 
 
 318828 
 319219 
 
 140024 
 127688 
 
 10 
 
 3 
 
 31961 
 31964 
 
 12336 
 9589 
 
 
 3197 
 
 2747 
 2557 
 
 
 32 
 
 190 
 160 
 
 
 3 
 
 30 
 
 28 
 
 (181) 
 
CUBE ROOT OF NUMBERS. 
 
 117 
 
 «. 
 
 |2.222222 +, Ans, 
 
 10.973,937 
 8 
 
 
 
 1200 
 
 2973 
 
 62 
 
 124 
 
 1324 
 
 2648 
 
 
 
 145200 
 
 325937 
 
 662 
 
 1324 
 
 146524 
 
 293048 
 
 
 
 147852 
 
 33889 
 
 666 
 
 132 
 
 147984 
 
 29597 
 
 
 
 14812 
 
 3292 
 
 7 
 
 1 
 
 14813 
 
 2962 
 
 
 1481 
 
 330 
 
 
 
 
 296 
 
 
 148 
 
 84 
 
 
 
 
 30 
 
 1 11.447400 66 -f, Ans. 
 1,500.101,520 
 
 
 
 300 
 
 600 
 
 31 
 
 31 
 
 331 
 
 331 
 
 
 
 36300 
 
 169101 
 
 334 
 
 1336 
 
 37636 
 
 150544 
 
 
 
 3898800 
 
 18557520 
 
 8424 
 
 13696 
 
 3912496 
 
 15649984 
 
 
 
 3926208 
 
 2907536 
 
 8432 
 
 2403 
 
 3928611 
 
 2750028 
 
 
 
 393101 
 
 157508 
 
 84 
 
 14 
 
 393115 
 
 157246 
 
 
 39313 
 
 262 
 
 
 
 
 236 
 
 
 39 
 
 26 
 
 
 
 
 23 
 
 (181) 
 
118 
 
 7. 
 
 RADICAL QUANTITIES. 
 
 | 1.051963 +, Ans, 
 1.164,132 
 
 1 
 
 305 
 
 1525 
 
 30000 
 31525 
 
 164132 
 157625 
 
 315 
 
 32 
 
 33075 
 33107 
 
 6507 
 3311 
 
 
 3314 
 
 3196 
 2983 
 
 
 331 
 
 213 
 199 
 
 
 33 
 
 13 
 10 
 
 RADICAL QUANTITIES. 
 (^47, page 183.) 
 
 1. V^=*^26x3=>/3, Ans. 
 
 2. f^8^«=f'49a« X 2 = 7af/2, Ans, 
 
 3. V\2x''y=VAx' x 3y=2a4^, Ans, 
 
 4. V547*=V27j;' x 1x=z^x\^2x, Ans, 
 
 5. 4^108 = 4^27 X 4 = 12V4, Ans. 
 
 6. f^x'-aV=*^a:»(a;-a')-a:V^^3^, Ans, 
 
 7. 6^32^= 6 Vsa' x 4:^12aV4, Ans. 
 
 8. 3f^2 8a V = 3f/4a V x 7a = QaaVia, Ans, 
 
 9. Va»+a'6«=>/a'(l +6')=aVr+6', ^»«. 
 
 (181-183) 
 
REDUCTION. 119. 
 
 10. {x-y)V2x*-^x'y + 2xy''=i{x-y)V2x{x'-2xy^-y^) 
 
 = {x-y)V2xx{X'-yy=(x-yyV2x, Ans, 
 
 11. (a - hy^d'h + ^ab^ + 26' = (a - 6)f'26(a' + 2ab + 6') 
 
 12. 66(6'-6')2=56[(6-l)6']^=56'(^-l)^ ^'i*- 
 
 13. (2a'6»-3a'6')5=[(2a«-36')a'5']^ = a6(2a'-36y, J»*. 
 
 14. ^(a*6' + a*6*)^=^[a»6»(a6' + a«6)]^=a'(a6' + a'6)^, Am, 
 
 15. ♦^8a'"'a^=V'4a'V» X 2 = 2a V'V2, Ans. 
 
 16. V^^= Va^V' X ar^=a'^c"'y/a'^c'^'^^ Ans, 
 
 1 11 
 
 1 7. (2a*»y- - 3a*"y»'»)*'= [a;»«y"(2 — 3a:"y«-)]- =ar'y(2 — 3a-"y"»)"», 
 
 1 1 1 
 
 18. a-*"c(a'^c'''— a'-^V) " =a-^c\ar''<r((r—a'^)] '* =€*(€"— oT) *, 
 
 (348, page 1.84.) 
 
 s.VT^=\/jj^=vir;rio=^VTo,Ans. 
 
 4. >/Y=Vi|z=iV75, Ans, 
 
 5, ^^=^H=^j*'t X 6=5*T^6, Ans. 
 
 6. 2|/|^2|/|=|i/6^, ^n,. 
 
 (183-184) 
 
120 RADICAL QUANTITIEa 
 
 8. -\ -T-;=-V -T-i x«*=-^^a^ ^W5. 
 
 (049, page 185.) 
 
 3.. {a'^cz)= [(a-c^)*]^ = (a*-4aV2: + 6aV«-4acV + cV)^, 
 
 Ans, 
 
 6. (a— 26)V^=*^2a(a-26)'=*^2a(a'-4a6 + 46'') 
 
 (950, page 187.) 
 
 4. The least common multiple of 1, 2, 3, and 4 is 12 ; whence by 
 Rule II, 
 
 »Va", *'V^, ">/^ 'V^«, ^n5. 
 
 8. The least common multiple of 2, m and n is 27nn ; whence, 
 •"Va-^^V^, '*V?y^, ""V^^^V^, Ans. 
 
 ADDITION OF RADICALS. 
 
 (051, page 188.) 
 
 1. ^l6a*x=4aVx 2. ^32= 4*^2 
 
 Vi^=2aVx Vn= QV2 
 
 QaVx, Am. f^l28 = _8l/2 
 
 18*^2, -4««. 
 
 8. V40= 2V5 4. Vl08= 3V4 
 
 Vl35= 3V/5 9V4 
 
 ^625= 5V5 Vl372= 7V4 
 
 ioV/6, ^n«. 19 V4, Aim. 
 (184-1S8) 
 
V. 
 
 ADDITION. 
 
 i; 
 
 V^ = ^V2 
 
 6. 
 
 Vv= |V3 
 
 V^=l^2 
 
 
 ^fi= ^V3 
 
 Vj\=iV2 
 
 
 ^fH= |V3 
 
 y2, Ans. 
 
 
 iiV3, ^n^. 
 
 Kl =K6 
 
 8. 
 
 3Va6»i'=3mf^ 
 
 if^=if/6 
 
 
 mV4ab = 2myab 
 
 i^U = T\^^ 
 
 V25abm'=5mVab 
 
 lif/6, Ans. 
 
 10wVa6, ^>w. 
 
 121 
 
 0. 2aVc*x--c'y=2acV'a:— y 
 
 3c Va'ar — a'y = 3ac l^a: — y 
 
 5 ^a^c^x — d*c^i/ =:5acVx—y 
 
 lOaci^a;— y, Ans, 
 
 10. 
 
 K20a'm— 20acm + 5mc' =f^5m(4a' — 4ac+c') =(2a--c)>^6; 
 
 V207»c' — 60acw + 45a'7» = V'5m(4c' — 12ac4-9a') = (2c— 3a)i^6»i 
 
 (c— a)l^5m, ^rw. 
 
 11. 3V^« = 3a:Vc 
 
 y/ax*=z Xy/a 
 
 2>^ax*z=2xVa 
 
 12. 
 
 3xv/c + 3a;Va=3ar(Va4-Vc), Ans. 
 2 j:(a'c? — aV) 3 = 2ax(rf — c) 3 = — 2aar(c — c?)^ 
 
 i 
 Zax(c—d)^^ Ans. 
 
 Note. — In transforming the second quantity, observe that the cube root of 
 a negative quantity ia negative. 
 
 (188-189) 
 
122 RADICAL QUANTITIES. 
 
 ' ai-b ' {a + by^ ' a + b 
 
 Hence the sum is, 
 
 (4_+-L.+°±:^y.-rr6-=("'-°^+°^:+'''--'^*Va.JI' 
 
 \a+b a—b a'—b' J \ a'—b' f 
 
 =2^^*^^', Am. 
 
 "• ♦'(^ +«)" -'^^S'(' +°)=TT^''i +« 
 
 nee the sum is 
 Vl+a l+a 1— a/ 1— a' 1— a 
 
 
 1-a 
 
 1, 
 
 SUBTBACTION OF RADICALS. 
 
 (asa, page 
 
 189.) 
 
 
 41^135 = 121^15 
 
 2. 
 
 ♦^75=51^3 
 
 ^Veo = 4V'l5 
 
 
 ♦/50 = 5*/2 
 
 8*^15, Ans. 
 
 6(V^3-f'2),-4n*. 
 
 (188) 
 
 
 
SUBTRACTION. 123 
 
 3 ^^= 3 aVb i^^W=i^Hfll=,\*/lT 
 
 (1 2a' - 3ayb, Ans. |f/lT, Ans, 
 
 5. ?i/l?_^-l'^V4^'.5^MV5 
 S'^ 338 3^^ 169 39 
 
 ^V36l_W36r:_19a 
 IS'^ 5 ~13^ 25 65 
 
 Hence the difference is 
 
 (14a 19a\ , a , , 
 39 65/ 16 ' 
 
 6. (aV-3c'a;)3=c(a'-3x)^ 
 2(aV-3rf*x)3 = 2rf(a'-3a;)i 
 
 (c-2rf)(a'— 3ar)3, ^^j. 
 
 1. (a'-a6' + a*6-6y={o'-6'+a6(a-i)}^ 
 
 = {(a« + a6 + 6' + a6)(a-6)i^=(a + 6)(a-fi)^ 
 
 (a'-3a'6+3a6»-.6«)^= (a-6)^ = (a-6)(a--6)^ 
 
 1 
 26{a— 6)2, ^w«. 
 
 8. a/^=„*|/^=V^=A^-"^ 
 a; — 1 '^ ar — 1 '^ (•^"'1) ^""^ 
 
 ' « + l ' « + l '^ (-e + l) *+l 
 
 ~Vl^l, An,. 
 
 «'— 1 
 
 (l89-i90) 
 
124 RADICAL QUANTITIES. 
 
 MULTIPLICATION OF RADICALS. 
 (3535 page 191.) 
 
 1. 6»/6 X 3*/8 = 15V^ = 15V^4 x 10=:30Vlo, Ans, 
 
 2. 4V'l2x 3^/2 = 12^24 = 12^4 x 6=24t/6, ^n*. 
 
 3. 3f^2 X 2f^8 = 6^^6 = 24, ^n5. 
 
 4. 2*^5 X 2*^10 x3f/6 = l 2*^300 = 12M0ax3 = 120f^3, Ans, 
 6. 2^^14x 31/4 = 6^56 = 6^8 x 7 = 12V7, Ans, 
 
 6. oc^/^X cVa' X V^' = 5cV^V X cVa* X V^=5c' V^V 
 
 7. (xy)^ X (^)* + (y^)^=(xy)^'^ X (xV)tV X (yV)!"? 
 
 8. (:r-y)5 x (x + y)^=[(ar-yr(x+y)']^^='V(^-y)>+y)' 
 
 = "^l(^-y)(^+y)}>+y) 
 
 = V(:r'-y')»(a: + y), ^n*. 
 
 9. Vl5 X ♦^10=^225 X Vi000=V225000, ^n«. 
 
 a « /a; yi/b' j /6? ay s /x' e /5* V^* 
 
DIVISION. 125 
 
 11. If we square the two first factors under the radical sign, V' , 
 we may extend the sign, ^/ , over the whole product ; thus. 
 
 
 
 DIVISION OF RADICALS. 
 (254, page 192.) 
 
 3 V20a^__ V400aV^ _ V 400aV' V 16a 
 
 4. (^qi = (a«6')^ = (a5)^ ^n.. 
 
 ^ (16a'-12aV)* .^ ^ .\ . 
 
 5. ^ — '— = (4a— 3ar)'^, Ant, 
 
 (4a')^ 
 ^ 45 3-5 3(5')^ .^ ■ 
 
 ^ 12c'(a-;r)* 3c[(a-xyy^ ^ . . • . 
 
 8. ^^ '—= — 5-^^ '-^^ — =:3c(a— ^)Ta, Ant, 
 
 4c(a-x)3 [(a_ar)']TV 
 
 i- JL 
 
 9. ^ ^ = i — i-L_= (^8»-"c— 3'«)«-, Ans. 
 
 {ac'y (a'^c"")"^* 
 
 (191-192) 
 
126 RADICAL QUANTITIES. 
 
 10. 
 
 Vx Va Va' 
 
 Va' 
 
 Va'b-ab' Va'b-ab' Va'bWb-ab') 1 w-— ~ 
 
 (a585 page 194.) 
 1. {V2^y=zV8^'=V8^,^^^=aV8^,Ans, 
 
 3. (3V4a^)*==:8lV266a»V==8ir ^256a"c*==8lVi6a'V 
 
 = 81 Vsa' X 2oc' = 1 62a' ^^20?, ^n*. 
 
 5. (Vl2^)'=t/l2^' = f^46' X 3a=26V3^, ^n*. 
 
 6. ("v/c(a_x)')'=(Vc(a-a;)')'=Vc»(a-ar)* = (a~a:)Vc'(a-ar), 
 
 8. (Vxy(x-y)y = Vxy{x-yy=xyVxi/{x-y)%Ans. 
 
 10. f-V96^') =^ ^96^"=^ Vs2x' xdcx=—V^^ Ans, 
 \x / x' x" X ^ 
 
 (^595 page 196.) 
 
 3. y2v^98=y v^8x98=:r v/784=V28, ^n*. 
 (192-196) 
 
POWERS AND ROOTS. 127 
 
 4. Putting the fifth power of the coeflScient under the radical sign, 
 we have 
 
 r v^^^486:=y v/64=V8, Ans. 
 
 6. Y 5V 5 =yVU5=y>^ 12~5=V 5^ Ans. 
 
 11 J- i 
 
 ~^j\ = \-r-if ; dividing out the factor 2 in the expo- 
 
 (ax \^ 
 -^-\ , Ans. 
 
 (aeO, page 197.) 
 8. Vl296=V Vl296=f/36 = 6, Ans, 
 
 4. Vl7797851516625=y ^17797851515625 
 
 =V421875=76, Am, 
 
 6. Vl9110297'6=y ♦/I91102976==v^l3824=24, -4iM. 
 
 6. V65536=y |/(65536)2=|/(256)^ = (l6)i=4, Ans. 
 
 7. We first take the square root, which is 
 
 a»_4a6 + 46'; 
 then the square root of this is, a— 25, Ans, 
 
 8. The square root of the expression is 
 
 and the cube root of this is a* + 5, Ans, 
 
 (196-197) 
 
128 RADICAL QUANTITIES. 
 
 GENERAL THEORY OF EXPONENTS. 
 (aei, page 199.) 
 
 6. a*b* xJb^=Jb^=a^y^, Ans. 
 
 7. Adding the exponents we have 
 
 aM 1-1 1-1 -» 1 /c'\i ^ 
 
 (jr*V *^ ^ 
 
 {x^y x^ 
 
 10. (a^-a^)(a^ + l)=a^4-a^--a^-a*=a*-a^, u4n*. 
 
 11. {2Vx'+yxy){SVx-yxy)=eVx'-2Vx'-yzy + 3Vx'yi^-xy 
 
 = 62;— 2Va;y + 3Va;y— a:y, ^n«. 
 
 12. a^-2a~* + a"* , , , ,. 
 
 13. a--b Va+Vb 
 
 J- a-i 
 
 1 
 
 a-hyablVa-^Vb, Am, 
 
 a*— 2a' + a * —Vab^b 
 
 — a" +2a~*— o~^ — Va6-6 
 
 a*— 3 4.3a * —a ^ , ^W5. 
 
 (199-200) 
 
THEORY OF EXPONENTS. 
 
 129 
 
 14. 
 
 15. 
 
 16. 
 
 Ill 
 J- J 
 
 a'-l 
 
 a —a", Ans. 
 
 - J+J 
 
 A i. 1 A. 18 .!_• 
 
 J-b^ 
 
 4 2 i. 3 4 1 !_• 
 
 a* + a^b^ -\-a'b^ +a*b' + ab^ -{-a^^ 
 
 5 2. i- i 11 1J» 
 
 -.a^b''~-a'b''-a*b'-ab^-a^b => -6* 
 
 a'—b*y Ans. 
 
 i. Ill 
 
 js+ar^a^+a^ 
 
 2. 11 a. 
 
 1 a. a. 1 
 
 If i F 
 111 !• 
 
 / Irirf^S y ld=31/5 4-15±6f/5 ^. . . 
 \~2~/ "^ 8 =2±|/5, ^w.9. 
 
 19. Raising each factor of the numerator to the power of nr^ and 
 each factor of the denominator to the power of ws, the expression 
 becomes 
 
 cT'y.cT'x oT' ~~ c'^+'-"«- ~\c) 
 (200) 
 
 Ans. 
 
130 BADICAL QUANTITIES. 
 
 ( 2f3-2Vl08 ) ^ j 2t/3-6V4 | ' ft' /i\^ /2 ' /72?\* 
 
 • llV^sWf ^leWaFsl =(31/9; =l3V ^3)^ 
 
 * (iV5 + |V2) ( i(2l/5 + 5V2) f 
 
 (21/5 + 5V2) 
 
 =? ( (2l/5-5V2)(2V5 4-5V2 ) ) * 
 ~5( 2V/5 + 6V2 f 
 
 = |(2V5-5V2)^, ^»». 
 
 22. Performing the multiplications indicated in the numerator 
 and denominator, the expression becomes 
 
 a-h 
 
 Va-n 
 
 —Va-\-Vhy Ans. 
 
 23. Omitting the exponent, the numerator is 
 
 (f/5 + 2)(V5+i'2)(V5-»/2) = (*/5 + 2)(i/5'-2) = 5-4 = l; 
 and the denominator, 
 
 (♦'i3 + 3)(VT3+f^3)(VT^-3)=:(n3 + 3)(*1i-3)=13-9=4; 
 and taking the square root, i, Arts, 
 
 IMAGINARY QUANTITIES. 
 
 ( 268, page 203.) 
 10. 
 
 (a + V^c)*=a* + 4ay=:^ + 6a'(»/~c)V4a(V'irc)'+(f/— )\ 
 
 =a* + c' — 6a'c + (4a*— 4ac)f^^, Am. 
 
 (200-203) 
 
QUADRATIC SURDS. 131 
 
 11. a^V-^ci) a'-f ^^^a {a-\-V —a—\^ Ans, 
 
 a — ay — a 
 
 aV^^a + a ^ 
 
 -a + ^/Hi 
 12. The equation can be written, 
 
 From (!807) we must have 
 
 a + y=c + ar, (1) 
 
 and xVc=yVa, (2) 
 
 Whence, by substituting in (1), 
 
 Vc 
 X — —x=ia—c 
 
 x=^ — —, — =:a-\-yac I 
 In the same way, y=c-\- ^ac ) 
 
 (275, page 208.) 
 4. Here wehave a=ll, Vb-QV2, 6 = 72; hence, 
 
 ll4-f^l21-72 ^ 
 x=. =9, 
 
 ll-*^21-72 „ 
 v= =2 ; 
 
 f/a;+|/y=3+i/2, Ans, 
 5. o=7, f6=4V'3, 6=48; hence, 
 
 7 + *'49- 
 
 -48 
 
 ~ 2 
 
 7-n9- 
 
 -48 
 
 y= = 3; 
 
 y 2 
 
 ^x—yi/ = 2—yd, Ans, 
 
 (204-208) 
 
132 RADICAL QUANTITIES. 
 
 6. a=7, Vh=2VlO, 6=40; hence, 
 
 7 + */49- 
 
 -40 
 
 2 
 
 7-^9- 
 
 -40 
 
 2 
 
 Vx--Vy=V5-V2, Ans. 
 
 Y. a=94, Vb=42V5, 6=8820; hence, 
 
 94 4- ♦^8836 -8820 _ 
 ar= _ —40 
 
 94 — ♦^8836 -8820 , 
 y= =45 ; 
 
 Vx+Vy=1 +^45 = 7 + SV5, Ant. 
 
 8. a=28, f^6=10V^3, 6=300; hence, 
 
 28+*^784 — 300 
 x= = 55, 
 
 28— ♦^784 — 300 
 
 y= n =3; 
 
 Vx+Vt/=5+ys, Ans, 
 
 9. az^np-h 2m', Vb = 2m*'njo + m', 6 = im'np + 4m* ; hence, 
 ^_ n/? -f 2m' +»'ny + 4m'n/? + 4m* — 4m^np— 4m* 
 
 2 
 =np + m''; 
 
 _np + 2 m' — ^w'y + 4m'n;7 + 4m*— 4m'np— 4m* 
 y _ 
 
 =m' : 
 
 f'a;— V'y=Vwp-fm' — m. Ant, 
 
 10. a=6c, f'6=26<^6c-6', 6=46'c-46*; hence, 
 
 6c+*^6V-46'c + 46* ^ „ 
 ar= =6c-6', 
 
 6c-i^6V-46'c + 46* „ 
 
 y= 2 =^^ 
 
 i^a?+f/y=*^6r— 6' + 6, ^n*. 
 (208) 
 
QUADRATIC SURDS. 133 
 
 11. a=Y, Vb=30V-2, 6 = - 1 800 ; hence, 
 7+*^49 + 1800 
 
 2 
 
 7—^49 + 1800 
 
 = 25, 
 = -18; 
 
 2 
 
 12. a=16, f/6=30V^-l, 6=-900; 
 
 16+^256 + 900 
 :r=: = 25, 
 
 16— K256 + 900 
 
 y= 2 =~9; 
 
 ♦^ar+V'y=5 + 3VCri 
 also, i^ar-»^y= 5-3»^iri 
 
 10, Ans, 
 
 13. We add the answers of examples 4 and 6, and have 
 
 3 +1^5, Ans. 
 
 14. a=:31, VA=12»^^, 6=-720; hence, 
 
 31+V961 + 720 
 x= = 36, 
 
 31-1^961 + 720 
 
 y= , =-5. 
 
 Again, a=K—l, V6=4f^^T, &=— 80; hence, 
 
 — l+»^l + 80 
 x= = 4, 
 
 — i-vr+80 
 
 y= 2 -=-'' 
 
 Therefore, 6 +^^ + 2 +f^=8 + 2VC:6, Ans. 
 
 (208) 
 
134 BADICAL QUANTITIES. 
 
 RATIONALIZATION. 
 (280, page 211.) 
 
 2. Multiply both terms of the fraction by |/6. 
 
 3. Here the factor is V«'. 
 
 . XTT 1 ^2x^81 ^2x\^2i~xS V8xV9 ^72 , 
 
 4. We have = = =— — - , Ant, 
 
 9 9 3 3^ 
 
 5. The factor is f/7—f'3. 
 
 6. The factor is j^a-\-\^c. 
 
 , Ans, 
 
 7. The factor is VTT+f^5, and we have 
 
 11 4-2*^55 + 5 _8 + »/55 
 6 3 
 
 8. The factor is VTT-V'3. 
 
 9. The factor is VTo+*/6 ; whence, 
 
 4 
 
 10. Multiply both terms of the fraction by t^6— V^^, and we 
 have, 
 
 6-2f^^irTg-3 1-t^^^^ , 
 
 s =—4—^ ^'^*- 
 
 11. The factor for rationalizing the denominator is (5 -\-\^5)(V3 — 1); 
 whence 
 
 (3+»^3)(3+»^5)(f5-2)(5+t^5)y3-l) 
 40 
 The product of the first and last factors of the numerator is 2 f^3 ; 
 of the others, 4V5. Therefore, 
 
 SVTE ,/— , 
 =1V15, Ans. 
 
 40 
 
 (211-212) 
 
RADICAL EQUATIONS. 135 
 
 12. Multiply both numerator and denominator by 1+a+^l— «'; 
 whence, 
 
 (l-i-«)' + 2(l+a)»^T^^' + l -a' 2(l+a)4-2(l+a)»^r^' 
 (1 +«)' — !+«' "■ 2(l+a)a 
 
 , Ans. 
 
 13. Comparing this with example 2, (^TO), we have a=5, x=2 ; 
 and the complete multiplier is 
 
 (^/5 + V2)(5 +V2) = 5f/5 +VlO + 5V2 + V8, An», 
 
 14. By (3) the factor is 
 
 x*—y' a*—b* L „i 14 ,, 18 u> 
 --=-1 ^=a«+a'63+a263-fafe'4-a='63+6 3 
 
 Hence, j— ^^ ? , Ans, 
 
 RADICAL EQUATIONS. 
 (a82, page 214.) 
 
 1, Transposing Vxy and squaring each member of the equation, 
 
 x-\-1=z4d—Uyx-\-x, 
 UVx=iA2, 
 
 Vx=Z ; ir=9, Ans. 
 
 2. Squaring each member, 
 
 a:' + 6a; + 9=ar'— 4a; + 69, 
 
 10^ = 50 ; x=5y Ans, 
 
 3. By squaring, Vx + 48 = 2^x; 
 
 squaring again, 3ar=48 ; x=lQy Ans, 
 
 (212-214) 
 
136 RADICAL QUANTITIES. 
 
 4. By cubing, |/j; + 2V^a + a;=a— ♦^a + a;. 
 
 Squaring this result, x + 2Va + a; = a' — 2aVa-\-x-{- a+x, 
 2(l+a)*^a+^=(l+a)a, 
 
 a' a'— 4a ^ 
 
 a-f-a;=— ; «= — - — , AnS, 
 
 4 4 
 
 5. Multiply by c f'ar, then 
 
 ac + az=c V7i; or x=c(Vn'-a\ Ans. 
 
 6. Multiply by f^l —a:', the least common multiple of the denomi- 
 nators, and we have 
 
 y^- -^ L + l^x=3x, 
 
 1 — X 
 
 (l—x)-\-(l+x)=3x; hence «= |» -4n«. 
 
 V. Multiply by Vc-l-x, 
 
 c+ar=Va4-p, 
 
 squaring, c'+2car+^'=a + ar; hence ^=""5 — , -4»«. 
 
 2(7 
 
 6. Transposing x and squaring, 
 
 9 + a^a;'— 3 = 9-6a; + a;', 
 
 a;' — 3 = 36 — 12ar + a;', 
 12a:=39 ; or «=3{, Ans. 
 
 9. By transposition, 2V« — 32 = 2 (\^x—ys), 
 
 ar— 32=a; — 2V^+8, 
 V^=20, 
 8a:=400; hence a;=60, ^n*. 
 (214) 
 
Ans, 
 
 RADICAL EQUATIONS. 137 
 
 10. The least common multiple of the denominators is «— 4, or 
 (yx^1i){i/x-{-2) ; multiplying by this, 
 
 ayx-}-2a—a — c=c\^x—2Cy 
 {a~cyx=—ai-c ; 
 
 therefore, ar = — I — — J , 
 
 11. Factoring the first member, we have 
 
 Vm(yx—l)_ Vx + m 
 ?c(f^x-l) — Vx + c' 
 VmX'\-eVrn=Vcx-\-mVc, 
 
 therefore, xz=mcy Ans. 
 
 12. By cubing, a*'-da'x + x*Vda—x=a*'-3a*x-{-dax*—x'; 
 
 V3a—x=:3a—Xf 
 or 1 z=Vda—x ; or «=3a— 1, Ans, 
 
 13. Clearing of fractions, 
 
 xVc* — aa: + c' — ax = c*, 
 
 c'—ax=:a' ; or «== , Ans, 
 
 a f^ 
 
 14. Squaring, i+^-^l^+lf^, 
 
 transposing, <fec., - + 
 
 X 5 5 X 
 
 
 1,41,4 1,1 
 squaring, _ + _._ + _=_4.- ; 
 
 11^ 
 
 -=^ ; hence, a;=20, ^n*. 
 
 15. By squaring, a— a;=i^a+V 
 
 a*—2ax-{-x*=a-{-x\ 
 
 2aa;=a'— a ; hence, ar=———,^«#. 
 
 (214-215) 
 
138 RADICAL QUANTITIES. 
 
 16. Clearing of fractions, 
 
 VA-\-5x-[-x*z=zV4—Xy 
 
 V4 + 5x-^z^ = 4—x, 
 4-\-5x-{-x^ = lQ — Sx + x*\ hence, a;=||, Ans, 
 
 17. By transposition, 
 
 v/5-x=ViO-V5+3-, 
 cubing, 5-x=l0- (3VIoo)(V5Ti) 4- (sVlo) (V(5+ar)') -5-ar, 
 canceling terms, (v^io) C>/(5 T^) = (3 Vioo) {VJ+x), 
 dividing by (3VlO)(V/5 + a?), v/6+a:=Vio, 
 
 5+x=10; hence, a: =5, ^n«. 
 
 Note. — If we had transposed the term Vs — x and solved, we should have 
 found a; = — 5 ; and it is evident that either value, x = ± 5, satisfies the origi- 
 nal equation. 
 
 18. Multiply hy^a+x, 
 
 Vax + x*+a-{-x=2x, 
 
 yax-\-x*=a—Xj 
 tzx-\-x*=a* — 2ax-\-x*^ 
 
 therefore, x=-jAn8, 
 
 3 
 
 19. Squaring, x^ + 2ax-\-a*=a^ -h^b*x'-\-x\ 
 
 ir' + 2aar=V6V + a:*, 
 X* + 4ax* + 4a V = 6 V + x\ 
 4<M; + 4a'=6'; 
 
 therefore, x= , Ans, 
 
 4a 
 
 20. Clearing of fractions, 
 
 24ar-8*^6x + 6*^6^— 12=24a:-9*^+8*^6^— 18; 
 whence, 
 
 ^6^=6; x=6y Ans, 
 
 (216) 
 
RADICAL EQUATIONS. 139 
 
 21. Cubing, we have 
 
 4-f-z 
 or, 64-\-x*—8z=l6 + Sx+x\ 
 
 I6xz=4:8; hence, x=df Ans. 
 
 22. Multiplying by ^5 +^, 
 
 VdX-^X^zzzlO — X, 
 
 by squaring, 5ar— ar' = 100 — 20a;4-a?*, 
 
 252:=100; hence, x=:4, Ans, 
 
 23. Multiplying by the denominator Vx-\-Vx^ 
 
 x+Vx-Vx^-x-^Vx', 
 dividing by ^x and transposing, 
 
 Vx-l=Vx-i, 
 squaring, x—l=x —Vx + j, 
 
 -f/arcri; hence, x=\^^ Ans, 
 
 24. Clearing of fractions, we have 
 
 3aa; + 56f^— 36Vo^— 56' = 3(Mr—25Va^ + 36Vai— 26'; 
 whence fai=36, 
 
 aa;=:96' ; 
 
 therefore, a;=— , ^n5. 
 
 a 
 
 25. Clearing of fractions, 
 transposing and dividing by 2, 
 
 4t/4ar+l=5l^, 
 Squaring, 6 4ar + 1 6 = 1 00 j*, 
 
 36ar= 16 ; hence, x=z^^ Ans, 
 
 (215) 
 
140 QUADRATIC EQUATIONS. 
 
 26. Clearing of fractions, 
 
 95f/x=190, 
 
 Vx—'l ; hence. ar=4, Ans. 
 
 27. Multiply the numerator and denominator of the left hand 
 member by the numerator, 
 
 a a?— a* 
 
 Multiply by a, and take the square root, 
 
 yx+i^x-a=± 
 
 Vx-a 
 
 mnltiplying by Vx—a, 
 
 squaring and transposing, we have, T=^a!^ —, Ant, 
 
 PURE QUADRATICS. 
 
 (^865 page 21 Y.) 
 
 , 5. aar' + l = (a-a-)(a + x), 
 
 By expanding =:a'— ar', 
 
 or, (a + l)a:'=a»-l, 
 
 a;'=a— 1 ; hence, a:=±f^a— 1, Ant'- 
 
 6. Clearing of fractions, 
 
 3(a; + 4)' + 3(a:-4)» = 10(a;'-16), 
 or, 62:' + 96 = 10a;'— 160, 
 
 4a:' = 25 6, 
 a;'=64 ; hence, a:= ±8, Ans, 
 (215-216) 
 
PURE QUADRATICS. 141 
 
 7. As before, Q{x + 2y + 6(x-2y = lS{x'-4), 
 or, 12i;' + 48:=13a:' — 52, 
 
 a;'=:100; hence, ir=dblO, -4«5. 
 
 8. Clearing of fractions, 
 
 (a; + a)'-f(ar-a)' = 7(a;'-a') 
 or, 2a;' + 2a'=:7a;' — 7a», 
 
 5x* = 9a\ 
 
 , 9a' 3a . 
 
 X = — ; hence, ar= ±-r-- , ^ns, 
 5 r 5 
 
 2! q 
 
 9. Omitting - and - on each side of the equation it becomes 
 
 l_x 
 
 ^""12' _ 
 
 or, «'=12; hence x ±V I2y Ans. 
 
 10. This equation has the same form as example 9; but as we can 
 not unite the letters as we did the numbers in 9, we clear of frac- 
 tions; whence, 
 
 ex* -f CL*c = ax* -\- ac*y 
 x*(a—c)=ac(a — c), 
 
 x^=iac\ hence ir = ±V^, ^n*. 
 
 11. Clearing of fractions, 
 
 «' — 8 = 6 + 6f^5, 
 whence, ar'=14 + 6f^5 ; 
 
 andby(ar5), . x=z±(d+V5), Ans, 
 
 12. Multiply by 6, then 
 
 2x*— 4f/2-3a;» + 9 = 6-6f^2; 
 whence, a;'=3 + 2l/2; 
 
 and by ( 375 ), x=db{l +f^2), Ans. 
 
 13. This equation may be written, 
 
 whence, ar"=:9; or, «=±3, Jn«. 
 
 (217-218) 
 
142 QUADRATIC EQUATIONS. 
 
 It is evident the the value a:r=--2 will satisfy the equation. If 
 we clear the equation of fractions it will be 
 
 a:' + 2x'~9J--18=rO, 
 
 a complete cubic equation in which x has the three values, 
 
 ar= + 3, ar=— 3, a:= — 2. 
 
 14. Clearing of fractions and uniting terms, 
 
 6z'=6, 
 ar' = 1.2; hence a;=dbl.095445 + , -4n5. 
 
 15. Transposing, we have 
 (x-b)(x + b) 
 
 = -H(^-4), 
 
 ar + 4 
 or, 12(j;»-25) = -ll(a:»-16), 
 
 23i:*=476; hence «= ±4.64924 + , An8. 
 
 AFFECTED QUADRATICS. 
 (aOO, page 220.) 
 
 The answers to the fourteen first examples can be written down 
 immediately, according to the rule under (^890). 
 
 15. Divide by 3, thus 
 
 a;'— 5x=: — 4 ; whence ar=4 or 4-1, An8» 
 
 16 and 17 are reduced to the general form by division and trans- 
 position. 
 
 18. Multiplying, we have 
 
 6«"— 6;r-.10a;+10=:2a:* + 30, 
 4ar'-16ar=20, 
 or, a:' — 4ar=5; x^zzb or -^1, Ans, 
 
 (218-220) 
 
AFFECTED QUADRATICS. 143 
 
 19. This reduces to 
 
 lOx' — 16a; + 10a:— 16=:5a;' + 4ar + 5a; + 4, 
 6a;'— 15ar=20, 
 or, x*—dxz=z4: ; 
 
 whence, a: =4, or —1, Ans, 
 
 20. We have 9a:' + 24ar4-16 = 54a: 
 or, X — 3 X — J , 
 
 whence, ar= y ± 1 ; ar=| or |, Ans, 
 
 21. This is already of the general form, and we have 
 
 ar= +Tj=^U ; whence ar=| or —j\, Ant, 
 
 22. Dividing by 15, 
 
 , 2a- 1 
 
 a:=— j*j±f| ; whence ar=| or —f, Ans, 
 
 23. Dividing by 4, 
 
 , 13a?_ 5 
 * 28 "72* 
 ar=i^±W'V ; whence x=j\ or — /j, ^n^. 
 
 24. Multiplying by 4(j:'' — 1), the least common multiple of the 
 denominators, we have 
 
 2(a:+l) + 12=a;» — 1, 
 or, a;'— 2a:=15 ; whence x=5 or —3, Ans, 
 
 25. Clearing of fractions, 
 
 4(aj + 2)(ar + 3)4-5(a:+l)(ar + 3) = 12(a: + l)(ar + 2); 
 or, 
 4ar' + 12a: + 8ar + 24 + 6«' + 16a? + 5a;4-15 = 12a;' + 24ar-M2a? + 24. 
 
 3ar'— 4ar=:15, 
 ar' — Aa;= 6; 
 whence, ar=3 or — |, Ans, 
 
 (2»1) 
 
144 QUADRATIC EQUATIONS. 
 
 (a91, page 223.) 
 
 1. Multiplying by 5, and adding 2', the complete equation is 
 
 25j:' + 20.r + 4=1024, 
 
 52: + 2r=±32; x=Q or -^j*y Ans. 
 
 2. Multiplying by 5, and adding 2', 
 
 25x' + 20^ + 4 = 1369, 
 
 5x + 2 = rt37; x=1 or ~y, Ans, 
 
 8, Multiplying by V, and adding 10', 
 49jr'~1402r+ 100 = 324, 
 
 7a?— 10= ±18; x=1 or —^^ Ans, 
 
 4. Multiplying by 24, and adding 16', 
 
 144x' + 3602: + 225 =441, 
 
 12jr + 15 = ±21; x=^ or ^3^ Ans, 
 
 5. Multiplying by 8, and adding 5'. 
 
 16x' — 40j: + 25 = 961, 
 
 4i:--5=±31; ar=9 or - y, ^n*. 
 
 6. Multiplying by 21, and adding 146', 
 
 441a;'-292x21a: + 21316 = 10816, 
 
 2l2r-146=±104; ar=llif or 2, ^n.f. 
 
 *!, Multiplying by 24, and adding 13', 
 
 144a;'-13x 24j:+ 169 = 25, 
 
 12.r — 13=±5; ar=f or f, ^n*. 
 
 8. Multiplying by 28, and adding 3'. 
 
 1 96a:' ~842r + 9 = 4489, 
 
 14a:— 3= ±67; xz= 5 or —^^, Ans. 
 
 9. Multiplying by 12, and adding 53', 
 
 36a:' — 53 X 36j- + 2809 = 2401, 
 
 6j:-53=±49; a:=l7 or f, ^n». 
 ^223) 
 
AFFECTED QUADRATICS. 145 
 
 10. Multiply by 4, and add 13', 
 
 4a:' + 52j: + 169 = 729, 
 
 2jr + 13=:db27; x=1 OT --20^ Ans, 
 
 11. Multiply by 3, and add 4', 
 
 9j:' — 24:r + 16 = 31 + 124/3, 
 3^-4 = ^(2 + 3^/3), 
 whence, ar=2+f3 or |—f/3, u4n«. 
 
 12. Multiply by 4, and add IT, 
 
 4j:« + 44jr + 121=441, 
 
 2x + ll = =t21 ; x=5 or —le, Ans, 
 
 13. Clearing of fractions and uniting terms, 
 
 2l2:»-292x= — 500, 
 which is example 6. 
 
 14. Clearing of fractions and uniting terms, 
 
 a:'— \7ir=-252, 
 4a:«-148jr+ 1369 = 361, 
 
 2a;— 37= ±19; «=28 or 9, -4n«. 
 
 15. Clearing of fractions and uniting terms, 
 
 2.r'--13:ir=— 6, 
 16a:' — 104a;+ 169 = 121, 
 
 4a:— 13 = ±11; a:=6 or ^, -4n«. 
 
 16. Clearing of fractions and uniting terms, 
 
 4a:'-lla:=-7, 
 64a;*-l76a: + 121= 9, 
 
 8a;— 11 = ±3; xz=i or 1^ Ans, 
 
 17. Clearing of fractions and uniting termp, 
 
 21a:' — 94a:=-13, 
 441a:'-94x21a: + 2209 = 1936, 
 
 21a:— 47 = ±44; a:=J/ or |, ^/la. 
 
 (223-224) 
 
146 QUADRATIC EQUATIONS. 
 
 (aOS, page 225.) 
 
 1. Here we have ^=|, and 
 
 4a: + f=±l; a?=i or —\yAns, 
 
 2. /zzrV'^; 
 
 whence, ar = 1 1 or — ^V» -4««' 
 
 3. <=|; 
 
 whence, a:=| or ■^\y Ans. 
 
 4. <=^; 
 
 49 , 6ar 9 , 
 
 whence, ^=if or — il* -^^*» 
 
 5 t^5 ' 
 
 IH^'-¥^ + 25=^36, 
 
 whence, «= W or —Hy ^»«- 
 
 6. Clearing of fractions and uniting terras, 
 49x' — SOjr^-Se. 
 Hence <=S% 
 
 whence, x=2 or — i|, -4««. 
 
 (394, page 226.) 
 
 1. Put 2a=7, and we have 
 
 x^—20x=2a + l, 
 ar*— 2aa: + a' =a' + 2a + 1, 
 a;— a=±(a + l), 
 
 a;=2a + l, or —1 ; 
 whence, a:=:8 or --l, An9, 
 
AFFECTED QUADRATICS. 147 
 
 2. Put 2a=ll, and add a' ; 
 
 x* + 2ax + a' = a' 4- 4a + 4, 
 a: + a=±(a + 2), 
 
 a;=-(2a + 2) or + 2 ; 
 whence, a;=~13 or 2, ^»«. 
 
 3. Put 2a =17, and add a* ; 
 
 a;' — 2a + a'=a' + 6a + 9, 
 a:-a=±(a + 3), 
 a:=2a + 3 or —3; 
 whence, ar=20 or -3, Ans, 
 
 4. Put 2a=21, and add a* ; 
 
 ar' + 2a 4- a' =«' + 4a + 4, 
 a; + a==b(a + 2) ; 
 whence, ar=2 or -23, ^n*. 
 
 5. Put 2a =75, and add a' ; 
 
 a;' — 2ac + a'=a'4-2a4-l, 
 ar-a=±(a + l); 
 whence, a^='^6 or -1» ^'W- 
 
 6. Put 2a =72, and add a' ; 
 
 a:' + 2aa; + a' =a' + 10a 4- 25, 
 a;+a=±(a + 6); 
 whence, a:=5 or -77, ^W5. 
 
 7. Put 2a =325, and add a' ; 
 
 a;'— 2a«+a'=a' + 20a + 100, 
 x-a=±{a + 10); 
 whence, a;=335 or -10, Ans. 
 
 (228-227) 
 
148 QUADRATIC EQUATIONS. 
 
 EQUATIONS IN THE QUADRATIC FORM. 
 («965 page 232.) 
 
 In solving these 
 
 examples, we shall put y 
 
 equal to the lowest 
 
 power of the unknown quantity. 
 
 
 1. Put 
 
 y=^'; 
 
 y«_34y=-225, 
 y«— 34y + 289= 64, 
 
 y= n±8. 
 
 
 
 
 x* = i/=. 25, or 8 
 
 •; 
 
 whence. 
 
 
 a;= 
 
 = ±5 or ±3, Ans, 
 
 2. 
 
 y=*'; 
 
 y«-35y=-216, 
 
 
 
 4y' 
 
 '-140y4-1225=: 361, 
 2y-35=±19, 
 
 
 
 
 x*=y= 27 or 8 
 
 » 
 
 whence. 
 
 
 
 ar=3 or 2, Ans, 
 
 3. 
 
 y=^'; 
 
 y'-4y + 4 = 625, 
 y-2 = =b25. 
 
 
 
 
 a;'=y=:27 or - 
 
 ■23; 
 
 whence, 
 
 
 x= 
 
 :3 or V-.23, Ana, 
 
 4. 
 
 y«'; 
 
 y' + 31y = 32 
 4y'-f31y + 961 = 1089, 
 2y + 31 = ±33, 
 ^•=y=l or-32 ; 
 
 
 whence. 
 
 
 
 x=l oPiC, Ans. 
 
 5. 
 
 y-^; 
 
 y'-y=56, 
 
 
 
 
 4y'-4y + 1 = 225, 
 
 
 
 '■ 
 
 2y-l = ±15, 
 a;^=yz=8 or —17, 
 a:^ = 2orV_7; 
 
 
 whence, 
 
 
 (232-233) 
 
 r=4 or V49, Ans. 
 
AFFECTED QUADRATICS. 149 
 
 6. y=af; 
 
 y'-2y+l=9, 
 
 y-i = ±3, 
 
 af=y=4 or —2; 
 whence, a;=V4 or VTr2, ^n«. 
 
 J. 
 
 20/-31y= — 12, 
 1600y«-2480y + 961= 1, 
 40y— 31=±1, 
 
 ar'^=y=A or -1; 
 whence, a;= (f)-, or (f )••, ^n«. 
 
 8. y=Vx; 
 
 3y«-103/=-3, 
 36y'— 120y + 100= 64, 
 
 Va:=y= 3 or | ; 
 whence, a:=27 or ,^, -4n«, 
 
 9. 
 
 y=»'^+ 
 
 5; 
 
 y*-y=6, 
 
 4y'-4y4-l = 2525, 
 
 
 
 
 whence. 
 
 
 Va: + 5=y=3 or —2; 
 
 ar=:4 or - 
 
 -1, 
 
 Ans, 
 
 10. y= 
 
 = (^+12)^ 
 
 1 
 
 y'+y= 6, 
 
 4y" + 4y + l = 25, 
 
 
 
 
 whence, 
 
 
 (a; + 12)^=y= 2 or -3, 
 a;+12 = 16or 81; 
 
 ar=4 or 
 
 69, 
 
 Am, 
 
 11. y= 
 
 =(a. + a)^; 
 
 
 
 
 
 (x+ay=y=b or —36; 
 whence, ar=6*— a, or 816*— a, Ans, 
 
 (233) 
 
150 QUADRATIC EQUATIONS. 
 
 12. Transposing x and squaring, 
 
 or «'— 21ar=— 64, 
 
 which gives ar=18 or 3, Ans, 
 
 13. y=V9x + 4; 
 
 y' + 2y + l = 16, 
 V9x + 4=y = S or —5, 
 
 9j; + 4=9or25; x=^ or I, Ans. 
 
 14. y=Vio+«; 
 
 y'-y= 2, 
 
 4y'-4y-l= 9, 
 Vio + a;=y= 2 or —1, 
 10 + a; =16 or 1 ; 
 whence, x=6 or —9, Ans, 
 
 16. y={x-5)^; 
 
 y•-3y + f=J-|^ 
 
 3. 
 
 (i— 5)*=y=8 or —5, 
 ar— 5 =4 or v^(-6)' ; 
 
 whence, ar=9, or 5 + ^{—o)\ Ans, 
 
 16. y=(l+ar-ar')^; 
 
 16y'-8y4-l= i, 
 whence, y= i or — ^. 
 
 Hence, l-f a;— a;'=|, or l+x—x*=:-^^j, 
 
 x'-x + }=Uy x'-x-^i = \\ 
 
 x-r=:±,±Vn, ar-i = db-it/lT, 
 
 whence, x=±±iVli, x=idbi^Tl, Ans, 
 
 (233) 
 
AFFECTED QUADRATICS. * 151 
 
 17. y=Va; + 16; 
 
 y=f±i, 
 
 '^a?+16=5 or —2; 
 Whence, x=9 or —12, Ans. 
 
 1j3. Clearing of fractions and uniting terms, 
 
 81a;* — 82a;'= — 1, 
 Six*— 82x' + -!-fi-»-= J-f^Ji-, 
 
 whence, a;=dbl or ±i, ^ws. 
 
 If we add 1 to each side of this equation, we shall have perfect 
 squares ; thus, 
 
 81a:' + 18 +3= 100, 
 or 
 
 and 9ar + -=dbl0; 
 
 X 
 
 hence two quadratics for ar, and the same values as before. 
 
 1 9. Clearing of fractions and uniting terms, 
 
 225a;* — 901a;»=:-4, 
 226ar*-901a:' + iU_._8_oj.=AYJ._«j.,. 
 15x* = 9J>^^db3-Y^ 
 x'=4: or ^ij ; 
 whence, x—dt.2 or iy'j* -^^*« 
 
 Here, if we add | to each side, we have 
 ^^ , 20 , 4 961 
 
 2 .31 
 
 whence two quadratics for x, 
 
 (233) 
 
152 QUADRATIC EQUATIONS. 
 
 20. x*-\-2x'-1z^ — 8x-{-l2 {x*+x 
 
 2a;* 4- x 
 
 — 8a;'— 8a?, 
 
 Hence, by taking the square root thus far we see that the ex- 
 pression may be written, 
 
 (a;' + a;)'-8(a:'4-a-) = -12; 
 or if y=a:* + a', y' — 8y=: — 12, 
 
 y'-8y + 16= 4, 
 y-4=±2. 
 
 Hence, a:'— ar— 4= 2, or a;' + a;— 4 =— 2, 
 
 x'-\-x+{= V-, a;' + a: + i= |, 
 ar + |=±f, a;+i = ±f, 
 
 and a;=2 or —3, ar=l or —2,Ans, 
 
 21. Multiplying by a*, the equation is 
 
 a;* — 8ar* + 19a;'— 12a;=:0, (a;'— 4a? 
 .a;* 
 
 2a;'— 4a; —8a;' 
 
 — 8x'4-16a;' 
 
 Hence, 
 
 Sa;*— 12x, 
 
 (a;'-4x)' + 3(a;'-4x) =0, 
 (a:«_4a;)' + 3(x'^4a;) + | = |, 
 
 a;' — 4a;=0 or —3; 
 ar»-.4a; + 4 = 4, a;'-4a;-f 4 = 1, 
 
 and a;=0 or 4, a;=l or 3, -4»#. 
 
 22. a;«-10a;' + 35a;'-50a;+24=0, (a;'— 5a; 
 
 a;* 
 
 2a;'— 5a; — 10a;' + 35a;' 
 — 10a;= + 25a;' 
 
 10a;'— 50a; 
 (233) 
 
AFFECTED QUADRATICS. 153 
 
 Hence, 
 
 (a;'-5a:)« + 10(x«-5ar) =-24, 
 
 {x^—5xy -\-10(x* ^5x) + 25= 1, 
 
 x^ — 5x= — 4: or —6 ; 
 
 and x=l or 4, a;=2 or 3, An». 
 
 23. ar* — 8aa:'4- 8aV + 32a'ar— 9a* = 0, (a;'— 4aa; 
 
 a;* 
 
 2x' — 4aj; — 8aa:»+ 8a V 
 
 — 8aa;' + 16aV 
 
 — 8aV + 32a'a:. 
 
 Hence, {x* — 4cp;)' — 8a' (a;' — 4ax) = 9a* 
 
 (a;' — 4ax)' — 8a'(ar' — 4a;r) + 1 6a* = 25a* 
 
 «' — 4aar =9a" or —a* 
 
 a;'— 4aar + 4a" = 13a' or a?' — 4aar + 4a'=6a' ; 
 
 and ar=a(2±^13) or a;=ia(2dbV3), ^»». 
 
 24. Performing the multiplication indicated, 
 
 y*-2cy' + cy-2y« + 2cy=c«, 
 or, (y«-cy)' - 2(y'-cy) =c«. 
 
 whence y =i ± /|- 4- 1 ±*^r-f?\ , An«, 
 
 PROMISCUOUS EXAMPLES IN QUADRATICS. 
 
 1. Completing the square, we have 
 
 4a;' + 44a; + 121 =441, 
 2a; + ll = i:21; 
 whence, a;=5 or —16, Ans, 
 
 (233-234) 
 
154 QUADBATIC EQUATIONS. 
 
 2. Clearing of fractions and uniting terms, 
 
 whence, x=4 or — 1, Arts, 
 
 3. Clearing of fractions and uniting terms, 
 
 whence, x=2 or —3, Ans. 
 
 4. Multiplying by 8(ar' — 1), the least common multiple of the de- 
 nominators, we have 
 
 12 — 2a; + 2=a:»— 1, 
 or, ar' + 2ir + l = 16; 
 
 whence, xz=S or —5, Ans, 
 
 5. Clearing of fractions, 
 
 {2x*+x—5)(x + l9) = (x^-\-4x + 3){2x + l5), 
 or, 16x*—52x=:li0; 
 
 ICar*— 52ar + iA»^=ip, 
 
 whence, ar=6 or — f, Ans. 
 
 6. Clearing of fractions, 
 
 or, ai' + 27a;= 28, 
 
 x*+2lx-\-i\a.= ifJL^ 
 
 whence, a;=l or —28, Ans, 
 
 Ex. Y and 8 need no solution here. 
 
 9. Add mn to both sides, ( 393 ), 
 
 mx* — 2 mxVn + mn = nx', 
 
 Vm • x—^mn = dc^n ' x ; 
 
 , Vnm . 
 
 whence, x= . , . , ^n*. 
 
AFFECTED QUADRATICS. 155 
 
 10. Add A to each side, ( 293 ) ; 
 
 4^ 8^ 4_ 64 
 49 21 9~ ~9' 
 
 whence, 
 
 
 :=1 
 
 or -111, 
 
 Ans, 
 
 11. Add 36 to each side, ( 293 ), 
 
 
 
 
 
 x' \2x ^ 
 
 
 
 
 
 ^-6 = ±2, 
 
 
 
 
 whence, 
 
 
 x=' 
 
 162 or 76, 
 
 Ans, 
 
 12. Transposing, 
 
 , we have 
 
 16 - « ,1- 
 
 0, 
 
 
 ^ 
 
 
 (2a;-4)* (2a;-4)''^ 
 
 
 
 ^ 1_ 
 
 0, 
 
 
 
 
 (2^-4)' - 
 
 
 
 (2.:-4)' = 
 2x-4c =: 
 
 4, 
 ±:2; 
 
 
 
 whence, 
 
 
 
 x=zd or 1, 
 
 -4n5. 
 
 13. Put y^Vx^ 
 
 + 11; 
 
 
 
 y'+y=42, 
 y'+y+i=-^P» 
 
 
 
 
 
 y=6 or —7, 
 
 
 
 
 
 ar' + ll = 36 or 49; 
 
 
 
 
 whence. 
 
 x= 
 
 :±5, 
 
 , or±f^, 
 
 ^»5. 
 
 14. Add 5 to each side of the equation, and put y=:Vd;'— 2a; + 5 ; 
 then, 
 
 y' + 6y=16, 
 y* + 6y4-9 = 25, 
 
 y= 2or-8; 
 ar" — 2a: + 5 = 4, or ar' — 2ar + 5 = 64, 
 
 ar«-2a:+l=0, a:' — 2a: + 1 = 60, 
 
 whence, ar=l, a?=:l±24^15, Ans. 
 
 (234) 
 
156 QUADRATIC EQUATIONS. 
 
 15. Add JyV-c' to each side, (^93). 
 
 x' + -'^'-x*+-\Y-^' = \Y^' 4- 34ar + 1 6, 
 
 xxcuuc, 
 
 a:'= 4, or 
 
 a:' + V^=-4, 
 
 and 
 
 ^=±2, 
 
 ar= — 8, or — 1, ^n*. 
 
 16. This equation may be "written, 
 
 ._ 2(^^x + l). 
 
 dividing by Vx-J-l, we have 
 
 whence, x=zl or 4, .4n». 
 
 17. Clearing of fractions, 
 
 2x + 2Vx= 16— a-, 
 ^x-{-2Vx= 16, 
 36ar + 24V'a: + 4= 196, 
 6f';r + 2 = dtl4; 
 whence, ir = 4 or 7|, Jn«. 
 
 18. By squaring, 
 
 \ 2f/2/ f/g ' 
 
 or, 
 
 ar'(12-a;') _ (2 4-ar')* 
 2f^2 2V2 ' 
 
 Hence, x*—4tx*= — 2y 
 
 z*= 2^=1^2 ; 
 
 and «=±(2dbf/2)^, ^n«. 
 
 (234-235) 
 
AFFECTED QUADRATICS. 157 
 
 19. This equation may be written, 
 
 ^ ^ ' X ^ X X 
 
 which is divisible by y ; whence, 
 
 / — r y^^- 4 ^^ 
 
 Bysquaring, a; + 1-2^:^+^1 + 1==^=; ^^1: V;^^/-/^^t/X^ -^ 
 
 squaring again, and clearing of fractions, / "yi ^- X — '' 
 
 4a;' + 4x' =a;* + 2j;' 4- 3j;' + 2a; + 1, 
 or, a:* — 2x«— a:" + 2a; + l=0; /^y'^" -^ f^J^ 
 
 taking square root, a;'— a;— 1 =0, ^/ - / / (/j^ 
 
 ar-i^rfciV'S; ^ -- t l/TtW^ 
 
 whence, x=\{\±Vb\ Ant, 
 
 20. Multiply both terms of the first fraction by 1 +4^1— a;', of the 
 
 second by 1— f^l— ar' ; then 
 
 
 l+t'l-ar' l-f'l-a:' VZ 
 
 
 ar' a:' a:" 
 
 
 or, 2f'l-ar«=V3, 
 
 
 squaring, x*=\\ 
 
 
 whence. 
 
 a;=d=|, Ans, 
 
 21. This may be written, 
 
 
 (U/)'-T' 
 
 
 / 1 \^ ♦^2a: 
 U+a:/ "" 12 
 
 
 
 1 2a? 
 
 
 l+ar^Hi* 
 
 
 «» + a?=72, 
 
 
 4a:' + 4ar + 1=289, 
 
 
 whence, 2a; + l = d:l7; 
 
 a;=8 or —9, Am. 
 
 (285) 
 
 
158 QUADRATIC EQUATIONS. 
 
 22. Multiply both terms of the fraction by x+Vx^—Qy and take 
 the indicated square root ; then 
 
 x-h^x'-9 
 
 f^a:'— 9 = 2a;— 6, 
 a:*-9 = 4ar'-24a: + 36; 
 whence, a:' — 8a: = — 15, 
 
 ar»_8a; + 16 = l; 
 and ^=5 or 3, Ans, 
 
 23. Completing the square, we have 
 
 x^ =27 or -28, 
 
 x^ =3 or v/- 28; 
 
 whence, a; =243 or V(_28)', ^rw. 
 
 Since the exponents are odd, we may write the last value of a? 
 
 A L 1 
 
 thus, —(28)* = -(2 X 2 X 7)=' = -(32 x 32 x iy 
 
 = -(512 X 2 X Vy = -8(2 X 7') 
 
 ^^^- 
 
 1 
 
 t5\3 
 
 1 
 
 = -8(33614)^ 
 
 24. Completing the square 
 
 144a;"»— 312a:" + 169 = 25, 
 12a:"— 13= ±5; 
 whence, a:=V| or V|, ^iw. 
 
 25. Transposing 2a: aT)d squaring, we have 
 
 2 + 2a; = c' — 2c'a: — 4ca: + c^x* + 4ca;' + 4a:', 
 or, (c4-2)V-2(c + l)'a:=2-c«; 
 
 by (a»3), (^ + 2)V-2(c + ira: + |^;=(g^', 
 
 ^ ^ c + 2 c + 2 * 
 
 c'— 2 
 whence, a:=l or . .^ , Ans, 
 
 [C + Z) 
 
 (235) 
 
AFFECTED QUADRATICS. 159 
 
 26. By expanding, 
 
 {a + xy= x*-\-5x*a+l0x'a'' + l0xW -\-5xa* + a\ 
 - (a— a:)»= —x' + 5x*a — 10x^a^ + 10xW — 5xa*-\-a\ 
 
 Therefore we have 
 
 10aa;* + 20aV + 2a*=352a', 
 
 x*+ 2a V= 35a*, 
 
 x*-^ 2a V+ a*= S6a\ 
 
 x^i- a'r=±6a'; 
 
 whence, x= ±:a V5y or ±aV—1^Ans. 
 
 27. Clearing of fractions and uniting terms, we have 
 
 a*cx^ — a(6 + 2c)x = — (6 + 2c) ; 
 (A + 2c)«_6'-4c« 
 
 by (a03), a'cx^-^aib + 2c)x+- 
 
 4c 4:C 
 
 whence, x==—(b+ 2c±»/6'— 4cO, ^n«. 
 
 28. Clearing of fractions, 
 
 abx =(a + b-\-x)bx-^(a + b-{- x)ax + (a + 6 + x)ab ; 
 
 multiplying and arranging terms, 
 
 {a + b)x*-{-(a-^byx=-^ab{a + b), 
 x* + (a + b)x= — ab, 
 
 a+b a—b 
 
 «=-- 2-±-2-; 
 
 whence, xz=z— a or —b^ Ans. 
 
 29. Clearing of fractions and transposing, we have 
 
 X* + 8ar» + 1 6x' — a V — Sa'x — 1 6a' := 0. 
 
 Extracting the square root according to (!896), this equation 
 roay be written, 
 
 {x'-{-4xy-a'(x + 4y=0, 
 or, (ar' + 4ar)'=a'(;r + 4)', 
 
 x^-\-4:X=±:a{x-\-4). 
 Dividing by a; + 4 a;=dba, Ans, 
 
 (235) 
 
160 QUADRATIC EQUATIONS. 
 
 30. Multiply both terms of the left hand member by the cUnoiRU 
 
 2a ~ 2a 
 
 Omitting the factor 2a, and squaring the left hand member, 
 
 2x— 2Vx' — a''=a?, 
 
 3 ' 
 whence, x=zdo2a^^\^ Ans, 
 
 31. Dividing out the factor a—x in the second term, and clear- 
 ing of fractions, we have 
 
 a' -f x' + a' + 2aa; + a;' = 4a' + 4az, 
 OP, x'— aar=a', 
 
 ar«-ax+-=-a'; 
 whence, a;=-(l±|/5), Ans, 
 
 it 
 
 32. Clearing of fractions and transposing, we have 
 
 )/2ax + x^ =ah -|- bx— a—x\ 
 squaring and uniting terms, 
 
 (26-6>' + 2a(26 -6>=a' + a'6« - 2a'6, 
 
 a;'-f-2ax: 
 
 a»-{-a»6'^2a'6 
 a' 
 
 ^. + 2ax+a'=2^-^,, 
 
 
 V26-6' ' 
 
 whence, x-=. ±a ■? — : > Ans, 
 
 (235) 
 
TWO UNKNOWN QUANTITIES. 161 
 
 83. Squaring the left hand member as indicated, 
 4 + 4x + x\ 2b-\-t t ^ 
 4 — 4x + x^~ 2b " ' 
 clearing of fractions and uniting terms, 
 
 ex* — 4cx* + 4cx=l 6bXf 
 
 , . ,166 
 
 or, a:'— 4ar + 4 = — , 
 
 c 
 
 whence, ar=2l l±2y -I, -4n«, 
 
 Note, — The value x=0, satisfies the original equation, and also our reduced 
 equation after uniting terms. 
 
 EXAMPLES OF SIMULTANEOUS EQUATIONS. 
 (301, page 243.) 
 
 1. From the second equation, we have x—2y*, and by substitu- 
 tion in the first, 
 
 16y'-8y + l = 121, 
 
 4y=12 or —10, 
 y= 3 or - 2n 
 2y«=ar=18or 12^ j ^'**' 
 
 2. The second equation gives by transposing y and multiplying 
 
 ary=lly-|-; 
 
 and substituting in the first equation, 
 
 3y'-f 22y=240, 
 9y' + 66y4-121=:841, 
 
 3y= — lldb29; 
 "= 6or-13i, ^^_ 
 
 2a?=22— y; x= 8 or l7f 
 
 (235-243) 
 
 i\ 
 
162 QUADRATIC EQUATIONS. 
 
 3. 
 
 By substitution from the second equation, 
 4y' + 9y = 100, 
 
 2y + f=±V, 
 
 and 
 
 reducing, 
 
 
 y=4 or - 
 
 - 61 
 
 ) 
 
 
 9y 
 
 
 , >Ans. 
 
 
 x=-f; x=9or- 
 
 -14, 
 
 M 
 
 4. By adding the equations, and dividing by 5, 
 
 ar" + a;=12, 
 
 x= 3 or — 4 ) 
 y=25— 5ar; y= 10 or —45 f * 
 
 5. Multiplying the second equation by 3, and adding, 
 
 13x' = 52, 
 a:'=4, 
 
 x=±2) . 
 
 y=±3[^'"- 
 
 6. From the second equation, 
 
 y=40 — 4ar, 
 a:y = 40a; — 4x'; 
 
 hence, the first gives / 
 
 , a:'— |0a;= — 336, 
 ar*— ^0;c + 400= 64, 
 
 ar=+20±8, 
 x= 28 or 12 ) . 
 y=40-4ar; ^=-72 or - 8 j ^'**- 
 
 *l. The second equation gives ^=-^ ; whence, 
 
 V=252, 
 y'=36, 
 
 y 
 
 a?=fy; x 
 
 (243) 
 
 ±15 H'**- 
 
TWO UNKNOWN QUANTITIES. 163 
 
 8. From the second equation, x=^i/^ 
 
 |ly» + 4y' = 181, 
 
 181y' = 181x25, 
 
 9. Adding the equations, 
 
 ar' + 2a;y4-y'=36, 
 
 ^ + y=±6, (1) 
 
 •ubtracting equations, a:'— y'= — 12, (2) 
 
 dividing (2) by (1), a:-y= qF2, (3) 
 
 from (1) and (3), ""= ^^ I Ans 
 
 10. Adding the square of the second equation to the first, we 
 have 
 
 2a;' = 5, 
 
 ft n AA/^ I ^nS, 
 
 11. Put af=vy; 
 
 vy + vy'=56, or 3/'= -5-— ; 
 ' v^ + v 
 
 v/ + 2y'=60, ory'=--— ; 
 
 whence by equating the values of y', 
 
 v=s <>*• -I- 
 
 With v=|, the second equation gives, 
 
 10y'=:180, 
 
 ±4*^2 [^'**- 
 Withv=— J, 3y'=300; 
 
 y=±io I . 
 
 :r=^14H'*'- 
 (243) 
 
164 QUADRATIC EQUATIONS. 
 
 12. Let y=vxy 
 
 CO 
 
 3x*-\-vx*=z 68, or a:'=- ; 
 
 3 + v 
 
 160 
 
 4i;V + 3i;a;' = 160, or a?"=T-i — S" » 
 
 whence, 68v"4-llv=120, 
 
 or, 4i;' + Uv=Vt% 
 
 , . 11 . /11\' 32761 
 
 '' +17^+168/ =wr 
 
 and, i;=f or — f^. 
 
 With the value v=|, the first equation gives 
 
 I7ar« = 68x4, 
 ar' = 16; x=±4 \ 
 y==h5 
 
 The value of t> = — «^|, gives x=±: — - j An*. 
 
 16 
 
 13. Let y=var; 
 
 » 1+v 
 
 rar" — 2i;V= 1, or aj' = — -, ; 
 
 ' V — 2v' 
 
 whence, 24v*— llv= — 1, 
 
 i;=i or \. 
 
 Hence, if v=|, 4x'=:36, a;=±3 
 
 y=±i 
 
 If v=|, 9x'=96 
 
 (243) 
 
TWO UNKNOWN QUANTITIES. 165 
 
 14. Put y—vx\ 
 
 X*— va:«-fvV= 21, 
 
 vV — 2var'= — 15, 
 whence, 
 
 21 
 
 , -15 
 or x =-= — — ; 
 v'— 2v' 
 
 36y'-67t;= — 15, 
 
 v= f or 1. 
 From the second equation, 
 
 jf^=i, /-6y'=-15; y=±Vd ^ ^"*- 
 
 ar=±34/3 
 
 15. Add 1 to the first equation, and take the square root, then 
 
 ar + y=i: + 10 or —12, 
 xy—y*= 8. 
 Taking ir + y= +10, the second equation gives, by substitution, 
 
 y'_5y=_4; 
 whence, y=4 or 1 ) . 
 
 x=6 or 9 J 
 Taking ar + y= — 12, we have 
 
 y« + 6y=-4j 
 whence, y= — 3±V^5 ) . 
 
 ar=-9::pf^5 i" 
 
 16. Put y=var; then 
 
 6x* + 2v*x* - 5vx* =12, or «' = ^ ^ 
 
 3vV— 3a;"— 2vx'= 3, orx' = — r 
 
 6 + 21;"— 6v' 
 3 
 
 Hence, lOv'— 3v=18, 
 
 t;=i or -I, 
 If we take v=f, the first equation gives 
 3x' = 12, 
 
 (244) 
 
 whence. 
 
166 QUADRATIC EQUATIONS. 
 
 If t;=- 
 
 -|, we have 
 
 372x«=r300, 
 
 ,_300_3*4-25 ^ 
 *"~372 3- 4 -31' 
 
 
 wbence, 
 
 
 . 5 ) 
 V31 1 
 
 1 
 
 y ^n*. 
 
 
 
 1 
 
 IV. Multiply the second equation by 2, and add the product to 
 the first; then 
 
 a:« + 2xy + /= 121, 
 subtract it ; a:' — 2xy + y' = 9, 
 whence, ar + y ==fcll, 
 
 a?— y =db 3; whence, a: ==t 7 or db4 ) 
 y=db4 or ±7 ) 
 
 18. Squaring the second equation, and subtracting the square 
 from the first, we have 2ary = 80 ; adding this to the first, and taking 
 the square root, 
 
 ar + y=±13, 
 
 K^y=- 3 ; whence, a:=4-8 or —5 ) . 
 y= + 6 or — 8 ) 
 
 19. Dividing the first equation by the second, 
 
 a;'- xy + y^= 273, 
 
 square of second, a;' + 2ry + y'= 324, (1) 
 
 3.ry = 61, 
 xy = 17. 
 
 Subtract 4a:y from (1), 
 
 a:'— 2ary + y'= 256, 
 hence, ar-y = =fcl6, 
 
 a:4-y= 18; whence, ar= 17 or 1 \a 
 y= 1 or 17 3 
 (244) 
 

 TWO UNKNOWN QUANTlTiESr^^ss^s:^^^^''^ 167 
 
 20. Adding three times the second equation to the first, 
 
 a:' + 3a;V + 3x/ + y'= 729, 
 
 taking the cube root, x + y= 9. (1) 
 
 The second equation gives, xy(x-\-y) = 180'y (2) 
 
 dividing (2) by (1), a-y= 20. (S) -i 
 
 Subtracting four times (3) from the square of (1), 
 
 ar«_2xy + y'= 1, 
 or, x—y=±l, (4) 
 
 From (1) and (4) we have x=5 or 4 ) . ^ 
 
 and y=4 or 5 ) 
 
 21. Performing the multiplications indicated, we have 
 
 x'--x'y+ xf-f= 13, (1) 
 
 ar'y- xy^ =6; (2) 
 (1)— twice (2), a:'— 3a;'y + 3ary'— y'= 1, 
 
 taking cube root, ar— y = 1, (3) 
 
 from (2) divided by (3), xy — 6, (4) 
 (3) plus four times (4), a;' + 2a:y + y'= 25, 
 
 a: + y=±5; j(o) 
 
 hence, from (3) and (5), a;=3 or —2 ) . 
 
 y=2 or —3 J 
 
 22. Dividing the first equation by a; + y and transposing, we have 
 
 ar' — 2a;y + y' = 0, 
 
 ar-y =0, 
 
 second equation is ar + y =4 ; a;=2 ) 
 
 y=2) 
 
 Arts, 
 
 23. The first equation is a quadratic in xy ; and completing th< 
 square, we have 
 
 56i:y-24a?y4-4= 16, 
 6a:y— 2 = =fc4, 
 
 xy= l,or-i, 
 and since ar=2y, 2y'= 1 or — | ; y— rhi|/2 or =t^ 
 
 ar=db ♦/2orzbif^. 
 
 Ans, 
 (244) 
 
 
168 QUADRATIC EQUATIONS. 
 
 24. Dividing the first equation by the second, we have 
 
 squaring the second, ir' + 2 ry + y' = 1 44 ; (l) 
 
 hence, 3ary ==144 — |ry, 
 
 or, a-y = 32, (2) 
 
 (1) — four times (2), a:' — 2i:y + y'= 16, 
 
 whence, a;— y=:d=4, 
 
 .+y= 12; .=8 or 4) 
 y=:4 or 8 ) 
 
 25. We make use of formula (C), page 242, in which 
 
 *=8, 5' = 64, «* = 4096. 
 
 2p'— 256;? + 4096 = 2402, 
 
 p'-\2Sp=— 847, 
 
 y— 128;> + 4096= 3249, 
 
 jr>-64 = rfc 67, 
 
 xy=:pz=. 121 or 7. 
 
 Taking a-y=7, and combining it with x4-y=8, we get 
 
 .-y=±6; x=7 orl) ^^_ 
 
 y=l or 7 ) 
 
 If iry=121, ar— y = rbi^— 420 = ±2*^— 105 ; 
 whence, ir=4rfcV-105 "I ^^^ 
 
 y=4::^i^-105 J 
 
 26. Dividing the first equation by the second, we have 
 
 z'*-{-x_x(x-\-\)_2 
 jr' + l a:' + l "~3* 
 X 2 
 
 or, 
 
 x'—x+l 3' 
 whence, 2x* — 52r = — 2, 
 
 .= J±j; x=2 or |)^^^_ 
 y=2 or 16 ) 
 (244) 
 
TWO UNKNOWN QUANTITIES. 169 
 
 27. Dividing 
 
 the first equation by .r+y. 
 
 
 
 
 x*— xj/ + y^ = 2xi/y 
 
 
 
 
 x'-2;ry + y'=:a:y-16; 
 
 
 
 
 a--y = ± 4, 
 
 
 
 
 x' + 2xi/ + y'= 80, 
 
 
 
 
 x-\-yz=dz4y5; 
 
 whence, 
 
 
 
 x=db2V5±2 
 y=dz2V5zp2 
 
 28. Put 
 
 J= 
 
 p, 
 
 y* = Q ; then the equations become, 
 
 •whence, 
 
 
 
 P+^ = f^a + 6, (] 
 
 and 
 
 
 
 P'-Q'=a-b, 
 
 and 
 
 
 
 --^'-^^ (^ 
 
 by (1) and 
 
 (2). 
 
 
 2P=^+Va + b, ' 
 
 Ant. 
 
 Va + b 
 2« = '^*+.S' 
 
 and wo obtain finally, P=^ . or -=(j~)\ 
 
 29. Dividing the second equation by the first, we have 
 
 x-y-l- ^' 
 
 or, (ar-y)«— 4(ar-y) = -4, 
 
 (^-y)*-4(ar-y) + 4= 0, 
 
 hence, x—y= 2. 
 
 Hence the second gives a; + y= 8; 
 
 x=z5 ) 
 
 y=3 3 
 
 Ans, 
 y=3) 
 
 (244-^245) 
 
170 QUADRATIC EQUATIONS. 
 
 8^. Adding and subtracting twice the second equation, 
 
 ar«— 2jry + y'=a — 26; 
 
 x-y=±Va--2b; 
 
 ^ , ' , > Ans, 
 
 y = rfcif^a + 26^lVa--26 J 
 
 il. We have x=— ; whence, 
 
 y 
 
 y* + 2ay'=46', 
 y*4-2ay* + a'=a' + 4J'; 
 
 whence, y=:i={ -a±f^o' + 46« }^ ) 
 
 in the same way, x= dz | a±^a' + 46' [ ^ ) 
 
 32. Squaring the first equation, we have 
 
 ^'> Vr^y% V^)- (-f fsj y'.4-_y = 333, (1) 
 
 a)ividmg (1) by (2), we have a quadratic in a:y, 
 
 ' - -^ ^v^eTc^ xy= 36 or 3^/^^ ^ ' ^^ ^' ' 
 
 /Substituting a?y=36 in (1), 37y=333 ; y= 9 
 
 ;l4>J^_/." ^y=3Vi«(l)» 37y=333-36; y=324 
 
 33. Adding the equations. 
 
 ^»*. 
 
 x-\-2Vxy + y=a-^h^ 
 
 Vx+Vy^V^^'^ (1) 
 
 (245) 
 
TWO UNKNOWN QUANTITIES. 171 
 
 subtracting them x—y=a—h, 
 
 Hence, from (1) and (2), x— 
 
 ''-''=^.- (^) 
 
 a-{-b 
 
 34. Put a:^=P, y' = §, then 
 
 P + ^ = 6. 
 
 The solution is the same as that of 19. 
 
 35. Clearing of fractions, the second equation gives 
 
 the first is, x-\-y=lO\ 
 
 whence, a:y = 16. 
 
 Combining the two last equations, x=8 or 2 ) ^ 
 
 y=2 or 8 j 
 
 36. Dividing the second equation by the first, v ^-^y ^ - >9> "" - 4y ^ 
 
 whence, a;*4-y^ = ±8, (1) 
 
 a;— V= 16, 
 
 by di vision, x^ -fy* = ± 2 . (2) 
 
 Taking the positive signs in (0 and (2), we have 
 
 x^=/b, y*= 3; 
 negative signs, a:*=— 5, y* = — 3; x=:25 \ . 
 
 yzx: 9 
 
 (246) 
 
 \ 
 
172 QUADRATIC EQUATIONS. 
 
 37. By transposition, and squaring the second equation, we have 
 
 a. 3 
 
 a. 
 and substituting this value of y^ in the first equation, 
 
 3. 
 
 x-Vx=2, 
 
 ♦/j- = 2or— I; a;=4orl 
 From the second equation, y=S 
 
 > Ans. 
 
 38. Adding twice the second equation to the first, we have 
 (i:Vy*)' + 2(a;^ + y^) = 35, 
 
 ± 1 
 From this quadratic in ar' +y^, 
 
 a;*+y'i= — l±6, =5 or ~7; 
 
 combining this with the value of a;^y', 
 
 x'-y= = ±l, or ±5; 
 
 whence, r x=2l, 8,-1,-216) ^ 
 
 .^~-fi^^.^r~^r^-/ y= 8,27,-216, -l[^^'- 
 
 1 1 
 
 39. Put Pzzzx^^ ^=y*, and the equations become, 
 
 P'+Q' + P+Q = 2e, (!) 
 
 PQ= 8. (2) 
 
 Adding twice the second to the first, 
 
 (^+cr+(^+«=42, 
 
 P+^==6or-7, (3) 
 
 P'-2P^+ ^' = 4 or 17, _ 
 
 P-Q=^±2 or rtV^17; (4) 
 
 hence from (3) and (4), P=^=» = 4, 2 or { {-1±^), 
 
 ■ Q=y^= 2,4ori(-7±m); 
 
 whence, x=±%, ±2V2 or ±\^(-1±y\^) }H ^^ 
 
 y= 32,1024or S^( -7:pVl7) ! * ) 
 (245-246) 
 
TWO UNKNOWN QUANTITIES. 173 
 
 40. The first equation is a quadratic in -; and adding 4 to each 
 
 side, and taking the square root, 
 
 V^_3 _11 
 fV-2 ""^ 2 ' 
 4a:=:9y or 121y. 
 From the second equation. 
 
 ar=9 or f ff ) 
 y=:4or^VTi 
 
 Ans, 
 
 41. Here P=x^, g=y* ' ^^^ ^ ^ ~ ^ 
 
 P'Q'=2Q\ -yi^\^ (1) 
 
 SP-Q=U; V^-^ (2) 
 
 from(l), P'=2(2, rl~^ xK//^ (3) 
 
 " (3) and (2), P'--16P=-28, ^ J^^ _/^^~^'^^ ^ 
 
 hence, a:^=P=8±6 = 14 or 2, 
 
 y^-Q— 98 or 2; 
 
 whence, ic=2744 or 8 ) 
 
 y=9604or4[ ^^*- 
 
 42. LetP=ar*, §=y*; then 
 
 P' + PC+C'= 1009 =fl, (1) 
 
 P* + P'^'+^* = 582193 = 57Ya, 
 Add 7^' C',and take the root, P' + ^' = VsTT^+P'V (2) 
 
 subtract (1) from (2), a-- P q-VWu^^'q\ 
 
 a*-2aPQ-\-P'Q^=5l1a + P^Q\ 
 2PQ=a-5l7, 
 PQ = 216, (3) 
 
 Adding (3) to (1) and ) r> '0-4... 
 
 taking the root, ) ^+V-±35, 
 
 subtracting three times ) __ 
 
 (3) from (1). ) /'-«=±19; 
 
 (246) 
 
174 QUADRATIC EQUATIONS. 
 
 ^ .jhence, / / «* =:P— 8 or —27, 
 
 ^^ Y'* - :i). -^ , <r^ -^y A3.^-±Sr:^l9y =^=-27 or 8, 
 ^ whence, Wt:i=/C,i-^f, Li-^. -/6 ^=81 or 16 ) 
 
 43. Complete the wjuare of eacn equation Dy/adlaing 16 ar to t 
 
 first, and x to the second ; then 
 
 y— 2ar^y'+ ar= 4+ar; 
 taking the roots, y— 4f/a;=±4V4+^, (1) 
 
 i^y-Vx=±Vl+i; (2) 
 
 (1) minus four limes (2), y— 4V'y= ; 
 
 i^rhence, y=16 ) ^^^^ 
 
 x= 2| ) 
 
 44. These equations may be written, 
 
 ary(ar + y)= 30, (1) 
 
 — ^= -. (2) 
 
 xy 6 ^^^ 
 
 Dividing (1) by (2), xy= 36 ; 
 
 ^ac^ /'/^v ^^^y - ^V , y -^ y ^yv ^y=±6, (3) 
 
 ^from (2)W^, / (x + y=±:5, 
 
 ' V/ ^^ v^ r^ ^' -^'^ ^' ' ^ /*=3» 2, 1, or -6 ) ^ 
 
 / ' ' V ^ — — -^ 
 
 ' ~' . -?• 7 '4^ Dividing the first equation by the second, we have 
 
 ^V= 16, 
 a'y = it:4. 
 Hence, a^4-2ary+A'= 16 or 0, 
 
 a:"— 2xy+y'= or 16 ; 
 ar + y = ±4 or 0, 
 x—y=. or ±4 ; 
 whence, ^=±2) ^^ i^=±2) 
 
 y=±2j (y=:F2) 
 
 (246) 
 
TWO UNKNOWN QUANTITIES. 175 
 
 46. Dividing the first equation by the second, 
 
 x* + x'y+xY-hx/ + y' = lOSl, (1) 
 
 (x-y*)-x*^4z'yi-6xY^4xi/'-\-y*= 81 ; (2) 
 
 (1) minus (2), 5x'i/—5xY + 5xi/*= 960, 
 
 dividing by 5ary, x*—xy-\-y*— , 
 
 xy 
 
 (ar-y)*=a:»-2xy + y'= 9, (3) 
 
 ary + 9iry= 190, 
 
 ary= 10 or —19; (4) 
 
 from (3) and (4), a^+y=± V, 
 
 "whence, ar=5 or —2 ) ^ 
 
 ^ ^\ Ans, 
 
 y=2 or —5 ) 
 
 47. We have given 
 
 x*+xy-\-y^= 7, (1) 
 
 x*+xy + y*= 133. (2) 
 Adding xY to (2), x* +2xY + y* = 133+ xYi 
 
 or, a:' + y» =V^133+a:y. (3) 
 
 Taking (1) from (3), 1 -xy=yU3-hxY, 
 
 49-Uxy + x'y^= 133+a:y. 
 
 xy= -6. (4) 
 
 Combining (1) and (4), ar+y= ±1, 
 
 also • x—y= ±5. 
 
 The values o( x-\-y and x—y^ may be combined in four ways, as 
 follows : 
 
 x + y=-{-l; x + y=z-^l] x + y=-\-l; x + y= — l; 
 ar— y=4-5; x— y= 5; x—y=—5; x—y= — 5; 
 
 ^. . j a;=+3; a;=+2; ic=-2; a;=-3; 
 
 Wivmg I y=-2; y=-3; y= + 3; y=+2. 
 
 Or, condensing the expression, 
 
 y=T3 or ±2 ) 
 (246) 
 
 1 
 
176 QUADRATIC EQUATIONS. 
 
 48. Eliminating y from the last two equations, we have 
 52^' — 36^=— 5, 
 
 whence, x—\ or ^^ 
 
 y=-y-; y=i or f| ) ^n*. 
 
 z=i orH 
 
 PROBLEMS PRODUCING QUADRATIC EQUATIONS. 
 (314, page 258.) 
 
 1. Let ar= the greater part; then 14— ar= the less. Hence, 
 Oar 16(14-a:) 
 
 14-a: X 
 
 9ar' = 16(14~a:)', 
 3ar=4(14-ar), 
 
 2. Let a:= the number of persons ; then 
 
 350 350 
 
 + 20 = 
 
 8 and 6, An%, 
 
 X x—2 
 
 Hence, ar*— 2x=35, 
 
 x'— 2a; 4- 1=36, 
 
 a;— 1 = 6; x—*J,Ans» 
 
 3. Let ar= the number ; then 
 
 (22-ar)ar= 117, 
 
 or, x' — 22ar=— 117, 
 
 ar"-22ar+121= 4, 
 
 ar=ll±2; ar=13 or9, v4n«. 
 
 (246-259) 
 
PKOBLEMS PRODUCING QUADRATICS. 177 
 
 4. Let «= one part, and 18— ar=: the other; then 
 
 X* 25 
 
 X 5 
 
 18— a:"~i* 
 
 9.r=90; 10 and 8, -4w«. 
 
 5. Let x= one number, and ar— 4= the other; then 
 
 (2a;— 4)(a;-2)8 = 1600, 
 x* — ^x= 96, 
 
 ar=2=bl0 ; 12 and 8, Am, 
 
 6. Let x=z one number, and y= the other, the less ; then 
 
 ^^^ x-y : y :: 4 : 3 
 
 ^-^^r/^5^ Sar^Ty, (1) 
 
 ^^^, V^--^^ ary'=504, (2) 
 
 y"=216; whence y= 6) ^ 
 ar=14 J 
 
 7. Take 8a; and 6a; respectively, for the length and breadth of the 
 field ; then 
 
 8a;x5a; a;» 
 
 — -— — — r= the number of acres =— . 
 
 loO 4 
 
 Hence, - x 8«rl3 x 26a;, 
 
 a;'=13xl3, 
 a;=13; 8a;= 104, length 1 
 
 6a;= 65, breadth [ ^'**' 
 
 8. Take 5a; and 4a; for the length and breadth of the stack ; \» 
 will be the height. Hence there are 70a;" cubic feet in the stack, 
 and 20a;' square feet on the bottom. Therefore, 
 
 70a;*x4a;=224x20x», 
 
 a;'=16, 
 
 a;=4, 
 
 Length =5a;=20; Breadth =4a;=16; Height =Ja;=14, Ans, 
 
 (269) 
 
178 QUADRATIC EQUATIONS. 
 
 9. Let «'— 7 be the number; then from the statement of the 
 problem, we have 
 
 or, a:' + 9=81 — 18a: + ar", 
 
 x=4 ; ic' — 7=9, Ans, 
 
 18 
 
 10. Leta:=A's number of eggs; then — — =A*8 price per egg; 
 
 100— a:=B's " " ; ~ =B'8 " « 
 
 X 
 
 Hence, since they received the same amount of money, if we 
 multiply each man's number of eggs by his price per egg^ we have 
 the equation, 
 
 183r _ 8(1 00 -a:) ^ 
 
 100-ar~ X ' 
 
 or, 9ar'=4(100— a:)', 
 
 3ar=2(100~-a;), 
 a?=40; A, 40; B, 60, ^n*. 
 
 11. Here we have from the statement, 
 
 a; + y= 6, 
 x« + /=72, 
 «ind the solution is like that of 19, (301). 
 
 no 
 
 12. Let x=z the number of miles per hour ; then — = the num- 
 
 X 
 
 )>eT of hours. Whence from the conditions given, we have 
 36 36 
 
 or, a:' + ar=12, 
 
 x*+x + \ = *^; xz=z3^An8. 
 
 13. Let X and y be the numbers ; we have from the statement, 
 
 ar + y=100, (1) 
 
 Vx-yy=2; 
 squaring, a;— 2V^+y=4, (2) 
 
 from (1) and (2), _Va:y=48, (3) 
 
 four times (3) + (2), ar + 2 Viy + y = 1 9 6, 
 
 Vx+Vy=l4, 
 
 Vx=8\ ar=64,y=36, ^w». 
 (259-260) 
 
PROBLEMS PRODUCING QUADRATICS. 179 
 
 675 
 14. Let a?= the number of pieces; then = cost of one piece, 
 
 X 
 
 and from the conditions given, 
 
 48x=675+— , 
 
 X 
 
 48a;'— 6'75a:=675, 
 or, 16j;' — 225a;=:225; x—\b^Ans, 
 
 See solution of Exanaple 3, Art (J1805), 
 
 15. Let x=z the price of cloth ; then rT:r= gain per cent., and 
 
 a:' 
 
 — -= the whole gain. Whence, 
 
 a;»4-100ar=3900, 
 
 a:=— 50±80 ; a;=$30, Ans, 
 
 4x 104a; 
 16. Let x= the purchase money ; then a; + ---^= =the whole 
 
 104a; 
 cost, and 390 — 77^= the gain. By the statement we have also 
 
 the gaiu = ^^. -• — . Hence, i^ - ^^^ , . 1 . 
 
 ^'/^ ^/-' ^:j^ ' • ti^* ' 
 
 104.r 104a:' ^^ -^A^Z- 
 
 or, / /~/f-t^ ^ ^ ^ =z^^^^ 
 
 2j;» -. -^^^ 
 
 3000-8a;= . )L^^/ T-CHJ \l ~^/^^^;>^<5^ 
 
 300 r- »^ 
 
 Put a =300, and divide by 2 ; 
 
 a;' 
 6a — 4a;=— , 
 a 
 
 a:' + 4aa: + 4a' = 9a', 
 
 a-— a ; a: = |300, .^tj*. 
 
 (260) 
 
180 QUADRATIC EQUATIONS. 
 
 17. Observe that 396 — 216 = 180 miles, B's distance. Let x-=. 
 the number of days they traveled ; then = A*s rate, and 
 
 = B's rate. Hence, 
 
 X 
 
 _216_180 
 
 ~ X X * 
 
 a:«=36, 
 ar = 6 ; A, 36 ; B, 30, Ans, 
 
 11*. 'Vrith two unknown quantities, a:+y=60, «y=704 ; Seo 
 (15J99), Example 3. With one unknown quantity, 
 
 60.r— x' = '704, 
 ar'—60jr + 900 = 196 ; x=^4A or 16, Ans, 
 
 19. Let xz= price of sherry per dozen, 
 y= " claret 
 
 Then 1x + l2i/=50; 
 
 10 6 
 —=- + 3, 
 X y 
 
 lOy 
 or, x^ 
 
 ^ 3y + 6 
 
 By substitution^ VOy + 36y'4-'72y= loOy + 300, 
 
 9y' — 2y=V5; y =Sy x=2y Ans, 
 
 20. Let 19ar= the distance from C to D ; 
 then, x= B*s rate per day, and also his number of days, 
 
 ar'= B's distance, 
 '7x4-32= A's distance. 
 
 Hence we have, 
 
 ar' + 7a; + 32 = 19ar, 
 ar' — 12a;=— 32, 
 a;=6±2, 
 
 a;=8or4; ldx=zl52 or I6y Ans, 
 (260-261) 
 
PROBLEMS PRODUCING QUADRATICS. 181" 
 
 21. Let x^= the bushels of wheat, 
 a;+16= " " barley. 
 
 Hence, 
 
 24_ 
 
 X 
 
 = 24 1 
 a;+16 4* 
 
 (25 
 
 cents 
 
 =iofa 
 
 doUa 
 
 24a? + 16-24 = 
 
 =24.+^'-^;«^ 
 
 1 
 
 
 
 
 . -'*'" 
 
 = 384, 
 
 
 
 
 
 ^' + 4^^+16 = 
 
 = 400, 
 
 
 
 
 
 X 
 
 2~ 
 
 = -4±20; 
 
 
 32, 
 
 and 48, 
 
 Ans, 
 
 22. Let 2ar= the distance from C to D, 
 
 ., ,. 4(x— 18) .- 
 
 ar + 18= As distance, — •'=A8 rate per day ; 
 
 DO 
 
 ar-18= B's " -'^ = B's " 
 
 28 
 
 Tlie distance A traveled divided by his rate per day gives the 
 number of days he traveled ; and since B traveled the same number 
 of days, we have the equation, 
 
 63(2;+18) _ 28(a;-18) 
 4(j;-18)~ ar-flS ' 
 
 or, 
 
 9(ar + 18)'=16(ar-18)', 
 
 3(ar + 18) = 4(a;-18), 
 
 ar =126 ; 2jr=252, Ant, 
 
 23. Let «= one number; y= the other; then, 
 
 (;c-y)(x'-/)= 32. 
 {x + y){x'+y')=2l2\ 
 or multiplying, 
 
 «•— ary'— ar'y + y'= 32, 
 «' + ary'+a;V4-y'=272. 
 (261) 
 
182 QUADRATIC EQUATIONS. 
 
 Adding and subtracting these equations and dividing by 2, 
 
 a:' + y' = 152, (1) 
 
 X2/'+x'y = l20, (2) 
 
 (l) + 3-(2), x*-hSx'y + Sxy' + y' = 512, 
 
 taking cube root, x-\-y= 8. 
 
 Dividing the first equation by this, and taking the square root, 
 
 a:— y=±2; ir=5, y=Z, Ans, 
 
 24. Let x=z the number of horses B put in, 
 
 — = price of one horse a week, X - h» i , y- .J)e 
 
 4-18 * Y-it;/f .•■•^:y 
 
 1-18== rent of pasture ; Y~^ ^0 : 2-0 : 4 ,' y -^ 2- 
 
 when ar becomes a: + 2, yyy.^^. T^Oy^/Zfi 
 
 -= price of one horse a week. *L^-2. y^ - // p- 
 
 4-20 Y^y^2.// 
 
 ^-j-^+20= rent of pasture. y^^2_^y ^/fy =z /iZ- 
 Therefore, y-^^ &y =:. ^?_ 
 
 36 40 
 
 X x-\-2 
 x* + 6x=l2y 
 
 x= 5. Rent =30 shillings, Ant, 
 
 26. Let «= the figure of the ten's place, 
 v= " " unit's " . 
 
 I0x + y_ ^ 
 
 — Z7. — » 
 
 (1) 
 
 xy 
 
 
 I0x+y + 2l= lOy+ar; 
 
 (2) 
 
 9z-9y=-2l, 
 
 
 x-y-- 3, 
 
 
 x= jf-3. 
 
 
 (261) 
 
 
PROBLEMS PRODUCING QUADRATICS. 183 
 
 Substituting in the first equation, we have 
 2y'-.l7y=-30, 
 
 X— 3 ; 36, Ans, 
 
 26. Let xr=: the first number, which is the least ; y= the second, 
 and 2= the third. Then 
 
 y^/^oV, '^ a:-27+z=- 6; (1) 
 
 ' ,/ y a: + y + 0= 33; (2) 
 
 : ^ / ;r' + / + z'= 441. (3) 
 
 ^ ' ^ ^From (1) and (2), 3y= 39, 
 
 y= 13; 
 
 substituting in (1), x-\-z— 20, 
 
 " (3), a:' + 2r»= 272; 
 
 a;» + 22:2 + 2'= 400, 
 
 2x2= 128, 
 
 a;— 2=±12, 
 
 x= 4, 
 
 2= 16 ; 4, 13 and 16, An9, 
 
 27. Let x and y be the numbers ; then 
 
 «y= 24, (1) 
 
 «+y+a;'+y'= 62. (2) 
 
 Adding twice the first to the second, 
 
 (•^+y)'+(^+y)=iio, 
 
 ar + y= 10, 
 
 x—y— 2 ; *=6, y=4, ^w*. 
 
 28. Let «+y= one number, and x—y=: the other; then 
 
 «»-y'+2ar= 47, (1) 
 
 2x" + 2y' — 2a:= 62, (2) 
 
 4a;' + 2ar= 156, 
 
 «= 6, 
 
 y= 1 ; 7 and 4, Ans, 
 
 (261-262) 
 
184 QUADRATIC EQUATIONS. 
 
 29. Let x-=. one number, and y= the other; then 
 
 ar+yz= 27, 
 a:' + y' = 5103. 
 
 Solution the same as that of Example 19, page 244. 
 
 30. If ar= one number, and y= the other ; then 
 
 x^y = 9, (1) 
 
 ar* + y*=24l7. (2) 
 
 See Example 1, (301). 
 
 31. Let x + y= one number, and a;— y= the other; then 
 
 2(x«-y')(jr' + y') = 1248, (1) 
 
 4xy= 20. (2) 
 
 From (1), x*—y*=: 624, 
 
 eliminating y, ar"— -624x*= 625, 
 
 x*= 625, 
 rr= 6, 
 y = 1 ; 6 and 4, Ans. 
 
 32. Let ar= the number of days it takes one man, 
 
 a; + 10= " " " the other ; then 
 
 12 . 12 
 
 z //... ^ /7 ^»-14ar=120. 
 
 y ^-^ /<. y. ^/T-O 
 
 ar=7±13 ; 20 and 30, Ans. 
 
 33. Let ir=:A's stock, 
 
 1000-a;=B'8 " 
 1140— ar=A's gain, 
 640— (1000-*)= X — 360 =B'8 " 
 
 Now A's stock must be to B's stock, as A's monthly gain to B*8 
 monthly gain. That is. 
 
 >Ch -7 ^fO-O-O ,^^^ 1140— a; ar— 360 
 
 -f i 
 
 
 ;>V;/'U^y~>7--^ ^'^^^^^^^^^^ 
 
PROBLEMS PRODUCING QUADRATICS. 185 
 
 Or, by taking the products of extremes and means, 
 
 (3?— 360).r _ (1140-.r)(1000-ar) ^ 
 _ _ _ ; 
 
 whence, d{x^—360x)z=2{U40000-2U0x+x*), 
 a;' + 3200a:= 2280000, 
 ar + 1600 = 2200, 
 ar=600; 
 
 A's, $600 ; B's, ^400, Ans. 
 
 34, Let 2x*= the number in the first drove, 
 
 4 + 4t= '' second " 
 
 6ar' + 12ar+12= " third 
 
 3x' 4- 6^ + 16= " fourth " 
 
 Hence, we have 
 
 lla:» + 22:c + 32=:1121, 
 or, a:' + 2a:=:99, 
 
 x=9y 
 2.r' = 162; 
 therefore, 1st, 162 ; 2d, 40 ; 3d, 606 ; 4th, 313, Ans, 
 
 35. Let x=z one number, and y= the other ; then 
 
 Sxy-x'-y' = U, (1) 
 
 2ary-a:' + y'=14. (2) 
 
 Assume vx=y ; then 
 
 11 
 
 Svx* — a;' — v*x* — 1 1 
 
 
 X*- • 
 
 \j Vim^ •*/ t/ »*• -— ■ A A * 
 
 3t;-l-v" 
 
 2vx*-x^ + v*x^=U^ 
 By equating values of a:', we obtain 
 
 
 x«- ^^ 
 
 2v-l+v«* 
 
 25v'— 20v=- 
 
 -3, 
 
 
 If V=:i, 
 
 whence. 
 
 f 
 25; 
 
 ori. 
 
 5 and 
 
 36. If x= one number, and,y= the other ; then 
 
 x-hy= 20, (1) 
 
 a:y = 9216, (2) 
 
 xy = 96, 
 «— y = 4 ; whence, 12 and 8, Ans, 
 (262) 
 
186 QUADRATIC EQUATIONS. 
 
 37. Let 
 
 X aud 
 
 y be the parts ; then 
 
 xY = h, 
 xy=Vh, 
 
 
 (1) 
 
 (2) 
 
 whence, 
 
 
 «-y=:(a'-4V6)^; 
 
 ) Am, 
 
 This is a general solution of Example 36. 
 
 38. Let X and y be the numbers ; then 
 
 :r=aV, (1) 
 
 xy=h\ (2) 
 
 By division, 
 
 y"=-i ; or y=-, «=a5, Ans, 
 a a 
 
 39. Let X be the first of the consecutive numbers ; the number 
 sought will be ar(a:-fl)(2r + 2). Hence, 
 
 (x + l){x + 2) + x{z+2)-hx{x-hl) = lA 
 or, (y- f) %(j^^f] ar' + 2:r= 24, 
 
 . )fVy f y'^/-^ ^V>^ =7^ a:=4or -6; 
 
 ^u'^- Vrwhence, 4- 5-6= 120 | 
 
 ^ . y- or, (_6)-(-5)-(-4) = -120 [^~*- 
 
 40. Let a:= width of the engraving ; and 2x= the length; 
 lien 2x^= square contents of the engraving, 
 
 X^ C 18x4-4x3'=; " " margin. Hence, 
 
 ->^- f^ _ ^ 2a;'— 36 = 18a; + 36, 
 
 U^^77^-hI(;^ ^/. ^*-3'^^^^^-9a;=36, 
 
 Ly-^ _ jwheDce, a; =12, -4w*. 
 
 If^^^i^ :^ ^^'^C (263) 
 
PROBLEMS PRODUCING QUADRATICS. 
 
 187 
 
 41. In order that the two lots may be embraced in a single inclos- 
 ure of six sides, they must be placed as in the following diagram. 
 
 Let ir = a side of the 
 greater square, and y a side 
 of the smaller. Then x*-\-y* 
 will be the area of the two 
 lots, and 3(a?-f-y) + (ar— y) or 
 4a;-f-2y, will be the length of 
 the fence required to inclose 
 them. Hence 
 
 a;* + y'=:4100, 
 4ar + 2y=280; 
 from (2), y=140— 2:r, 
 
 substituting in (1), .a;' + (140--2ar)'=4100, 
 
 a;'— 112a;=— 3100, 
 a;-56=±6, 
 
 ar=:62 or 60 
 y=16 or 20 3 
 
 0) 
 (2) 
 
 An9, 
 
 42. Let rr= the first portion, 
 
 a— a;= the second portion, putting a=1300, 
 y= the first rate of interest, 
 r= the second " 
 Then 
 
 xy={a-x)z, 
 arz = 36, 
 (a— a?)y=49. 
 We have 
 
 (1) 
 (2) 
 (3) 
 
 49 
 
 V— 
 
 36 
 
 and eliminating y and z from the first equation, 
 49a:"=36(a-ar)', 
 a?=:600; 
 whenoe, y=7 per cent., ^=6 per cent., Am* 
 
 (263) 
 
/- r I ^ y 
 
 I V I 
 
 ^xL 
 
 ts 
 
 r't.e/ 188 
 
 QUADRATIC EQUATI0N8/^^^^^W'<^/- v 
 
 •L 2 r '*-?• ■'^^^ ^~ ^'* number of hours, y^ / J" ^ ,'/ 2, / y 
 
 »i= the distance from London to York; 
 
 then * 2i>^f,^f^ ^ y: -h^^y 
 
 I — = A's miles per hour : — = B's miles per hour; ^ 
 
 >/ * ,/ y ^^^Qc^^-^c^ 
 
 ^'^ ^^^.u:^:; ^L>^Z::) "^^^^^^^n ^^ z!n - ^- /^^ 
 
 a; y 
 ar— 25=y— 36, 
 
 or, ^ _+_=l, (1) 
 
 and ar-25=y-36, (2) 
 
 since they traveled the same number of hours before they met. 
 
 Hence, from (2), a:=y — 11, 
 
 and from (1), y' — 72y= — 396 ; 
 
 therefore, y=66, ar=55, Am, 
 
 Street. 
 
 44. Let ar= the side of 
 B's lot. 
 
 From the conditions we 6 
 have 
 
 36 + 6jr=(ar— 6)ar, 
 a;' + 12jr=3a, 
 
 x=6 + 6V'2; 
 whence, 6(1 +♦^2), Ans, 
 
 6 
 
 X 
 
 A'a lot 
 
 
 
 B'3 lot. 
 
 45. Let ar, y and ^ be the three numerical quantities. Then 
 
 Hence, 
 
 y'+«'+y+2=50. 
 
 «'-y« + 2r-y=10, 
 
 22' + 22 = 60, 
 
 2' + z = 30, 
 
 2=5 or —6 \ 
 y=4 or —5 V -4nj. 
 a;=3 or —4 ) 
 (263-264) 
 
 (1) 
 (2) 
 (3) 
 
PROBLEMS PRODUCING QUADRATICS. 189 
 
 46. Let x-=. the side of the cube ; then 
 
 a:'= the number of solid units, /x^7'' ^ ^'^ d^A^- ~i 
 V^=. the diagonal. t^'ly^Y'^ - ^<~^- ~ i/^y^ 
 
 Therefore, x^^Vzx'', 
 
 1 
 
 x« = 3x'; 
 
 
 whence. 
 
 x=X/Z, Am. 
 
 47. Let X and y be the numbers ; 
 
 
 x-\-y=xy\ 
 
 from (2) and (3), iry=3, 
 whence, x—y=±.^—Z 
 
 (1) 
 (2) 
 (3) 
 
 and, x=i(3±i^-3), y=i 
 
 (SrpVCrs), ^n5, 
 
 48. Let X and y be the numbers ; 
 
 x-^y=zxy, (1) 
 
 a:«-y»=ary; (2) 
 
 whence, x—y—\. (3) 
 
 Subtracting the square of (3) from the square of (1), 
 — T - / s^y*—Axy — 1 ; 
 
 ^ >-/^ X ""- / x-{-y=xy = 2±V5, 
 
 -^J^^^l x-y =1; 
 
 '/'z/r whence, ar=|(3=h»/5), and y=^(ld=f/5), ^n*. 
 
 49. Let X and y represent the numbers ; then 
 
 x'-f=zy, 
 x'-y'=x'^y\ 
 
 (1) 
 (2) 
 
 Put x=:vy] then the equations become 
 
 
 vy-y*=vy\ 
 vy^y'==vY-\-y\ 
 
 (3) 
 (4) 
 
 Dividing each equation by y', 
 
 (v«~l)y=vHl. 
 (264) 
 
 (5) 
 (6) 
 
190 
 
 QUADRATIC EQUATIONS. 
 
 From (5) we have, by the rule for quadratics, 
 
 2v=ldbV^5. 
 Multiplying (5) by v, v*—v=zv*\ 
 
 transposing in (5), v*=v-\-l ; 
 
 whence, by (8) and (9), v*— v=v + l, 
 
 or. 
 
 l=r2y. 
 
 (7) 
 
 (8) 
 (9) 
 
 (10) 
 
 Putting this value of (v*— 1) in (6), we have 
 
 2vy=v' + l=i; + 2, 
 or, 4i'y = 2v+4; 
 
 from (7), 2(litf^5)y==5±V^5; 
 
 whence, y=zt^V5 
 
 and x=v}/=^{l±:V5) - ^V5 ; or x=}(5±y5) 
 
 VAns, 
 
 
 ANOTHER METHOD. 
 
 
 
 «'—y'=^y» 
 
 (1) 
 
 
 ^_y«=ar' + y'. 
 
 (2) 
 
 From (1), 
 
 ^•-:ry=y'; 
 
 (3) 
 
 completing square. 
 
 .•-..?=¥• 
 
 
 or, 
 
 «-iy=±i»'s-y. 
 
 
 or, 
 
 x=i(l±Vo)y; 
 
 
 by involution. 
 
 ^'=i(3±»'5)y', 
 
 
 Substituting the values of x* and x^ in (2), we have 
 (2±f/6y-y'=i(3±*/5y + y', 
 
 (lztV5)y=^(5±y5) ; 
 
 x=^(5±y5) 
 
 y-^^'' \Ans. 
 
 (264) 
 
PROBLEMS. 191 
 
 PROBLEMS IN PROPORTION. 
 
 (334, page 276.) 
 
 6. Let X and y be the numbers. 
 
 ar + 4 : y + 4= 3 : 4, 
 ar — 4 : y— 4= 1 : 4. 
 Therefore, 
 
 Zy—4:Xz=z 4, 
 y-4a;=-12, 
 whence, 2y= 16 ; y=8, x=^b, Ans, 
 
 *1. Let X and y be the numbers. 
 
 a:4-y = 27, 
 xy : «' + y'=20 : 41. 
 
 Multiply the antecedents by 2, and we have, by composition and 
 division, 
 
 {x + yY : 2ary =81 : 40, 
 
 (x-yY : 2xy = 1 : 40; 
 whence, (^+y)' • (^— y)'=81 : 1, 
 
 or, x+y : x—y = 9:1, 
 
 or, 27 : x—y =9:1, 
 
 or, x—y = 3; 
 
 x + y =27; 
 whence, ir=15, y=12, Ant, 
 
 8. Let a;= the number of gallons of rum, 
 
 y= " " " brandy. 
 
 x—y : y =100 : a;, 
 
 a;— y : ar = 4 : y. 
 By Proposition XI, 
 
 {x-yY : iry=400 : ary, (co^^j 
 
 ox, X^-yY • 1 =^0^ = 1 ; 
 
 therefore, x—y= 20. 
 
 (276) 
 
192 
 
 PROPORTION. ff, ^ ,^ ; p / ; /,^ / jT y 
 
 Again, 
 
 1:^ = 25:-, 4^.,^,^C.^ 
 
 or, 
 
 1 : 25= y* : x\ y ~i-S- 
 
 
 1:5= y XX, 
 
 therefore, 
 
 6y= X', 
 
 ■whence, 
 
 y = 5, ar=25. Am, 
 
 9. Let a?+y= the greater, 
 X — y= the less. 
 
 {x + yY = x*^ 3ar'y + 3:ry« + y\ 
 lx—yY=x^ — dx'yi-3xy*—y\ 
 
 Hence, 
 
 6a:V + 2y': 8y'= 61 : 1, 
 
 
 3x' + y' : 4y'= 61 : 1 ; 
 
 
 therefore, 3^' + y'=244y', 
 
 
 a:'= 81y'. 
 
 
 But, «'-y»=320, 
 
 
 80y' = 320, 
 
 
 y= 2, 
 
 
 a:= 18; 
 
 
 whence. 
 
 20 and 16, Ans, 
 
 Note.— See Example 4 for a diflferent solution. 
 
 __^ 
 
 .^^ 
 
 -x.^^ . v~y 
 
 10. Let X and y be the numbers. 
 
 ^V - 6 ^ , y- - JJ 
 
 a-y : a:' + y» = 2 ; 5, 
 
 >^"7\'^r" ^y-^.'i ^-; :! 
 
 or, 22-y : a:' +y' = 4 ; 5 ; ^V^- ^Y '• -^v V ZyV : ^ / / 
 
 by Prop. VI, (i^ + y)': (^-yr= 9 : 1. ^f\^l\\);\r' 
 
 We have given, ar + y =60 ; ^^ ' ^ V •' •' -^ 'V 
 
 whence, by (1), ar-y =20 ; \^f^^f^ ^' ' ( 
 
 therefore, a?=40, y=20, Ans? 7 
 
 11. Let 3ar and 2a; be the numbers. 
 
 3x4-6 : 2ar--6 = 3 : 1, 
 whence, a;=8 ; 24 and 16, Ans, 
 
 12. Take 16a; and 9a; for the numbers. 
 
 16a; : 24 = 24 : 9a;, 
 
 x= 2; therefore, 32 and 18, Ans, 
 (276-277) 
 
PROBLEMS. 193 
 
 18. Let X and y be the numbei's. 
 
 x + y : x—y= 4:1; 
 by Prop. VI, x : y =5:3; 
 
 5i/=3x. 
 Again, a;' + y' : x =102 : 5, 
 
 «' + 5V^' : X =102 : 5, 
 or, 34^ : 25 =102 : 5; 
 
 whence, ar=15, y=y, -4w«. 
 
 14. Let X and y be the parts. / / ^^ _ y ///,'/ 
 
 a: : y= 9 : 1, v -/-^ - :^y 
 
 9y= a:; /^^ //" . 2^' V ^ '^ 
 
 a; + y = 20; i/ifyi^ = ^ 
 
 whence, y= 2, 
 
 a? =18. Hence, V'2ry=6, Ans, 
 
 15. Let 3j? and 2ar be the numbers. 
 
 32? + 6 : 2ar— 6= 9 : 4, 
 :r=13; 
 whence, 39 and 26, Ans, 
 
 16. Take ar, ary and xy* for the numbers. 
 
 a;'y : 2ry=ar : 2a'y^ 
 or, 1 : y« =1 : 2y; 
 
 whence, y=2. 
 
 By the conditions, ar + ary' = 300, 
 whence, 5a;=300; therefore, 60, 120 and 240, ^n5. 
 
 17. Let x and y be the two numbers, 
 
 ar' + y' : a;»-y' = 559 : 127; 
 
 Prop. VI, 
 
 2ar' : 
 
 2y»=:686 : 432, 
 
 or, 
 
 X* : 
 
 y» = 343 : 216, 
 
 or, 
 
 X : 
 
 y= 1 : 6; 
 
 whence, 
 
 
 6.r= 7.y. 
 
 By the conditions. 
 
 
 a:V = 294; 
 a;' = 7-49 ; whence, 7 and 6, Ann, 
 («77) 
 
194 PERMUTATIONS AND COMBINATIONS. 
 
 18. Let X and y be the numbers. 
 
 x' : f= S : 1, (1) 
 
 , , y' : ar'=96 : 1; (2) 
 
 from (3), 7 ^ ^^ y-^, 
 
 y : ^ : : 3 : / i a 
 
 _ ^ whence from (4), x'=S^'9Qx% 
 
 -^Vor, ir^= 3^-96, 
 
 or, ar»=3'-(32)' ; 
 
 therefore, a:=12, and y=24, ^n#. 
 
 19. By Prop. X, this proportion becomes 
 
 (^ + 1)' : (•^-1)'=2 : l; 
 whence, xi^2—^2=x-\-l, 
 
 and, ^=f^2^? ^'*'' 
 
 20. Putting the given equation in the form of a proportion, we 
 have 
 
 a-\-b + c-{-d : a—b + c—d—a-^b—c—d : a — b—c-\-d\ 
 by Prop. VI, and dividing by 2, 
 
 a-\-c : b-\-d=a—c : b—d, 
 or, a-\-c : a — c=2b + d'. b—d^ 
 
 Again by Prop. VI, a : c=b : rf, 
 
 or, a : b=zc : c?, Ans, 
 
 PERMUTATIONS AND COMBINATIONS. 
 (341, page 283.) 
 
 I.Here n — r + l = 10 — 4 + 1 = 7 ; hence, ^t>r7 
 
 10- 9- 8- 7=5040, ^n*. ' ^ 
 
 2. 6-5-4-3-2-l=V20, u4n«. 
 
 (277-283) 
 
PERMUTATIONS AND COMBINATIONS. 195 
 
 3. By formula (B), 
 
 10'9*8-7-6-5-4*3-2* 1 = 3628800, Ans, 
 
 4. Omitting the 0, the other four figures can be arranged in 
 
 4 • 3 • 2 • 1=24 ways. 
 
 Now we must reject every combination formed by placing the cipher 
 he/ore all the other figures. Hence, in each of the 24 combinations 
 of the figures 4, 3, 2, 1, the cipher may have 4 places. Therefore, 
 
 24x4=96, ^n*. 
 
 5. By formula ((7), 
 
 6. Formula (C) applies ; n = 16, r = 5j n— r+l = 12. 
 
 „ 16-15-14-13-12 _^„ . 
 ^= 5-4 3-ii-l -=-*^'^'-^"'- 
 
 ^ „ .» 20 -19 -18 -17 -16 -15 „_^„ . 
 
 V.Here 2^_______=38760, ^«, 
 
 8. Omitting the boy denied the privilege of the head, the others 
 can be arranged in 
 
 5 • 4* 3 -2 • 1 = 120 ways. 
 
 The omitted boy may occupy each of the five lower positions ; hence 
 all the ways will be 
 
 120x5 = 600, Ans, 
 
 9. The prime numbers below 40 are 1, 2, 3, 5, 7, 11, 13, 17, 19, 
 23, 29, 31, 37 ; or there are 13. Hence by ( 341), 
 
 n=13, r=!i^=6, n-r+l=8. 
 
 „ 13 •12-11 •10-9'8 __- . 
 
 Z= ; — =1710, Ans, 
 
 6- 5-4-3-2-1 
 
 (284) 
 
196 PERMUTATIONS AND COMBINATIONS. 
 
 10. By formulas {A) and (C^, we have 
 
 «(n-l)(»-2)(»-3)(n-4)=.120(fc^'^) ; ' 
 
 or, 11* — 7 n— 8, 
 
 whence, n= 8, Ans. 
 
 11. Heren = 8; and by ( 341 ), r=:- =4, n—r + 1 = 5. 
 
 Hence, « = -———— —7^ ; and -—=$0.60, Am, 
 
 4 * o * ^ * 1 70 
 
 12. By formula (C), if we let n— the number of horses, 
 
 n(n-l)(n-2) ^/«(»-l)\ 
 
 3-2-1 H 2-1 r ''^' ''"'^' *• 
 
 13. First omit 4 of the points which are in the same straight 
 line ; and considering the remaining 8, we find the number of com- 
 binations of 8 things taken 2 at a time to be 
 
 8-7 
 
 2-n = 28; 
 
 or 28 different straight lines. Now the four other points may be 
 joined each with the 7 points not in the same straight line, making 
 28 more different straight lines ; and adding the line containing the 
 five points, we have 
 
 28 + 28 + 1 = 57, Am. 
 Again, if no three of the points are in the same straight line, 
 there will be as many straight lines as there are couibinations of 12 
 
 things, taken 2 at a time, or =66. But since five of the 
 
 5 • 4 
 points are in the same straight line, —— = 10 of the combinations 
 
 are lost ; and adding the straight line containing the five points, we 
 have 66—10 + 1=57, Am. 
 
 Generally, if there are n points in a plane, of which J9 are in the 
 same straight line, there will be 
 
 n{n-\) pi^p-X) 
 2-1 2-1 
 
 different straight lines formed by joining the points. 
 
 (284) 
 
ARITHMETICAL PROGRESSION. 197 
 
 ARITHMETICAL PROGRESSION. 
 
 (353, page 288.) 
 
 Note. — In some of the following examples, we shall employ the formulas of 
 354, instead of substituting the given data in the primary equations. 
 
 1. /=7 + 35 -3 = 112, ^ns. 
 
 2. «=Y '280 = 6440, ^n«. 
 
 3. Here we have «, n and d given to find a and I. From formulas 
 (A) and (5), 
 
 l—a=.{n — \)d, 
 
 2* 
 l-\-a= — . Hence, 
 
 n 
 
 _2s—n{n—l)d 2s-\-n(n—l)d 
 
 and by substitution, a=:2, /=37, Am, 
 
 4. Substitute the value of / from formula (A\ in formula {B\ and 
 we have 
 
 .=l\2a + (n-l)dl =151(2 + ~)=2626, An>. 
 
 37 — 7 
 6. By formula ((7), (352), d~ — - — =6, and the terras are 
 
 5 
 
 7, 13, 19, 25, 31, 37, Arts, 
 
 6, We have a, n and 8 given, to find d and Z. By No. 5, (354), 
 
 7 ^440 ^ ,^, , (3720- 180) -2 ^ . 
 
 7. Here »= 11; hence, 
 
 5=V(9 + 109)=649, ^»«. 
 (288-289) 
 
198 SERIES. 
 
 8. By (352), rf=i^=5V, ^»«. 
 
 9. By No. 1, (354), 
 
 a=i|i{2+(365-l)2i=365'=$1332.25, ^n*. 
 
 10. By 9, (354), since we have given a, d and s to find n, 
 
 3_40±V'(37V + 8-3-438 -37^:^11881 — 37±109 ,„ 
 
 „= LJ _= = 6— =''• 
 
 Ans, 
 
 11. The last term will be n ; and by formula (B\ 
 
 «=^(l+n), ^n«. 
 
 12. Herea=l, rf=2, and *=^{2 + (n — 1)2J =n', Ans, 
 
 2 
 
 13. We have n, rf, and « given ; hence, 
 
 1900 — 24-25-3 1900 — 1800 
 
 60 60 
 
 2yAn8, 
 
 14. We have given a, rf, and « to find n. By No. 9, (354), 
 
 «- i - ^ ; 
 
 3 3 
 
 — 1 4- A4J. 
 
 or, w= — 5 3_ = i50, Ans, 
 
 T a. ' 
 
 ,^ 666-66 ^ , ,6666-666 ,^^^ 
 
 15. =5=rf; and =1200. 
 
 1 ^U o 
 
 Hence, w=1200 + 120 + 14 = 1334, Ans, 
 
 (289) 
 
ARITHMETICAL PROGRESSION. 199 
 
 PROBLEMS IN ARITHMETICAL PROGRESSION. 
 (355, page 291.) 
 
 1. Let (^— y), ^ and {x-{-y) be the numbers. 
 
 From first condition, 3x = 18, 
 
 X = 6 ; 
 from second condition, 3x' + 2y' = 1 5 8 ; 
 whence, 2y'=: 50, 
 
 y = 5. 1, 6 and 11, Ans, 
 
 2. Let(a:— 2y), (x—y)y «, (« + y), (ar -f 2y) be the numbers. 
 
 From Ist condition, 5x = 65, 
 
 a; = 13 ; 
 
 " 2d " 5a;' + 10y'=:1005, 
 
 a:' + 2y'= 201; 
 
 whence, 2y'= 32, 
 
 y = 4. 6, 9, 13, 17, 21, Ans. 
 
 3. Let (a;— 6), (a;— 2), (a; + 2) and (a? + 6) be the numbers. 
 
 (a:'-4)(a;«-36)=a;*-40a:' + 144 = l'76985, 
 or, a;*-40a;'= 176841, 
 
 a;«_40a;' + 400 = l77241, 
 a;» = 20i:421, 
 a;'=441, 
 a; =21 
 
 15, 19, 23, 27, Ans, 
 
 4. Let (a;— 3y), (ar— y), (X'{-y) and (a; + 3y) be the numbers. 
 
 2aj= 8, 
 
 or, ar= 4; 
 
 aj'-y» = 15; 
 
 whence, y'= 1, 
 
 y = 1. 1, 3, 5, 7, Ans, 
 
 (291-292) 
 
200 SERIES. 
 
 6. Let n= the number of days the first person travels, 
 d=lj and l=:l ■^{n — l)d=n. 
 
 . «=q(1 + n) = the whole distance. 
 
 and 
 
 15(w-6)= « " 
 
 hence, 
 
 in{l+n) = 15(n-6\ 
 
 or, 
 
 n' + w=30n— 180, 
 
 
 n'-29n=-180. 
 
 First person travels. 
 
 n = Y=t-V-= 9 or 20, 
 
 
 -6 -6, 
 
 Second « ** 
 
 3 or 14, Ans, 
 
 Explanation. — Call the first person A, and the second B. Now B overtakes 
 and passes A after A has traveled 9 days and B 3 days. But as A is increasing 
 his rate one mile per day, he finally gains on B, and overtakes and passes him 
 after A has traveled 20 days, and B 14. They are together after having trav- 
 eled 45 and 201 miles. 
 
 6. Let x=: one of the equal payments. 
 
 At the given rate per cent $60 will amount to $61 at the end of 
 60 days. As the rate of interest is ^-^^^ of the principal for a day, 
 the firet partial payment will amount to 
 
 the second to ^+3loo-^» 
 
 the third to ^+3lJo^, 
 
 and the last to x. 
 
 Hence the sum of the partial payments is, 
 
 Q0x-\-^j\^{59x-\-58x + 5lz. . . . +x), 
 or by summing the series, 
 
 60^ + tVo^' 
 Since the debt is to be canceled, we must have 
 60a- + T¥o^= 61, 
 T259ar=7320, 
 
 whence ^=^1t2J7» ^^' 
 
 (292) 
 
ARITHMETICAL PROGRESSION. 201 
 
 7. Let {x—Sj/y (j^— y), (^ + y)» and (x + Sy) be the numbers; then 
 
 2j:' + 18y'=66, 
 
 (1) 
 
 2«' + 2y'=61, 
 
 (2) 
 
 y'=h 
 
 
 y=i, 
 
 
 *=V; 
 
 4, 5, 6, 1, Am. 
 
 : notation as before, 
 
 
 4a;=24, 
 
 (1) 
 
 ■10j:y + 9y* = 945. 
 
 (2) 
 
 Whence, ar=6; 
 
 and from (2), y*— 40/= -39, 
 
 y'= 20±19, 
 
 y=ly 3, 5, 7, 9, -iw*; 
 
 9. Let (ar—y), a^, and (^ + y) be the digits. We have 
 
 I00(x-y) + 10x + x + y^^^^ 
 ox 
 100{x—i/) + l0x-hx + i/-{-l98 = l00{x + y) + lt)x + x—y\ (2) 
 from the second equation, y=l» 
 
 from the first, 3jr=9y, 
 
 x=3 ; 234, Ans. 
 
 10. Let n= the number of days. Then from No. 2, (3«54:), 
 
 -[6 + (n — 1)2}= the distance A travels, 
 2 
 
 ^{8 + (n-l)2}= « B " 
 
 The sum of these expressions is equal to the whole distance ; hence, 
 2»'+5»=102, 
 
 2n= — fit.3/; n=6, Ans. 
 
 11. Let a;~ the number of weeks if no one dies; 
 
 21a?= the pecks of corn distributed. , ' 
 
 In the second case the number of pecks distributed each week 
 will form an arithmetical series, in which 
 . (292) 
 
202 SERIES. 
 
 0=21, cf=--l, n=2aj; 
 and 
 
 S=zx(i2—'2x-\-l)=z the pecks of corn. 
 
 Equating the two expressions for the number of pecks, we obtain 
 43jr-2ar' = 2l3-, 
 2a:=22, 
 z=ll ; 21a;=231, Ans, 
 
 GEOMETRICAL PROGRESSION. 
 
 (364, page 296.) 
 
 1. Here a=l, r=2. 
 
 ^ 2»-l 512-1 ^^^ . 
 ^=-— ^ = — =511, Ans. 
 
 2. a =2, r=3 ; hence, 
 /=:2-3' = 2-2187 = 43Y4, ^n«. 
 
 3. a=l, r=§ ; hence, 
 ^="131-= ^-1-69049 ^ -59049 ' ^'^• 
 
 4. r=(VJ)^=2. Hence, 48, 96, ^w«. 
 6. r=(ifi)^ = (256)^ = 2. Hence, 
 
 6, 12, 24, 48, 96, 192, Ans. o ^^ 
 
 6. a=l, ^=f; hence, 
 
 *I, a=f, »*=f ; hence, 
 
 ^ s 
 
 (292-297) 
 
GEOMETRICAL PROGRESSION. 203 
 
 8. a=5, r=i ; hence, 
 
 1 3 
 
 
 JL?. 
 5- •"" 
 
 10. «=f'o'o> rrrr-j-i^ ; hence, 
 
 11. «=i» '*=— i; hence, 
 
 12. o=}, r=-i; hence, 
 
 X 
 
 13. o=l, r=- ; hence, 
 
 a 
 
 ^ I la, 
 
 , Ans, 
 
 X a—x a — X 
 a a 
 
 1 X* 
 
 14, o=-, r= — i; hence, 
 
 a a 
 
 1 
 
 a 1 a' __ a 
 
 ^4 
 
 15. By formula {B\ 
 
 1785= ^^^ ~ ^ = 255a. Hence, a='7, ^w.<r. 
 (297) 
 
204 SERIES. 
 
 16. By formula (5'), 
 
 7812^^' + " 
 
 or, /= 
 
 5-1' 
 31248 + a 
 
 By (B), Y812= "(f /^ =3906a. 
 
 5 — 1 
 
 Hence, o=2, and /=6250, Ans, 
 
 17. By formula (^), 
 
 1215 = 5r', 
 or, r'=243. Hence, r=3, Ans, 
 
 18. By formula (5), 
 
 2"-l 1024-1 ,^„„ . 
 /Sf=-— — -= = 1023, Ans, 
 
 PROBLEMS IN GEOMETRICAL PROGRESSION. 
 ( 365, page 300.) 
 
 3. Let the numbers be ar, Vxy, y ; then 
 
 X -\-V^ + y= 21, (1) 
 
 x'+ a:y + /=:189, (2) 
 
 The solution is like that of Example 1. We have a=21, 5=189. 
 
 441-189 ^ 
 i^=— ^^— =6; 
 
 and from (1) and (2), ar4-y=15, 
 
 x—y=z 9. Hence, 3, 6, 12, -4nf. 
 
 4. Let ar, ary, ary' be the numbers ; then 
 
 a;4-ary + a:y' = 210, (1) 
 
 xy^-x= 90. (2) 
 
 (297-300) 
 
GEOMETRICAL PROGRESSION. 205 
 
 Dividing by ar, and eliminating y', we have 
 
 120 
 
 *=:^ • 
 
 24-y 
 
 Substituting this value of x in (2), 
 
 12y'-9y=30, 
 y= 2, 
 ar=:30 ; 30, 60, 120, Ans. 
 
 5. Let Xj ary, ary', a?y" be the numbers ; then 
 
 
 x^xy+xf + xt/'=80, 
 
 
 
 
 (1) 
 
 
 xf _4 
 
 xy-\-X]/^ 3 * 
 
 
 
 
 (2) 
 
 From (2), 
 
 f _4 
 
 1+y 3' 
 
 
 
 
 
 or. 
 
 y=2; 
 
 
 
 
 
 whence by (1), 
 
 aj+2a: + 4a; + 8j;=30. 
 
 
 
 
 
 
 a: = 2; 
 
 2, 
 
 4, 
 
 8, 
 
 16, Ans, 
 
 6. Adopt the same notation as before ; then, 
 
 ar + a-y' = 148, (1) 
 
 ary + ar/ = 888. (2) 
 
 Dividing (2) by (1), y= 6, 
 
 from (1), 37a;=148, 
 
 x= 4 ; 4, 24, 144, 864, Arts. 
 
 7. Solution the same as in 1 and 3. 
 
 8. Let X, ary, a?y', xy* be the numbers ; then, 
 
 xy^-xy= 24; (1) 
 
 ar+ary* : ary-hary'= 7 : 3, 
 or, 1 + y' : y+ y"= 7 : 3, 
 
 dividing by y + 1, y'— y + l : y= 7:3, 
 
 3y'-10y=:-3; (2) 
 
 from (2), y= 3, 
 
 whence from (1), a;= 1 ; 1, 3, 9, 27, Ans. 
 
 (300) 
 
206 SERIES. 
 
 9. Let ar, jy, xy* be the numbers ; then, 
 
 « 
 
 
 a:+^y = 20, 
 
 
 (1) 
 
 xy*—xy=z^O. 
 
 
 (2) 
 
 ,. .. 1+y 2 
 r division, ^,_^- -, 
 
 
 
 ^y*-5y= 3; 
 
 
 
 y= 3, 
 
 
 
 ;r= 5; 
 
 5,15, 
 
 45, Arts. 
 
 10. Adopt the same notation as before; then, 
 
 . 
 
 
 a:y = 216, 
 
 0) 
 
 
 x^ 
 
 + ary = 328. 
 
 (2) 
 
 From (1), 
 
 
 xy= 6, 
 x^- '^• 
 
 • 
 
 from (2), 
 
 
 , 328 
 
 
 hence, 
 
 »/ 
 
 -82y'=-9; 
 y= 3, 
 
 
 
 
 ar= 2; 
 
 2, 6, 18, Ans, 
 
 11. Let XyVxy^ y be the numbers ; then 
 
 . ' a;+V^4-y=13, (1) 
 
 (:r + y)*^=30. _ (2) 
 
 Hence, a;4-y=13— Vary, 
 
 30 
 Vxy 
 ary— 13 V^=:— 30, 
 \^= 3, 
 xy= 9, 
 x-\-y= 10, 
 
 x^y= 8 ; 1, 3, 9, Ans, 
 
 (300) 
 
GEOMETRICAL PROGRESSION. 207 
 
 12. Let Xy xy^ a;y', be the numbers; then 
 
 
 xY= 64, 
 
 (1) 
 
 
 «' + a;yf a:y = 584. 
 
 (2) 
 
 From (1), 
 
 
 
 by substitution in (2), 
 
 16/-130/ = -16, 
 y= 2. 
 
 
 whence, 
 
 x= 2; 
 
 2, 4, 8, Ans, 
 
 13. Let ar, rry, a:y' be the numbers ; then 
 
 
 xy=i; 
 
 and, 
 
 xy—x : xy^—xy=5 : 1, 
 
 Of, 
 
 y-1 : y'-y=5 : 1, 
 
 or. 
 
 1 : y=5 : 1, 
 
 whence, 
 
 y=}; 
 
 from (1), 
 
 ary=l, 
 
 (1) 
 
 (2) 
 ar=5 ; 6, 1, ^, ^/w. 
 
 14. We might divide 120 into four parts, .which should be in 
 arithmetical progression, by assuming a sinyle unknown quantity for 
 the first term, or the common diflference ; but as we have two condi- 
 tions to satisfy, we take two unknown quantities. Let x be the first 
 term, and y the common diflference ; then 
 
 x + {x + y)-\-{x+2y) + {x-h3y) = l20, 
 or, 2a: + 3y=60. (1) 
 
 Again, we have by the second condition, 
 
 X, {x + y-12), (ar + 2y-12), (ar4-3y + 24), 
 
 for the geometrical series. Hence, by ( 363 ), 
 
 {x + y-l2y=x(x-^2y-l2), (2) 
 
 or, y'— 12a:~24y= — 144; 
 
 substituting from (1 ), y* — Qy— 216, 
 
 -?y,ic- V. v^;r. y^-?/ y= 18, 
 
 Ult/2'o,^z3ef ^^ 3. 3, 21, 39, 67, ^w*. 
 
 -/ 
 
208 SERIES. 
 
 15, Let ar, Va;y, y, be the numbers ; then 
 
 ar+^+y=31, (1) 
 
 a: + y = 26, (2) 
 
 V^= 6, 
 xy=25, 
 whence, jr— y = 24, 
 
 ar + y = 26; 1, 5, 25, -4yw. 
 
 16. Let Xj xy^ a-y', a-y*, ary*, ry*, be the numbers ; then 
 
 ar + ary + ry'4-ary'+a;y*+ary* = 189, (1) 
 
 xyJfxy'= 54, (2) 
 
 from (1), a;(l4-y + y')+^y'(l+y + y') = 189, 
 
 189 
 x-\-xy* = r, 
 
 ^ 1+y+y" 
 Multiplying this equation by y, and equating with (2), 
 
 189y 
 
 54: 
 
 1+y+y" 
 
 2y«-5y=-2, 
 y= 2, 
 ar= 3; 
 3, 6, 12, 24, 48, 96, Arts, 
 
 17. Adopting the same notation as before, we have 
 
 (a;+a5y) + (ar+ary)y*=189-36 = 153, (1) 
 (a; + a:y)y« = 36. (2) 
 
 1+y* 
 By division, — ^i— = y , 
 
 whence, y=2, 
 
 a;=3 ; 3, 6, 12, 24, 48, 96, Ans. 
 
 18. Let X represent one of the equal payments ; then 
 
 p{\ +r)—x^ due after 1st payment^ 
 
 ^(14-r)'-ar(l+r)-a;, « 2d 
 
 ^(I4.r)'-a:(l+r)'-ar(l+r)-a;, " 3d " 
 
 etc., etc., etc., 
 
 (801) 
 
DECOMPOSITION OF FRACTIONS. 209 
 
 or, ;>(l ■\-ry-x{l -hr)'^'- -x{l-\-r)-x, 
 
 due after the nth payment. And since the debt is to be canceled by 
 the nthy or last payment, we have 
 
 p{l-^ry—z(l+ry-'—x(l +r)'-'— — a-(l +r)—x=0 ; 
 
 or, ;>(l+r)"-{(l+r)'->+ (1 +r)'-'+ . . . . +(1 +r) + l(ar=0. 
 
 Summing the series in the parenthesis, ^ / /-/ r~/ y- 
 
 whence, X=^rf7^f » ^"*- 
 
 DECOMPOSITION OF RATIONAL FRACTIONS. 
 (369, page 308.) 
 
 1. 
 
 x'-^x-V\^ = {x-n){x-^\ 
 
 Hence we assume. 
 
 
 
 7^_24 _ A ^ B 
 
 --'. -s?t--i^.r»'''~' 
 
 (ar-7)(2:-2) x-1 x-2' 
 whence, 1x—24 = {A-h B)x—(2A + IB) 
 
 Equating coeflScients, 
 
 A+£= 1, -i</^ J^ . _ 
 
 whence, <i^A A- 5, £=i. ^ ^ / 'V 
 
 2. 2x^ + Sx--20 = (2x—5)(x-\-4), -^ ^.^J^ 
 
 . 20X + 2 ^ , ^ 7 
 
 ^^'"^ (2:r-5)(^ + 4) -27-r5-^JT4- 
 
 (301-308) 
 
210 SERIES. 
 
 then . 20x+2 = (A-h2B)x + (iA-BB), 
 
 Equating coefRcienis, 
 
 ^ + 2^=20, 
 4A-5B= 2; 
 whence, ^=8, i?=6. 
 
 + 7, -4n«. 
 
 2a;— 5 x + i 
 
 3. Assume 
 
 6ar«-22 + 18 _ A ^_ C 
 
 (x— l)(x' — 5j: + 6) x—\ x — 2 x— 3' 
 then 
 ea:*— 22a:4-18=^(x-2)(x-3)+^(x-l)(x-3)+C(x-.l)(«-2). 
 
 Hence we have 
 
 A-[-B+C= 6, 
 
 5^ + 4^4- 3 C= 22, 
 
 6^ + 3j5+2C=18. 
 Whence, .4=1, J?=2, C=3. 
 
 H ^ + r, Ans. 
 
 x—\ x—2 x— 3 
 
 4. Let _:_=:_+__- + 
 
 ar'— a? a; a: + l ar— l' 
 then ar + 2=^(x'-l) + ^(a:»-ar)4-C(a;« + a;). 
 
 Thence we have 
 
 A-\-B->rC= 0, 
 -^+C= 1, 
 -^= 2. 
 Whence, ^=—2, j5=:^, (7=1. 
 
 2^ 1 ^ 3 
 "x'^2(^Tr)"^2(^=T)» "*''*• 
 
 (308) 
 
BINOMIAL THEOREM. 211 
 
 6. We have 
 
 Hence we assume 
 
 10 A B C D 
 
 + ^ + — ^ + 
 
 a:*--13x' + 36 x^2 .r — 2 a: + 3 a:— 3 ' 
 from which we find 
 
 10= ^(a;' — 2.c' — 9^+18) + ^(x»+2ar' — 9a;— 18) 
 4-C(x'-3a:'-4a;+12)+i>(a;'H-3x«-4ar-12). 
 
 And equatinpj coefficients, 
 
 A + B+C-{-D= 0, 
 
 — 2^ + 2^-3(7+3i)= 0, 
 
 9^ + 9^ + 4(7+4i>=: 0, 
 
 18^-18^ + 12 C—12i>=:10. 
 
 Hence, combining the first and third equations, 
 A=—B, and C=—D, 
 
 From the second and fourth equations, u£i^'iztvci^ 
 
 2J-32>=0, . 
 
 18^ — 122>=5. 
 
 Hence, A=\, B=-\, (7=-i, D=\, 
 
 1111 
 
 + —f —. , An9» 
 
 2(« + 2) 2(.r--2) 3(i;+-3) 3(ic— 3) 
 
 BINOMIAL THEOREM. 
 (377, page 316.) 
 
 In examples 1 to 10 the exponents are whole numbers, so that 
 these examples need no solution here. In example 10 the numerical 
 coefficients are the same as those of example 2, of the illustrations ; 
 and we shall have the answer by simply changing +« to —x and 
 observing that the odd powers oi —x are negative. 
 
 (308-316) 
 
212 SEBIES. 
 
 11. Here we have 
 
 A = +1 ■ 
 
 B = Axn = +1 
 
 2 3 3 2 3 '6 
 
 _ „ n— 2 2 5 1 2-5 
 
 i>=Cx— -— =H •-•- = + 
 
 i: = j) 
 
 3 3-633 3-6-9 
 
 »-3 2-5 8 1 2-5-8 
 
 4 3-6 -9 3 4 3-6-9-12 
 
 The law of the numerical coefficients is now evident, and we may 
 write out the series, observing that the odd powers of — x are nega- 
 tive. Hence, 
 
 ,, A , X 2x« 2-6j:« 2-5-8x« 2-5-8-lla:' 
 (l~arf = l-- - 
 
 3 3-6 3-6-9 3-6-9-12 3-6-9-12-15 * 
 
 Ans, 
 
 12. A = +1 
 
 B = Axn = +1 
 
 n-1 13 1 3 
 
 C = Bx 
 
 2 4 4 2 4-8 
 
 J, ^ n-2 ,3 7 1 3-7 
 
 E=Dx 
 
 3 4-8 4 3 4-8-12 
 
 n-3 3-7 11 1 3-7-11 
 
 4-8-12 4 4 4-8-12-16 
 
 Hence, 
 
 ^ ^ 4 4-8 4-8-12 4-8-12-16 
 
 _ T/i , i 2 3-7 3-7-11 . 
 
 ""** ^ "^40 4^8^ 4-8-12a' 4-8-12- 16a*"^ * ' * *''' 
 
 Ans, 
 
 13. Since ?i is the same as in Example 11, the numerical coeffi- 
 cients will be the same. Hence, 
 
 / rA 4 a~^^ 2-a~36' 2 • 5a~'6' 2-5-8«~'3'6* 
 ^ ^ ^ ^3 3-6 3-6-9 3-6-9-12 
 
 \ ^^a 3-6a'^3-6-9a' 3 • 6 • 9 • 12a*^ /' 
 
 ^ (316) 
 
BINOMIAL THEOREM. 213 
 
 A = +1 
 
 B = A X n = ^1 
 
 C = Bx'^=(-l)x(-l) = +l 
 D=Cx''^ = ( + l)x(-l) = ^l 
 
 U=J)x !^ = (_i)x(-l)= +1 
 
 Hence, (a—b)-'=a-' + a-'b + a~'b'-\-a-'b' + 
 
 I b b' b' 
 
 = -+-7+-, + —+.... , Ans, 
 a a' a' a* 
 
 15. — — =cr(l— a;)"'. Omitting the factor a, 
 
 A =+l 
 
 B =A xn = ( + l)x{ — 2) =-2 
 
 C = B X _-=(-.2)x(-|)=.+3 
 
 j^ = /> X ^^^ = (-4)x(-f) =+5. Hence, 
 a(l — a;)-*^a(l + 2a; + 3x' + 4.c' + 5x* + 6u;' + ), ^n«. 
 
 16. The coefficients are the same as in Example 10. 
 
 17. A = + 1 
 B = Axn =(4.i).(4-f) =4-| 
 
 (316) 
 
214 SERIES. 
 
 ^ _, n-S I 2-4 \ / 1\ /1\ 2.4.7. 
 
 Hence, 
 
 ,xl I 2a~V 2a"3c* 2 • 4a~V 2 * 4 • 7a~^3^c' 
 (a — c')'=a' 
 
 3-6 3-6-9 3-6-9-12 
 
 """ \ 3a"~3-6a'""3-6-9a' 3* 6* 9 '120* /' ***' 
 
 18. ^ = =+1 
 
 B=Axn =( + l).(-i) :=z-^ 
 
 3 
 = +2^4 ■ 
 
 5_ 
 
 --¥=(-y(-=)(y 
 -=-"-^-(-i^y (-D o-^i^. 
 
 Hence, we have 
 ./ , ,^-i y , c-'x' 3c-V 3 • 5c-' j:' 3 • 5 • 7c-».r' \ 
 
 rf/ a:' 3j:* 3 • 5x' 3 • 5 • 7.c' \ 
 
 =cV"2? + ^^*-2"^T^c"^ + 2-4-6-8c«~- ' "h "^''** 
 
 _i 
 Note. — The exponent" of c in the first term is (c') ^ =:c ' ; and the ex- 
 ponents of c in the terms following diminish by 2 throughout. 
 
 19. ^ = +1 
 
 B = Axn = -3 
 
 i)=Cx^={ + 6)x(-5) --10 
 
 ^ = i>x^=(-10)x(-y=: +15 
 
 Hence the law of tlic coefficients is evident, and we have 
 
 (1 — a)-*= 1 + 3a + 6a' -f 10a' + 15a* + 21a' + 28a"' +....', Ans. 
 (316-317) 
 
' DESITYl 
 
 BINOMIAL Td^EOREM. ^ ^^ V*^''215? 
 
 20. A = +1 
 
 B==Axn =( + l).3 =4-1 
 
 _ 3 
 
 
 5 
 
 12 
 
 9 
 T6 
 
 Hence, 
 
 — ' —1 _i _ i_i 
 
 ,x?^ 4 3a ^a;' 3a ^x* S'5a *x' 3*5-9a ^ x' 
 
 (a'— ar )*=a* . . . . 
 
 ^ ^ 4 4-8 4.8-12 4-8-12-16 
 
 _ / 3a:' 3a:* 3* 5a;* 3*5 •Oa:" \ 
 
 ~ \ ~3a'~*4 • 8a'~"4 • 8 • 12a'~4 • 3 • 12 • 16a' ~ /' 
 
 Ans, 
 
 3 3 
 
 Note. — ^The exponent of a in the first term is (a') * =. a - ; the exponents in 
 the terms following diminish by 2 throughout. 
 
 21. A = +1 
 
 B=zAxn =(-{-i)x(-4) =-4 
 
 C = Bx'^=(-4).{-l) =+10 
 i)=(7x!^=( + 10)x(-?) = -20 
 JS = Dx^^= (-20)x(-^)= +35 
 
 F=zl!x~ = ( + 35)x(— ^) = -56 
 
 Hence, 
 
 (a + y)-*=a-*-4a^V + 10a-y-20a-y + 35a-y-56a-V + .. 
 
 1 4y lOy- 20y' 36/ 66/ . 
 
 =a*-"T* + -^— ^^-^^ ^+....,^«*. 
 
 (817) 
 
216 SERIES. 
 
 r -_i 
 
 22. We have = r(l— r)^; 
 
 V 1— r 
 
 A =+1 
 
 " = -•^ = (4) •(-!)■© -.^. 
 
 4 \ 2-3-5V V 5/ \4/ ^2-3-4-5* 
 
 Hence, 
 
 /, ^-i /, »• 6r' 6-llr* C'll'lCr* 
 r(l-r) ^=.(l+^ + 2-T^ + 2-^.+ 2^3T4-7^+.... 
 
 =»• + - + 
 
 6r» 6-llr* C-1116r' 
 
 5 2 
 
 ^• + 2^^T'-^^^3^.^+---^'^*' 
 
 23. Vi_a:* = (i_a-y\ 
 
 ^ = +1 
 
 n — 2 
 
 i>= Cx 
 
 ~ \ 2-15V * Vis) * W ~ 
 
 2 -IS" 
 
 14-29 
 
 2-315* 
 
 14-29-44 
 
 3 
 rr r. ^-^ f. 14-29 \ / 44\ /1\ 14-29-44 
 
 Hence, 
 
 »/ a;* 14j:" 14-29x" 14-29-44x'* . 
 
 VI «♦ — 1 >4«j? 
 
 ~ 15 2-15' 2-3-15' 2-3-4-15^ ....,^'**. 
 
 In all expansions of this kind, the chief thing to be aimed at, is a 
 simple and systematic method of calculating the numerical coeffi- 
 cients, and one which will clearly show the law of their formation. 
 
 (8X7) 
 
BINOMIAL THEOREM. 217 
 
 (378, page 318.) 
 
 1. (a~26)*=a»-3a'(26) + Sa(2by + (2&)* 
 
 =a' — 6a'6-fl2a6' + 86', Am, 
 
 2. (2a 4- ^xy = (2a)* + 4(2a)'(3a?) + 6(2a)'(3x)' + 4(2a)(3x)' + (3a:)* 
 
 = 16a* + 96a'x + 216aV + 216ai;' + Six*, ^na. 
 
 ,. (-iy=.-.©«6)'-,©V(jy 
 
 = 1— 2a + |a'— la' + j^^«*, ^ns. 
 
 4. (a'— tfj: + a:')* = («'— aa-)* 4-4(a' -ax) V + 6(a'— aj;)'a;* + 
 
 4{a*—ax)x*-\-x\ 
 Performing the operations indicated, 
 a'— 4a'a;+ 6a V— 4a V 4- a*x* 
 
 4- 4aV— 12aV + 12a*a:*— 4a V 
 
 4- ea*x* — 12aV4- 6a V 
 
 4- 4aV — 4ax'4-a;'' 
 
 a*— 4a'a:4- lOaV — 16ttV 4- 19a*x* — 16aV4- lOaV — 4aa;' +x% 
 
 Ans. 
 
 6. The numerical coefficients will be the same as those in Exan> 
 pie 12, (377). Hence, 
 
 (4a'-3x)^ 
 
 -(ia^\^ (^^')V ) 3(^^') M3 ^r 3-7(4a/') '^(3x)» 
 ^ ' 4 4-8 4-8-12 
 
 ') ^(3a;) 3(4a') ^(3 x)' 3 • 7<-^-^^ 
 
 ~~4 r^8 
 
 f 3x 3(3x)' 3''7(3i:)» 
 
 ^^a^ 3 1 j\ 
 
 A{2ay 4 -8(20)^ 4-8-12(2a) = 
 
 (318) 
 
218 SEBIES. 
 
 FRENCH'S THEOREM. 
 
 (379, page 320.) 
 
 b 5 
 1. Here we have n= 4, a=2, 6 = 5, -=-; hence, 
 
 
 
 c, = 
 
 (2)* 
 
 
 
 = 16 
 
 
 
 
 C, = 
 
 16" 
 
 \ 
 
 •i 
 
 == 160 
 
 
 
 
 0,= 
 
 160- 
 
 f 
 
 •f 
 
 = 600 
 
 
 
 
 C,= 
 
 600 
 
 •?- 
 
 •t 
 
 = 1000 
 
 
 
 
 ^, = 
 
 1000' 
 
 i 
 
 •f 
 
 = 625 
 
 
 Hence, 
 
 
 
 
 
 
 
 
 (2ar+ 
 
 5yy-- 
 
 = 16j:* + 160j:V + 600x' 
 
 y' + 1000jr/ + 625y*, 
 
 Am, 
 
 a. n=5, a=:2, 6 = 3, -=-. 
 
 ' a 2 
 
 C, = (2)* = 32 
 
 C, = 32 • i • f = 240 
 
 (7, = 240 • A . 1 = 720 
 
 C^ = V20 • f • f = 1080 
 
 C, = 1080 • i • I = 810 
 
 C, = 810 • i • a = 243 
 Hence, 
 
 (2a — 3x)' = 32a*~240a*j: + 720aV-1080aV + 810aa:* — 243ar*, 
 
 Ani^ 
 
 s. 
 
 n=6, a=3, 6=4, ^-=\. 
 
 
 C, = (3)' = 729 
 
 
 C, = 729 • f A = 5832 
 
 
 ^3 = 5832 • f -i = 19440 
 
 
 C^ = 19440 • f • i --= 34560 
 
 
 C, = 34560 • ^ • i = 34560 
 
 
 C. = 34560 • f • i = 18432 
 
 
 C, = 18432 •^- ^ = 4096 
 
 Hence, 
 
 
 (3+4:r')*: 
 
 =729 + 5832a;' + 19440x* + 345602:' + 34560a:' 
 
 
 + 18432a:'-'' 4- 4096a:'«,^n*. 
 
 
 (320) 
 
BINOMIAL THEOREM. 219 
 
 3 ;. ^ * 1^ 
 4, n=i, a=-, 6==^, -=-. 
 
 Ci 
 
 = a)* 
 
 = AV 
 
 c. 
 
 = AV 
 
 •fif= fJ 
 
 o. 
 
 = fj; 
 
 •fif= M 
 
 c. 
 
 = ir 
 
 • 1 • II = Hi 
 
 /7 1*2 • 1 • il ill 
 
 ^5 125 4 15 625 
 
 Hence, 
 
 /3a 4/-V 81 , 27 , 54 , , 192 , 256 . , 
 \4 5/ 256 20 25 125 626 ' 
 
 2,369 
 
 n=6, a=-, o=~. - = -. 
 
 ' 3' 2' a 4 
 
 0, = (!)• 
 
 
 = ^%^ 
 
 c, = AV 
 
 ■f 
 
 •f = H 
 
 C, = M 
 
 •f 
 
 • f = V 
 
 C. = V 
 
 'i 
 
 ■ J = 20 
 
 C, = 20 
 
 . 1 . 
 
 4 
 
 ■ J = XJ» 
 
 C. = xji . 
 
 •1 
 
 • f = n' 
 
 C, = iLll ■ 
 
 i 
 
 • f = w 
 
 Hence, 
 
 /2< 3rY 64 . 32^, 20^, , ^^^, , 135 , , 243 . 
 
 729 
 
 1 A 1 * * 
 
 «=5, «=j, 6=g, -=-■. 
 
 <7, = (J)' 
 
 
 — Tjir 
 
 <?, = tAt 
 
 •f 
 
 • t = Tk 
 
 C» = ih 
 
 •i 
 
 •i= T*T 
 
 C, = TiiT 
 
 •f 
 
 • J = 5i,r 
 
 0, = jh 
 
 •r 
 
 •}= IH 
 
 0, = ^h 
 
 •i' 
 
 • 1 = jtVi 
 
 Hence, 
 
 U^sj ~1024"256'^T60~200"^600~3T25 ' 
 (320) 
 
220 SERIES. 
 
 7. n=8, 
 
 Hence, 
 
 .4. 
 
 1 
 
 '~2' 
 
 b 
 a 
 
 = 1, 
 
 
 c, = 
 
 a)' 
 
 
 = 
 
 il» 
 
 c. = 
 
 lii- 
 
 r 
 
 1 = 
 
 iV 
 
 c, = 
 
 ?v 
 
 J 
 
 = 
 
 iv 
 
 c, = 
 
 /i- 
 
 1 
 3 
 
 = 
 
 3V 
 
 <^. = 
 
 /i- 
 
 i 
 
 = 
 
 Wt 
 
 c. = 
 
 t'sV- 
 
 i 
 
 = 
 
 3't 
 
 c, = 
 
 /J- 
 
 1 
 
 — 
 
 j'l 
 
 c. = 
 
 s'i- 
 
 t 
 
 = 
 
 -A 
 
 c.^ 
 
 j'i- 
 
 i 
 
 ^:^ 
 
 Tk 
 
 m' 1m' 
 
 7m' 
 
 + • 
 
 35 
 
 7 
 
 .+ 
 
 1 
 
 128 S27n* 64m* 32w* 
 
 1 
 
 256m' 
 
 i, ^n«. 
 
 380, page 323.) 
 
 :V8 + l = 2Vl-|-|. 
 
 
 ^ = +1.0000000 
 
 ^= + ^ •|-^= + 
 
 416667 
 
 C=-i-i-5 = - 
 
 17361 
 
 i>=-i-|-C= + 
 
 1206 
 
 JE=^l .|-2)=- 
 
 101 
 
 ^+ -HT^= + 
 
 9 
 
 1. 0400420 = Vl+| 
 
 
 2 
 
 V9 = 2.080084, Ans. 
 
 (320-323) 
 
 
DEVELOPMENT OF SURD ROOTS. 221 
 
 -/si =V27 + 4=3^1 + _4^. 
 
 
 A = +1.0000000 
 
 
 JB = + J- '^\' A=z + 493827 
 
 
 C = - ^ '^\'B = - 24387 
 
 
 i> = - I • aV ^ = + 2008 
 
 
 ^=~f j\'J)=- 198 
 
 
 ^=~n-iV^=+ 21 
 
 
 
 1.047l27l = Vi 
 
 +iV 
 
 3 
 
 
 V31 := 3.141381, Ans, 
 
 
 Vl00=:v/i25 — 25 = 5^1 _j. 
 
 
 ^ = +1.0000000 
 
 
 5=-i-l-^ = - 666667 
 
 
 C7=-i4-i?=- 44444 
 
 
 i>=:-A.j.C7=- 4938 
 
 
 ^=-|-}-i>=- 658 
 
 
 ^=-|i*i-^=- 97 
 
 
 G = - I 'i'F=:^ — 15 
 
 
 ^=-if4-^=- 2 
 
 
 0.9283l79 = Vr 
 
 -i 
 
 5 
 
 
 Vioo = 4.641589, Ans. 
 
 
 VllO=Vi25-15 = 5<^l-^3^. 
 
 
 A = +1.0000000 
 
 
 £ = -^ '^\'A= - 400000 
 
 
 C= — |-^j-^=- 16000 
 
 
 Drr— 1 .^j.e=-- 1067 
 
 
 ^ = -- f • /j • /> = ~ 85 
 
 
 ^=-H*i\-^=- ^ 
 
 
 6^=-^-jV-^=- 1 
 
 
 0.9582840 = Vr 
 5 
 
 ^^'V 
 
 VllO = 4.791420, ^n<f. 
 
 
 (323) 
 
 
222 SERIES. 
 
 5. ^297=^243 + 54=3^1+1. 
 
 A = +1.0000000 
 
 ^ = + 1 • I • .1 = + 444444 
 
 C=-f '%' B = - 39506 
 
 i>=-f-a-C=+ 5268 
 
 F— -\\'\' Ez= + 138 
 
 Q= -L 'X' F= - 25 
 
 /?= -If' I* 6^= + * 
 
 1.0409504 =Vn. I 
 3 
 
 V297 = 3.122851, Arts, 
 
 V60=V64 
 
 — 4i 
 
 = 2Vl- 
 
 -tV 
 
 
 A 
 
 
 
 
 = +1.0000000 
 
 B= - 
 
 -i . 
 
 tV 
 
 .^ 
 
 = — 
 
 104167 
 
 C = - 
 
 -^-.' 
 
 tV 
 
 ■J? 
 
 = — 
 
 2713 
 
 I)= - 
 
 -u- 
 
 tV 
 
 (7 
 
 = — 
 
 103 
 
 E=- 
 
 -u- 
 
 I'a 
 
 •i> 
 
 = — 
 
 5 
 
 0.9893012 =Vi_yy 
 2 
 Vei = 1.978602, Ans, 
 
 7. V4=V32-28 = 2Vl_i, 
 
 In this case the last term, |, is nearly equal to unity, and the 
 series will converge very slowly. We may give the calculation 
 another form by taking 
 
 4 = VaS and 1/4 = 1^128. 
 
 Whence, |Vl28=|V243-115=^v/l_|i|. 
 
 (323) 
 
DEVELOPMENT OF SURD ROOTS. 223 
 
 A 
 
 
 
 = 
 
 + 1.0000000 
 
 B = 
 
 - i- 
 
 iH- 
 
 A = 
 
 — 
 
 946502 
 
 C = 
 
 -tV 
 
 iif ■ 
 
 B = 
 
 — 
 
 179173 
 
 D = 
 
 —,%■ 
 
 Hi- 
 
 C = 
 
 — 
 
 50876 
 
 E = 
 
 -a- 
 
 iH' 
 
 D = 
 
 — 
 
 16854 
 
 F = 
 
 -H' 
 
 m 
 
 ' E = 
 
 — 
 
 6062 
 
 G = 
 
 -H- 
 
 m 
 
 ' F = 
 
 ~ 
 
 2295 
 
 H = 
 
 -If 
 
 iii 
 
 • G = 
 
 — 
 
 900 
 
 I = 
 
 -a- 
 
 iH 
 
 ' H = 
 
 — 
 
 362 
 
 J = 
 
 -n 
 
 •Hi 
 
 'I = 
 
 — 
 
 150 
 
 K = 
 
 -u 
 
 ■Hi 
 
 J = 
 
 — 
 
 62 
 
 L = 
 
 -a 
 
 Hi 
 
 ' K = 
 
 — 
 
 26 
 
 jr = 
 
 -If 
 
 ■Hi 
 
 • L = 
 
 — 
 
 11 
 
 N = 
 
 -If 
 
 ■Hi 
 
 • M = 
 
 — 
 
 5 
 
 = 
 
 -li 
 
 •Hi 
 
 •ir = 
 
 ( 
 
 2 
 
 
 }.8796720=v^l-iif 
 
 
 
 
 
 
 f 
 
 W = 1.319508, Ans. 
 
 >^3275=V3125 + 150=6Vl+_|_. 
 
 A = +1.0000000 
 
 ^ = + J * yf 5 • ^ = + 9^00^ 
 C= -f -tIj-^ = - 1843 
 
 ^ = + f • tIj • C' = 4- 63 
 
 1.0094208 = ^^l+yf 3 
 5 
 
 v^3275 = 5.047104, Ans, 
 9. v^T25=Vi28 — 3 = 2Vi_-.|^. 
 
 A 
 
 B = 
 = 
 i> = 
 
 (323) 
 
 + 1.0000000 
 
 — 33482 
 
 — 336 
 
 — 5 
 
 
 
 
 
 
 0.9966177 = ^1- 
 
 2 
 
 -Tie 
 
 
 1.993235, Ans, 
 
 
224 
 
 SERIES. 
 
 Note. — A table of logarithms affords an easy and practical method of finding 
 the roots of numbers, since we have only to divide the logarithm of the number 
 by the index of the root, and look out the number corresponding. See 40'ft. 
 
 To find ^4, we have 
 
 log. 4 = 0.6020600 
 
 -^^ = 0.1204120 
 o 
 
 Hence, V* = 1-319608 
 
 «8 in Example 7. 
 
 METHOD OF INDETERMINATE COEFFICIENTS. 
 (38a, page 327.) ^_^^ 
 
 l-2a; 
 l + dx 
 
 :A-hBx+Cx* + I)x*+Ez*-^ 
 
 0= A 
 — 1 
 
 x'+ B 
 
 x-¥ C 
 
 x*+ B 
 
 x^+ E 
 
 -ZA 
 
 ^3B 
 
 -3C7 
 
 -32) 
 
 + 2 
 
 
 
 
 «*+. 
 
 Therefore, 
 
 whence, 
 
 ^-1=0, 
 
 ^=1; 
 
 j5-3^ + 2 = 0, 
 
 ^=1; 
 
 (7-3^=0, 
 
 C=3; 
 
 2)-3C=0, 
 
 i> = 9; 
 
 JF-3i> = 0, 
 
 -£'=27, etc. 
 
 14-aJ + 3x' + 9j;' + 27x*4-81a;*+ , Am, 
 
 l4-2a; 
 \—x—x 
 
 ^=A-\-Bx-^Ca^-\-Dx* + Ex* + 
 
 0= A 
 
 x'-hB x-hC 
 
 x* + D 
 
 x*-hE 
 
 — 1 
 
 -A -A 
 
 -5 
 
 -C 
 
 
 - 2 -B 
 
 -C 
 
 -D 
 
 -Aj rr^f 
 
 i 323-327) 
 
 :% 
 
INDETERMINATE COEFFICIENTS. 
 
 225 
 
 Therefore, 
 
 ■whence, 
 
 A-l =0, 
 B—A-2 =0, 
 C-'A-£=0, 
 
 ^=1; 
 
 (7=4; 
 /)=7; 
 
 JE'=11; etc. 
 
 1 +3a;+4:r'4-Va;* + ll^* + 18^»+ , Ans. 
 
 Therefore, 
 
 whence, 
 
 1— a; 
 
 l_3x— 2a;' 
 
 ■.A-{-Bx-\-Cx'-{-Dx*-i(-Ex*-^,, 
 
 0= ^ 
 — 1 
 
 x'-\- B 
 -^A 
 + 1 
 
 x+ C 
 -2^ 
 
 x'+ D 
 -3(7 
 -2J5 
 
 a;'+ E 
 -SB 
 -2(7 
 
 «*+. 
 
 -4— 1=0, 
 
 B-3A-{- 1=0, 
 
 (7—35-2^=0, 
 
 i)-3 (7-25=0, 
 
 Jgr-3i>— 2(7=0, 
 
 ^ = 1; 
 5=2; 
 (7=8; 
 i>=28; 
 JS:=100;etc. 
 
 l+2a;4-8a;' + 28a:' + 100a;*4-356a;»+ , Ans. 
 
 J±^f^=A + Bx-\'Cx'+I)x*-{-Ex*-^ 
 1— 4z + 4a:' 
 
 Therefore, 
 
 "Whence, 
 
 0= A 
 — 1 
 
 x'-\- 5 
 -4^ 
 — 5 
 
 x-h C 
 -45 
 + 4^ 
 
 x'4- B x'+ E 
 
 -4(7 
 
 + 45 
 
 -42> 
 
 + 4(7 
 
 «• + 
 
 ^- 1=0, 
 5—4^— 6=0, 
 (7-45 + 4^=0, 
 2>-4(7+45=0, 
 ^-42) + 4 (7=0, 
 
 .4 = 1; 
 5=9; 
 (7=32; 
 i>=92; 
 ^=240; etc, 
 
 l+9a; +32a;' + 92a?' + 240x*+ 
 
 X 
 
 « + 9a;' + 32a:' + 92a:* + 240a:*+ , Ans, 
 
 (327) 
 
226 
 
 SEB1E6. 
 
 
 
 2 
 
 ■r — 
 
 .Ax-' ^Bx' -^ Cx^-Dx" ^ Ex' Jr , . . . 
 
 *• Zx-lx*- 
 
 0= 3^ 
 — 2 
 Therefore, 
 
 ar» + 3^ 
 -2^ 
 
 3^- 
 
 a; + 3(7 ar' + Si) 
 -2J? -2C7 
 
 2=0, ^ = 
 
 a;« + 3jr «* + 
 -22> 
 
 35-2^=0, 
 
 ^=i; 
 
 3(7-2J5=0, 
 
 c=^S\ 
 
 3i)-2C:=0, 
 
 ^=h; 
 
 
 3^— 2i>=0, 
 
 jg^= 
 
 iVV, etc. 
 
 Whence, 
 
 2 4 8a; 16a;' 32a;' 
 8i + 9 + 27 + ^ + 2i3+---^'^- 
 
 l+2a;' + 3ap 
 
 -^=Ax'-\'Bx-\-Cx*-\-Dx* + Ex'-\- ... 
 
 0= A 
 
 - 1 
 
 Therefore, 
 
 a;'+^ 
 
 a;+ C 
 
 a;'+ i> 
 
 a;'+ E 
 
 a;*+ i^ 
 
 a;'^+ e' 
 
 
 + 2 A 
 
 + 2 J? 
 
 + 2(7 
 
 + 22) 
 
 + 2^ 
 
 
 
 
 + 3^ 
 
 + 3^ 
 
 + 3(7 
 
 «•+ 
 
 Whence, 
 
 ^- 1 =0, 
 
 A= 
 
 Ij 
 
 B =0, 
 
 B= 
 
 0; 
 
 (7+2^=0, 
 
 (7=- 
 
 - 2; 
 
 2) +2^=0, 
 
 /)= 
 
 0; 
 
 3-4+^+2(7=0, 
 
 E= 
 
 1; 
 
 3^+i^+2i)=0, 
 
 F= 
 
 0; 
 
 8(7+(y + 2^=0, 
 
 G= 
 
 4; 
 
 3^+/+2(?=0, 
 
 /=- 
 
 -11; 
 
 3Gr4.ir+2/=0, 
 
 /{= 
 
 10; 
 
 3/+jlf+2^=0 
 
 M= 
 
 13, etc. 
 
 (7) 
 
 l-2ar' + a;* + 4a;«-lla;' + 10a;" + 13a;"— , Arts. 
 
 Note. — It is obvious that that the alternate equations may be omitted ; this 
 is done from the sevenUi onward. 
 
 (327) 
 
INDETERMINATE COEFFICIENTS. 
 1 + 2ax + a V 
 
 227 
 
 0= A 
 — 1 
 
 x'+ B 
 -{-2a A 
 — 1 
 
 x+ C\x'+ D 
 + 2aB\ +2aC 
 
 x'+ E 
 
 x*-\- F 
 
 -\-2aI> 
 
 + 2a£ 
 
 + a'C 
 
 + a'J9 
 
 x'+. 
 
 Therefore, 
 
 ^-1=0, 
 
 B-{-2aA — l=0, 
 
 C+2aB + a'A=0, 
 
 D'}-2aC-\-a^B=0, 
 
 E-\-2aD->t 0^0=0, 
 
 A=l; 
 
 ^=:-f(l-2a); 
 (7 = -(2a-3o'); 
 i) = 4-(3a'-4a^); 
 ^=-(4a' — 5a*); etc. 
 
 Whence, 
 1 +(l-2a)a:-(2a-3a>' + (3a'-4a')a;'-(4tt'-5a>*+ . . . ., 
 
 Ans, 
 
 8. \r[:^=A-\-Bx-\-Cx''-\-Dx'-\-Ex' + 
 
 Squaring both sides, we have 
 
 l-x^zA' + AB 
 +AB 
 
 X +AC 
 -hAC 
 
 Therefore, 
 
 A'-l = 0, 
 2AB-\-l = 0, 
 
 2AC-{-B* = 0, 
 
 2AI)-\-2BC = 0, 
 
 2^^+C" + 2^i> = 0, 
 
 Hence, 
 
 x' + AD 
 + BC 
 + BC 
 + AJ) 
 
 x' + AE 
 + BD 
 + C' 
 -hBD 
 + AE 
 
 ^-2-4' 
 
 E = 
 
 x'-^,. 
 
 2-4-6' 
 
 3-5 
 2-4-6-8 
 
 ; etc. 
 
 X x* 
 
 ri-«=l-;T-;r^- 
 
 Zx"" 
 
 ^■bx* 
 
 2 2-4 2-4-6 2-4-6'8 
 (327) 
 
 . . . , Ans* 
 
228 
 
 SERIES. 
 
 9. Assuming the same form of development as in Example 8, we 
 shall have, 
 
 ^'-1 = 0, ^ = 1; 
 
 2AB-S = 0, -5 = 1; 
 
 2AC+B'-5 = 0, C7 = V; 
 
 2AD + 2BC-1 = 0, ^ = H; 
 
 2AE+C-{-2BI)-9 = 0, H = HI ; etc. 
 
 "Whence, 
 
 , Sx liar' . 23x' . Il9x* . 
 
 2 8 16 128 ' 
 
 10. 
 
 — ; ? — i i— ^ rs = Ax'-\-Bx'-^ Cx' + Dx*-\- 
 
 l+a;'-fa:* + x« + a:'4-a:'°+ 
 
 0= A 
 -1 
 
 x' + B 
 
 ■\-A 
 + 2 
 
 + A 
 —3 
 
 a:« + i> 
 
 + A 
 + 4 
 
 a:« + ^ 
 + i> 
 + C 
 -¥B 
 + A 
 — 5 
 
 x*-\-F 
 
 Whence, 
 
 ^-1 = 0, 
 
 B+A-^2 = 0, 
 
 C+B + A-3 = 0, 
 
 i) + CH-^ + ^ + 4 = 0, 
 
 l—dx*+5x*—7x'+9x'—lW + 
 
 + E 
 + i> 
 
 + (7 
 
 4-6 
 ^= 1; 
 ^ = -3; 
 C = +5 ; 
 2> = —7 ; etc. 
 
 r" + ..., 
 
 ., Arts. 
 
 REVERSION OF SERIES. 
 (383, page 330.) 
 1. Here we have 0=1, i=l, c=:l, etc.; whence by formula (J^), 
 
 Hence, 
 
 ^=1, ^=-1 C=l, D=-\, E=\, etc. 
 
 *=y— y'+y'— y*+y'— — » ^'i** 
 
 (328-330) 
 
. . etc. 
 
 5 
 
 REVERSION OF SERIES. 229: 
 
 2. a=l, 6=3, c=5, etc. 
 By formula (T), 
 
 A=zl, B=-3, (7=13, i)=— 67, -^=381, etc. 
 Whence, 
 
 «=y-3i^' + 13y'-67/ + 381/-...., Ans. 
 
 3. a=l, 6=-^, c=i, rf=-i, c=i, etc. 
 By formula {F)y 
 
 A=l, B=-, C=—, i>=— i_, B=z , 
 
 ' 2' 2-3' 2-3-4* 2-3-4-5' 
 
 Whence, 
 
 4. a=l, 6= — 1, c=l, c/=— 1, e=l, etc. 
 By formula (G), 
 
 A = l, B=l, C=2, i>=5, J^=14, etc. 
 whence, 
 
 ar=y + y' + 2y* + 5/ + 14y'+....,^n«. 
 
 5. a=2, 6=3, c=4, rf=5, « = 6, etc. 
 
 By formula (6^), 
 
 A=i, B=-j\, C=V/3, i>=-TV/4,etc. 
 Whence, 
 
 y 3/19/ 152/ 
 ''-2""l6"^T28 ""1024 + • • • " ^''^• 
 
 6. a=2, 6=4, <'=6, c?=8, c=10, etc. 
 By formula (F), 
 
 A=^, B=z-i, C=f, D=-h ^=Hf etc. 
 "whence, 
 
 _x x* 5x' 7a;* 21a;" 
 ^""2""^ T" "s" le" 
 (830) 
 
230 SEKIES. 
 
 (384, page 331.) 
 In the following examples, let s be the sum of the series. 
 
 1. By formula (F), 
 
 1 4,2-8, 4-16 , 8-32 , 
 6 25 ^125 ^625 ^3125 
 
 Substituting the value of «=}, 
 
 ^=1(1)*+ (!r+2(i)'+4-.(i)'+8 •«)■"+.... 
 
 This is a geometrical series in which r=2(f)'; and by 361, 
 
 x= ^ ^^^; ,, =0.11764706 + , Ans, 
 1-2(1)' 
 
 2. By formula (F)y we have 
 
 f ~3«*' ""^To-* "■alVo'" 
 
 Substituting the value of *=|, we find 
 
 « = -f 0.500000 
 
 ~|s»= -0.041667 
 
 — ^V«'=-0'003472 
 
 . -^|^«*= -0.000231 
 
 __ji^_5»=, -0.000010 
 
 Therefore, ~ ar= 4- 0.454620, ^>w. 
 
 3. By formula (G), we have 
 
 Hence, 
 
 « =+0.200000 
 + i5»= +0.001333 
 
 Therefore, ' * «r= 0.201369, Ans. 
 
 (331) 
 
RECURRING SERIES. 
 4. By formula (F), 
 
 x=s 4- !«' + ^V*' + 4^8** + 3 if f o«* + 
 Hence, 
 
 s =+0.250000 
 
 |«»= +0.023438 
 
 ^^«'= +0.001139 
 
 j|i^5*= +0.000074 
 
 9lffo«'= +0.000004 
 
 Therefore, 
 
 ar= .274665, Am. 
 
 Mi 
 
 RECURRING SERIES. 
 
 (391, page 335.) 
 1. We have by formula (P), 
 
 whence, 
 
 By formula (g), 
 
 m + Sn=4, 
 8m + 4n=1 ; 
 m=l, 
 n=l. 
 
 1+3^;— ar l + 2a; 
 
 o=- =- , Am, 
 
 l—x—x* 1— ar— ar'* 
 
 2. By formula (P), we have 
 
 6m + 12»=48; 
 m— 6, 
 n= 1. 
 
 whence, 
 
 By formula (§), 
 
 l + 6.r— a; _ l+5a; 
 l_a:-6a;'~"l-a;-.6a; 
 
 ^ , -^w*. 
 
 (331—385^ 
 
23*2 SERIES. 
 
 3. By formula (P), we have 
 
 m + 2n=— 5, 
 
 2;/i — 5n=4-26; 
 
 whence, w= + 3, 
 
 »=— 4. 
 Hence by (^), 
 
 l4-2ar-f-4ar_ l4-6a; 
 
 4. By formula {T)t wo have 
 
 m + 4n + 3r=— 2, 
 
 4wi + 3» — 2r=+ 4, 
 
 3m— 2» + 4r= + l7; 
 
 whence, wi= + 3, 
 
 n=— 2, 
 
 r=+ 1. 
 Hence by ( F), 
 
 l-jr + 2ar'-3x' l-ar4-2ar'-3«' * 
 
 6. By formula (P), we have 
 
 wi4-3w= 5, 
 
 3m + 5/1= 7; 
 
 whence, m= — 1, 
 
 n=+2. 
 
 By (C), 
 
 „ l4-3i;--2a; 1+ir 
 
 6. By formula (P), we have 
 
 wi4-w= 5, . 
 m + 5n=13 ; 
 whence, wi= 3, 
 
 n= 2. 
 
 By (C), 
 
 ^_ l+g— 2g _ 1—3? . 
 
 ^"-l^2x-dx'''l--2x^Zx** "^""^ 
 (335) 
 
RECURRING SERIES. 233 
 
 7. By formula (T), we have 
 
 m + 4n + 6r= 11, 
 
 4m + 6n + llr= 28, 
 
 6m+llw + 28r= 63; 
 
 whence, m=-f-3, 
 
 n=-l, 
 r=+2. 
 
 By(r). 
 
 l-f4a; + 6a;' — (l-f 4a:)-2a;+a;* 
 1 — 2a: + a;'— 3a;' 
 
 _l4-2a;-2j:' + ig '_(l+a?)'-2.g* 
 l_2x + a;'-3a;»~(r^a:)'-3«' ' 
 
 8. 
 lei 
 
 By 
 
 ice, 
 
 formula 
 
 (n 
 
 we 
 
 have 
 2 
 
 7n_ 
 
 ■ 2 ~ 
 
 m= 
 
 n= 
 
 1 
 2' 
 
 10; 
 
 3, 
 2. 
 
 By (§) we may now sum the series, but we must notice that 
 equations (2), (389), will now have the form, 
 
 I>=mBx* + nCx^^ etc.; 
 or that we must change a; in (Q) to x'. Hence, 
 
 -+X*'-X' 
 
 „_ 2 _ X 
 
 ^-l-2x'^dx-2-4x^-6x*' "***'• 
 
 (335) 
 
234 SEBIES. 
 
 DIFFERENTIAL METHOD. 
 (395, page 339.) 
 
 1. a = l, rf,=3, c?3=l, d^ = 0. 
 By formula (A), 
 
 8*7 
 r, = l+3*8H — — = 53, ^n«. 
 
 2. a = l, rf,=3, rfa=3, rf^^l, rf^=0. 
 By formula (A)^ 
 
 3. a=l, c?,=5, rfj=10, ^3 = 10, c?^=5, dj.=0. 
 
 Hence, ^s^'^^l, and ^^ = 1231, Atis. 
 
 4. a=l, rf,=7, ^3=12, ^3 = 6, rf,=0. 
 
 r,,=l + 7'19 + ^^-12 + ^^'\^'^^ -6=:8000,^y... 
 2 2 3 
 
 5. a=l, rf,=2, ^3=1, c?3=0. 
 
 2 + 4n— 4+n' — 3rH-2 n(n4-l) . 
 
 = = -^ — - , Ans, 
 
 2 2 * , 
 
 (339) 
 
DIFFERENTIAL METHOD. 235 
 
 a=l, rf,=.3, d^=S, c?3 = l, d^=zO. 
 
 m , «/ ,^ (n — l)(n-2) (n-l)(n — 2)(n^3) 
 T,= l+3(/i~l) + '^ ^^ -^-3 + ^ 2^"^ 
 
 6 
 n* + 3n' + 2n_n(n + l)(n + 2) 
 
 6 6 
 
 , Ans. 
 
 7. a = l, e?,=4, «?a=6, d^=4, d^=l. 
 
 ^ w ,v (n—l)(n—2) ^ (w— l)(w-2)(n~3) , 
 7;=l+4(n-l) + ^ ^ ^'6 + ^ 2-3 "^^'*" 
 
 { n-l){n-2){n-S)(n-4) 
 2-3-4 
 
 __ _ n*4-6n' + ll»'4-6n n(n4-3)(n'4-3»4-2) 
 Or, T^= 24 = 24 
 
 n{n + S)(n + 2){n + l) 
 = 24 ' ^ • 
 
 8. a=l, c?,=2, rf2=l, <f3=0. 
 
 By formula (i?), (394), 
 
 « «^ 20 • 19 ^ 20 • 19 • 18 ,^^^ . 
 
 5=20 + — - — -2+ — — =1540, Ans. 
 
 2 2 * u 
 
 9 a=l, rf,=4, ^3=5, ^3 = 2, rf^=0. 
 
 ^ ,^ 12-11 , 12 • 11 • 10 ^12 -11 -10 -9 „ 
 5^12 + __. 44— 2T3— • 5 4—yTyT-r- • 2; 
 
 or 5=2366, Ans. 
 
 10. a=l, rf,=3, ^3=6, d^ = 9, d^=0. 
 
 10-9 10 •9-8 ^10-9-8-7 ^ 
 
 5=10 + ^- . 3 + -2T3- • 6 + -2T3T-r- • 9; 
 
 or 5=2755, Ans. 
 
 (339) 
 
236 SERIES. 
 
 11. a=2, <f,=4, ^3=2, ^3=0. 
 
 r* « . w(n — 1) , n(n-l)(n— 2) ^ 
 ^=2n4--^--2— ^ • A+-^—^ ^ • 2 ; or 
 
 ^^ n' + 3;i' + 2^ 4n4-2)(n4-l) ^^^ 
 3 3 * ' 
 
 12. a = 6, </,=18, rfj,=18, <f3 = 6. 
 
 o ^ »(«-l) ,o n(n-l)(n-2) , n(n-l)(n-2)(n-3) ^ 
 
 ^_n' + 6n*4-lln' + 6» n(n + 3)(n + 2){n + l) 
 
 ^— ^1^ = , Ans. 
 
 13. The series in examples 13, 14, and 15 may be summed by 
 formula (B)^ (389) ; but the following method by indeterminate 
 coefficients is easier. 
 
 Assume 
 
 l' + 2'+3'+ +n* = A + Bn+Cn*-\-Dn'i-JSn*+ (l) 
 
 Change n into n 4- 1 ; then 
 
 l« + 2' + 3«+....+n' + (n + l)«=^ + ^(n + l)+C(» + l)' 
 
 + i)(n + l)' + ^(n + l)*+.... (2) 
 
 By. subtraction, 
 
 n'+2n + l=B+C{2n-{-l) + I>(Sn* + Sn + l) 
 
 + E(4n' + 6n^ + 4n + l) + 
 
 or, by arranging terms with reference to the powers of n, 
 
 »' + 2» + l=4^»' + (3i> + 6^)n' + (2C7+3i) + 4^)n 
 
 + (5+C7+i> + ^) + .... (3) 
 
 Now by (368, III.), we have 
 
 iF=0, or JS:=0; 
 and the same value for all coefficients beyond JS, 
 
 Equating coefficients of like powers of n in (3), omitting the 
 terms containing JE', we have the following equations of condi- 
 tion: ^ " 
 
 (339-340) 
 
DIFFERENTIAL METHOD. 237 
 
 3i>=l, whence i>=|; 
 
 2(7+3i> = 2, " (7=1; 
 
 i?+C7+i>=l, " jB=i. 
 
 And by substitution in (l), 
 
 l. + 2' + 3'+....+„'=^ + ^ + |' + J, 
 To determine A, put n=l ; then -4=0, and we have 
 
 ^=6 + 2 +F= 6 = 6 ■ ' ^"'- 
 
 In Example 14, we shall have ^=0, and the same for every 
 coeflScient beyond F. The equations of condition arc 
 
 4ii'=:l, whence -fi^=i ; 
 
 3Z> + 6^=3, " D= 
 
 a 1 
 2C'4-3i> + 4jE'=3, " = 1'^ 
 
 Hence, 
 
 ^ n* n' n' (^'-fn)' . 
 ^=T+2+4=^-T^»^"" 
 
 In Example 15, we have G=0, and the same for every coeflScient 
 beyond G. The equations of condition are 
 
 5F = 1, whence /"=}, 
 
 4^+10/^=4, '' ^=1, 
 
 8i) + 6J^+10i^ = 6, " i>=|, 
 
 2^(7+ 32) + 4^+ 5i^=4, " (7 = 0, 
 
 ^+(7+i> + ^+/' = l, " J?=-^V. 
 
 Hence, . 8=.- + - + - --, Ans. 
 
 In this solution the symmetry of the equations that determine 
 Fj F, Dy «fec., should be noticed. Taking the numbers in vertical 
 columns, they are the binomial coeflScients. We can easily extend 
 this method to the fifth and sixth powers ; and so on. 
 
 (840) 
 
238 
 
 SERIES. 
 
 16. Performing the multiplications indicated, we shall have the 
 two series, 
 
 m(l+2 + 3+ +n) 
 
 and, l' + 2' + 3'+....+7i'. 
 
 Hence, 
 
 or, 
 
 -, «(» + !) n(n + l)(2n+l)^ 
 
 INTERPOLATION. 
 
 (397, page 341.) 
 
 For the first, second, and third examples, we have 
 a=2.V58924, rf, = +.043115, rf^ = -.001287, ^3= +.000091. 
 
 For the fourth, fifth, and sixth, 
 a=2.802039, rf, = +.041828, e?^ =.001196, (/,= +.000083. 
 
 By substitution in the formula, we obtain the following results ; 
 
 1. 1st term, +2.758924 2. 1st term, +2.758924 
 
 2d " + 14012 2d " + 37726 
 
 3d " + 141 3d " + 70 
 
 4th " + 6 4th "+ 2 
 
 2.773083, Ant. 
 
 2.796722, An\ 
 
 Isttcrm, +2.758924 
 2d " + 19695 
 3d " + 160 
 
 4th " + 6 
 
 2.778785, Ans, 
 
 1st term, +2.802039 
 2d " + 10457 
 3d " + 112 
 
 4th « + 6 
 
 2.812613, Arts, 
 
 5. Isttei-m, +2.802039 
 2d " + 28611 
 3d " + 129 
 
 4th « + 4 
 
 2.820783, Ans. 
 
 6. 1st term, +2.802039 
 2d " + 31371 
 3d " + 112 
 
 4th " + 3 
 
 (340-341) 
 
 2.833525, Ans. 
 
INTERPOLATION. 
 
 23Sr 
 
 (398, page 342.) 
 
 FUNCTION. 
 
 d. 
 
 
 d. 
 
 d. 
 
 66° 6' 38" 
 
 
 
 
 
 72 24 5 
 
 + 6° 17' 
 
 27" 
 
 
 
 Y8 34 48 
 
 + 6 10 
 
 43 
 
 -6' 44" 
 
 ■ 
 
 84 39 4 
 
 + 6 4 
 
 16 
 
 -6 27 
 
 + 17" 
 
 90 37 18 
 
 + 5 68 
 
 14 
 
 -6 2 
 
 + 25 
 
 96 29 57 
 
 + 5 62 
 
 39 
 
 — 5 35 
 
 + 27 
 
 Note. — In' forming the differences we may use as a check to guard against 
 numerical errors, the rule, " The sum of the differences in any column, plus the 
 first term of the preceding column, is equal to the last term of the preceding 
 column." 
 
 Observe that for the 1st, 2d, and 3d examples, 
 . a=66°6'38'; rf, = + 6** 17' 27'; d^ = '-6'W\ d^=+ir\ 
 For the 4th, 5th, and 6th, 
 
 a = 72°24'5", rf, = +6® 10'43", c?3-6'27", fl?3 = + 25". 
 For the 7th, 8th, and 9th, 
 
 a = 78°34'48", rf, = +6° 4' 16", d^ = -^Q' 2'\ (£3 = + 27". 
 Again, we have 
 
 for examples 1st, 4th, and 7th, **=? j 
 
 " " 2d, 5th, and 8th, n=^ ; 
 
 •* " 3d, 6th, and 9th, n=^. 
 
 1. 1st terra, +66° 6' 38' 
 
 2. 1st term, +66° 6' 38" 
 
 2d " 
 
 + 1 34 22 
 
 2d " 
 
 + 38 43.5 
 
 3d " 
 
 + 38 . 
 
 3d " 
 
 + 50.5 
 
 4th " 
 
 + 1 
 
 4th " 
 
 + 1 
 
 
 67° 41' 39", ^n5. 
 
 - 69° 16' 13", 
 
 
 (342) 
 
 
 
240 SERIES. 
 
 3. 1st term, +66* 6' 38" 4. 1st term, +720 24' 6" 
 
 2d "+ 4 43 5 2d " + 1 32 41 
 
 3d " + 38 3d " + 36 
 
 4th " + 1 4th " + 1 
 
 10^*50' 22", Ans, 73^67' 23", Ans, 
 
 6. 1st term, +72° 24' 5" 6. 1st term, +72° 24' 5' 
 
 2d " +35 21.5 2d " + 4 38 2 
 
 3d " + 48.4 3d " + 36 
 
 4th " + 1 4th « + 1 
 
 75** 30' 16", Ans. 77° 2' 44", Ans. 
 
 7. 1st term, +78° 34' 48" 8. 1st term, +78° 34' 48" 
 2d "+ 1 31 4 2d " + 3 2 8 
 
 3d " + 34 3d « + 45 
 
 4th " + 1 4th " + 2 
 
 80° 6'27",^rw. 81°37' 43",^iW. 
 
 9. 1st term, +78° 34' 48" 
 2d " +4 33 12 
 3d " + 34 
 
 4th " + 1 
 
 83° 8' 85", Ans. 
 
 LOGARITHMS. 
 (416, page 355.) 
 
 2. log. 104=2.017033 3. log. 73 = 1.863323 
 
 " 5=0.698970 " 2 = 0.301030 
 
 2.716003, Ans. 2.164353, Ans, 
 
 (842-356) 
 
LOGARITHMS. 
 
 241 
 
 4. log. 50 = 1.698970 
 
 « 29 = 1.462398 
 
 5. log. .53 = 
 3 = 
 
 3.161368, Ans. 
 
 -1.724276 
 0.477121 
 
 0.201397, Ans. 
 
 6. log. 1017 = 3.007321 
 " 2=0.301030 
 
 7. log. 10.91 = 1.037825 
 " 7=0.845098 
 
 
 
 
 3.308351, Ans, 
 
 
 
 1.882923 
 
 
 8. 
 
 log. 
 u 
 
 .01005: 
 2: 
 
 = -2.002166 
 = 0.301030 
 
 9. log. 
 ns. 
 
 .91 = 
 .42 = 
 
 -1.959041 
 — 1.623249 
 
 
 
 2.303196, A^ 
 
 — 1.582290, 
 
 Ans, 
 
 10 
 
 1, 
 
 
 log. 103 = 
 « 15 = 
 " 11 = 
 
 = 2.012837 
 rl. 176091 
 .1.041393 
 
 
 
 
 
 4.230321 
 
 
 (41T, page 356.) 
 
 log. 10720 = 
 405 X 0.4 = 
 
 4.030195 
 162 
 
 
 4.030357, Ans. 
 
 log. 10.85 
 
 = 1.035430 
 
 400 X 0.39 
 
 = 156 
 
 
 1.035586, Ans. 
 
 log. 1021 
 
 = 3.009026 
 
 425 X 0.56 
 
 = 238 
 
 
 3.009264, Ans, 
 
 log. 5.6 
 
 = 0.748188 . 
 
 :. 101.5232 
 
 = 2.006565 
 
 3.754753, Ans, 
 (365-356) 
 
242 SERIES. 
 
 1. log. 3 = 0.477121 
 
 log. 1081.333 = 3.033960 
 
 3.511081, Ans. 
 
 8. log. 3.6 = 0.556303 
 log. 101.4601 = 2.006295 
 
 2.562598, Ans, 
 
 9. log. 1.3 = 0.113943 
 log. 101.977 = 2.008502 
 
 2.122445, Ans, 
 
 10. log. 5.6 = 0.748188 
 log. 101343.04 = 5.005794 
 
 5.754982, Ans. 
 
 11. log. 2.5 = 0.397940 
 log. 103.48 = 2.014856 
 
 2.412796, Ans, 
 
 12. log. 1.2 == 0.079181 
 log. 1.08 = 0.033424 
 
 0.112605, Ans. 
 
 13. log. 57 = 1.755875 
 log. 101.4737 = 2.006353 
 
 3.762228, Ans, 
 
 (356) 
 
LOGARITHMS. 243 
 
 . EXPONENTIAL EQUATIONS. 
 (418, page 357.) 
 
 1. Taking the logarithms, 
 X log. 7 = log. 8 
 
 locr. 8 0.C03090 , 
 -— ^— — —1.06862, Ans, 
 
 log. 7 0.845098 
 
 2. Taking the logarithms, 
 -(log. 5) = log. 30. 
 
 X 
 
 2 log. 5 1.397940 ^^,,„,, , 
 a^=i — ^ = 7-77^777777=0-946395, Ans. 
 log. 30 1.477121 ' 
 
 3. X log. a — 2 log. 6 + 3 log. c 
 
 2 loff. 6 + 3 logr. c ^ 
 
 •whence, ir= 2_j 2 — , ^w*. 
 
 loor. a 
 
 4. Clearing of fractions and transposing, we have 
 
 ah' :=.din -\- c 
 log. a + ar log. 6 = log. (c?w + c) 
 
 loff. (c?m + c) — log. a . 
 
 whence, x^-^-^ — -, '- ^— , Ant, 
 
 log. 6 
 
 5. log. m + -{log. a) = log. 6, 
 
 whence, x—-, j^^ , Ans, 
 
 log. 6— log. m 
 
 6. Adding and subtracting the equations, 
 
 a'=:c + (/, and6''=c— 6?; 
 
 log. (<: + c?) log. (c— c?) 
 
 whence, a; = -V^ -, y= ^, ^ , — ^ , -/^ w5. 
 
 log. « log. 6 
 
 (357-368) 
 
244 
 
 PROPERTIES OF EQUATIONS. 
 
 V. Taking the logarithms, 
 
 8. 
 
 whence, 
 
 9. 
 
 whence, 
 
 _log. 729_log. 3'_6 log. 3_ 
 log. 3 log. 3 log. 3 ' 
 
 (log. 216) = log. 12, 
 
 _3 log. 216_3 log. 6'_9 log. 6 . 
 '^~ log. 12 1^12"~log.l2 ' ' 
 
 (log. 616)=log. 12; 
 
 __3 log. 516 _ 3 log. 43 + 3 log. 12 
 
 log. 12 
 
 log. 12 
 
 3 log. 43 „ , 
 - »— +3,^n,. 
 
 log. 
 
 10. Taking the logarithms, 
 
 X log. 6 = 6 log. 24 + ^ log. 17 — log. 71 
 
 18log.24 + loor. 17 — 3loff.7l , 
 
 whence, x= — r^ — , Am. 
 
 ^ 3 log. 6 ' 
 
 PROPERTIES OF EQUATIONS. 
 (428, page 366.) 
 
 1. Factors, < ^ ^ 
 Product, a;'— X— = 0, Ans. 
 
 2. By (4^3), the 1st term is ar* ; 
 
 " (434, 1) " 2d " 6a?»; 
 " (434, 2) " 3d " 2x; 
 " (434, 5) " 4th " -8. 
 Hence, a:' + 5i;' + 2.c— 8 = 0, -4n*. 
 
 (358-366) 
 
PROPERTIES IN EQUATIONS. 240 
 
 Factors, 
 
 ar— 3=0 
 
 ic + 2=0 
 a;+l=0 
 x + 5 = 
 
 Product, ar* — 5x' — Tar' + 29^ + 30=0, ^n«. 
 
 Factors, 
 
 ar-(l+f/--5)=:0 (1) 
 
 ;r_(l_|/_5)=0 (2) 
 
 X-V5 =0 (3) 
 
 x + V^S =0 (4) 
 
 Product of (1) and (2), x*—2x =6 (5) 
 
 (3) - (4), x'-5 =0 (6) 
 
 " (5) " (6), x*-22r» + a;' + 10a;-30=0, ^ns. 
 
 _ 5. By (423,) The 1st term is «'; 
 
 « (4a4, 1), " 2d " -4x*; 
 
 ** (4^4, 2), " 8<i " Ox'] 
 
 " (404, 3), " 4th " 22a;'; 
 
 « (424, 4), " 5th " -26a:; 
 
 " (4«4, 5), " 6th » -42. 
 
 Hence, the required equation is 
 
 a;> — 4a:* + 22a:' — 25jr— 42 = 0, Ans, 
 
 Or thus : 
 
 ar+l=0 ' (\/ 
 
 ar + 2=0 (2) 
 
 Factors, } ^3=0 (3) 
 
 ;r_(2+^-3)=0 (4) 
 
 x-{2'-i^s)=0 (5) 
 
 Product of (4) and (5), a:' -4a; + 7=0. 
 
 Hence, 
 
 (x'-4ar + 7)(ar + l)(.r + 2)(a;-3)=a;»-4.r*±0a;' + 22a;'-25ar-42=0, 
 
 Ans, 
 
 (366-367) 
 
246 PROPERTIES IN EQUATIONS. 
 
 6. {x'^5x^ + ldx—2l)-r-(x-3)=x^-2x-{-1=0j Ans, 
 
 1, (x*-\-2x'—34x* + 12x-{-35)-^(x-\-1)=x'-5x*-\-x + 5=z0j Ans, 
 
 8. (x — 2){x-\-3)=zx* + x—6',hencey 
 
 (x*-3x'—4x^-\-30x-3Q)-T-{x^+x-Q)=x*'-4x+6=0, 
 
 the depressed equation, which may be solved thus : 
 x'—4x=-6, 
 
 ic=2+*^^, or 2—^^, 
 
 COMMENSURABLE ROOTS. 
 
 (432, page 373.) 
 2. 
 Divisors, 6, 3, 2, 1, —1, —2, —3, —6. 
 
 -6 
 Quotients, —1, —2, —3, —6, 6, 3, 2, 1. 
 
 Add 11 
 
 10, 9, 8, 5, 17, 14, 13, 12. 
 
 2d quotients, 3, 4, 5, —17, —7, —2. 
 
 Add -6 
 
 — 3, -2, -1, —23, —13, -8. 
 
 3d quotients, —1, —1, —1, 23, 
 
 There are three final quotients equal to — 1 ; and the correspond- 
 ing divisors are 3, 2, and 1. Hence the given equation has three 
 commensurable roots, 1, 2, and 3, Ans, 
 
 (367-373) 
 
COMMENSURABLE ROOTS. 
 
 247 
 
 O P j^ 
 
 H 
 
 H4» »^ 
 
 i ^ 
 i I 
 
 § - 
 
 S" 
 
 89 
 
 J* 
 
 p 
 
 ^ 
 
 in- 
 
 
 
 
 e^ 
 
 
 
 1 
 
 
 1 
 
 O 
 
 J-» 
 
 
 
 o 
 
 ■ 
 
 o 
 
 1 
 
 
 JO 
 
 (ft 
 
 
 
 
 I 
 
 
 
 1 
 
 p 
 
 1 
 
 CO 
 
 aq 
 
 -• 
 
 Bi 
 
 e 
 
 <' 
 
 p 
 
 f 
 
 e- 
 
 § 
 
 g 
 
 ^ 
 
 «r»- 
 
 S9 
 
 CD 
 
 o 
 
 rt) 
 
 to 
 
 ►o 
 
 " 
 
 P 
 
 
 P 
 
 1 
 
 
 
 — '. 
 
 
 O* 
 
 >• 
 
 P 
 
 
 »-!• 
 
 1 
 
 CO 
 
 1 
 
 O 
 
 JO 
 
 P^ 
 
 
 
 so 
 
 *< 
 
 p 
 
 o 
 
 a- 
 
 -^ 
 
 
 1^ 
 
 1 
 
 CO 
 
 p 
 o 
 
 EJ". 
 O* 
 
 P 
 
 |> CO 
 Cl, ^ 
 
 p 
 o 
 
 P 
 o 
 
 o 
 
 > c> 
 
 I I 
 
 ^^ to 
 
 + I I 
 
 (—1 
 rfi. J35 JO 
 
 I I 
 
 CO «o 
 
 03 CO 
 
 to jf^ 
 
 H-l CO 
 
 I I 
 
 h-t CO CX) 
 
 p 
 
 g 
 
 55' 
 o 
 
 1 
 
 1 1 
 
 OS J-" 
 
 00 
 
 1 
 
 JO 
 
 1 
 
 1 
 
 1 
 
 ►o 
 
 1 
 
 1 
 
 lO 
 JO 
 
 1 
 
 J3S 
 
 1 
 
 to 
 
 CD 
 
 1 
 JO 
 
 1 
 
 I—" 
 
 to 
 
 1 
 
 t— • 
 JD 
 
 J35> 
 
 1 
 JO 
 
 4^ 
 
 1 
 
 J^ 
 
 1 
 
 to 
 
 en 
 
 t—i 
 
 (373) 
 
248 PROPERTIES OF EQUATIONS, 
 
 4. 
 
 Divisors, 21, 7, 3, 1, —1, —3, —7, —21. 
 21 
 
 Quotients, 1, 3, 7, 21, —21, -7, -3, — 1. 
 Add -16 
 
 — 15, —13, -9, 5, -37, -23, —19, —17. 
 
 2d quotients, —3, 5, 37. 
 
 Add -6 
 
 — 9, -1, 31, 
 3d quotients, —3, —1, —31. 
 
 Add ±0 
 
 —3, —1, -31. 
 4th quotients, —1, —1, 31. 
 
 The final quotients show that the given equation has two com* 
 mensurable roots, 3, and 1, Ant. 
 
 6. 
 
 Divisors, 10, 5, 2, 1, -1, -2, -5, -10. 
 
 — 10 
 
 Quotients, —1, —2, -5,-10, 10, 5, 2, 1. 
 
 Add 2 
 
 1, 0, -3, -8, 12, 7, 4, 3. 
 
 2d quotients, 0, —8,-12. 
 
 Add 5 
 
 5, -3, -7. 
 
 3d quotients, 1, — 3, 7. 
 
 Add -6 
 
 -5, -9, 1. 
 
 4th quotients, —1, —9, — 1. 
 
 (373) 
 
EQUAL ROOTS. 249 
 
 The final quotients show that 5 and —1 are the commensurable 
 roots of the given equation. Hence, 
 
 x-b=Q, (1) 
 
 ir + l=0. (2) 
 
 Dividing the given equation by the product of (1) and (2), we 
 have 
 
 a;'— 22'+l = -l, 
 Therefore the four roots are 5, —1, 1 ■\-V~^\^ and 1— i^^, Arts, 
 
 EQUAL ROOTS. 
 
 (435, page 379.) 
 2. We have given 
 
 The first derived polynomial of the first member is 
 
 X,=5a:* + 8x' — 33a:'-16a; + 20. 
 
 The greatest common divisor of this, and the first member of the 
 given equation found by (1045), is 
 
 D=x'—x-1 = {x-2){x + \), 
 
 Therefore x—2 is twice a root of the equation, also x=—\ is 
 twice a root ; and the equation has two roots, each equal to 2 ; and 
 two roots, each equal to —1. 
 
 Dividing a;' + 2.c*-lU'-8j:'' + 20.r + 16=0, by (:r-2)'(« + l)'=:0, 
 we have ar + 4 = 0, and x— — i. Hence the roots of the given equa- 
 tion are, 2, 2, —1, —1, —4. 
 
 (377-379) 
 
250 PROPERTIES OF EQUATIONS. 
 
 3. We have given 
 
 whence, 
 
 X, =5x*-8j;» + 9j;'— 14a;+8=:0. 
 
 By (105) we obtain 
 
 D=x^—2x-{-l = {x-iy=0. 
 
 Therefore, since x=l is twice a root of i>, by (4:3«5, II) it is 
 three times a root of the given equation X. 
 
 4. X = x* — 2x'—Ux* + l2x + S6=0; 
 X^=4x'-6x^-22x + l2 = 0. 
 
 Whence, by (105), 
 
 D=x^-x—6 = {x—S){x + 2)=0, 
 
 Hence x=3 is twice a root of X, and also x= — 2 ; and as the 
 equation is of the fourth degree, it has only four roots. Therefore, 
 
 3, 3, —2, —2, Ans. 
 
 TRANSFORMATIONS. 
 (437, page 384.) 
 
 5. Put x=y-\-2] then x'=2. Whence, 
 X' =(2)*-4(2)»-8(2) + 32, 
 X,= 4(2)'-12(2)'-8, 
 
 X\_ 3-4(2)' 2-12(2) 
 
 2 ~ 2 2 ' 
 
 X3_ 2-3-4(2) 2-12 
 2-3~ 2-3 2-3 ' 
 
 X, _ 2-3-4 
 2-3-4~' 2-3-4' " 2-3-4 
 
 Therefore the transformed equation must be 
 
 y + 4/ — 23y=0, Am, 
 
 (379-384) 
 
 or 
 
 X = 
 
 0; 
 
 or 
 
 X, = - 
 
 -24; 
 
 or 
 
 2 
 
 0; 
 
 or 
 
 2-3 
 
 4; 
 
 or 
 
 ^4_ 
 
 1. 
 
DETACHED COEFFICIENTS. 251 
 
 6. Putar=y— 3; thcna;' = — 3. Whence 
 X' =:(-3)* + 16(-3)=' + 99(-3)' + 228(-3) 
 
 + 144, or X' = 0; 
 
 -X",=4(-3)' + 48(-3)' + 198(-3) + 228, or X',=:-42; 
 
 -X-,_ 3-4(-3r 2.48(-3) 198 ^^ X',_ 
 
 X^3_ 2-3-4(- 3) 2-48 X'3 _ 
 
 2-3 2-3 ■^2-3* ^^ 2.3~ ^' 
 
 X\ 2-3-4 X'. 
 
 2-3-4"2-3-4' ^^2.3.4- ■^' 
 
 Therefore, the transformed equation must be 
 
 y* + 4y' + 9y' — 42y = 0, Ans, 
 
 v. By (437), put a:=y + |, then 
 
 x'-. 
 
 =2. Whence, 
 
 X' =(2)*-8(2)'4-(2)' + 82(2)- 
 
 60 
 
 = 60; 
 
 X\=4(2)'-24(2)»+2(2) + 82 
 
 
 = 22; 
 
 X\ 3-4(2)' 2-24(2) 2 
 2 2 2 "'"2 
 
 
 = -23; 
 
 X'3 2-3-4(2) 2-24 
 2-3- 2-3 2-3 
 
 
 = 0; 
 
 X', 2-3-4 
 
 
 = 1. 
 
 2.^.4 2-3-4 
 
 Therefore, the transformed equation must be 
 
 y* — 23y' + 22y + 60 = 0, Arts. 
 
 MULTIPLICATION AND DIVISION BY DETACHED 
 COEFFICIENTS. 
 
 ( 
 
 44O5 
 
 page 390. ) 
 
 
 3-2-1 
 
 
 5. 
 
 3-5-10 
 
 4 + 2 
 
 
 
 2-4 
 
 12 — 8-4 
 
 
 
 6—10-20 
 
 _,-6-4-2 
 
 
 
 -12 + 20 + 40 
 
 12-2-8-2 
 
 
 
 6 — 22d= + 40 
 
 
 (38£ 
 
 1-391 ) 
 
 
252 
 
 PROPERTIES 
 
 OF 
 
 EQUATIONS. 
 
 6. 
 
 1+1+1 
 
 1. 
 
 l_44-5-2 
 
 
 1-1 + 1 
 
 
 1+4-3 
 
 
 1 + 1 + 1 
 
 
 1_4_|. 5- 2 
 
 
 -1-1-1 
 
 
 + 4-16 + 20-8 
 
 
 +1+1+1 
 
 
 - 3 + 12-15+ 6 
 
 
 ld=0 + l±0 + l 
 
 
 l±0-14 + 30-23 + 16 
 
 SYNTHETIC DIVISION. 
 (44^5 page 394.) 
 
 -1+2—2+2 
 
 1 — 2 + 2 — 2 + 2, etc. 
 Hence the quotient is 1 — 2^ + 2a;'— 2z' -|-2ar*— , <fec., Ans, 
 
 -1+1-1+1 
 
 1 — 1 + 1 — 1 + 1, &c. 
 Hence the quotient is 1 —x + x* —x* + x* — , <fec., Ans, 
 
 3. 1-5 + 10-10 + 5-1 |2-1 
 
 + 2— 6+ 6-2 
 
 - 1+ 3-2+1 
 
 1 — 3+ 3-100 
 Henee the quotient is a' — Sa^x -\-Sax*— x*, Ans, 
 
 4. 1-5 + 15-24 + 27-13+5 | 2-4 + 2-l 
 
 + 2— 6 + 10 
 
 — 4 + 12-20 
 
 + 2— 6 + 10 
 — 1+ 3-5 
 
 1-3+ 5 0. 
 Hence the quotient is x^—3x-\-5, Am. 
 
 (391-394) 
 
CARDAN'S RULE. 253 
 
 6. Idb0d=0db0=b0±0±0-1 |1 
 
 1 + 1 + 1 + 1 + 1 + 14-1 
 
 1 + 1 + 1 + 1 + 1 + 1 + 1+0. 
 Hence the quotient is x* + x^y-^-x^y* + x^y* + x*y* 4-ary' + y*, Am, 
 
 CARDAN'S RULE FOR CUBIC EQUATIONS. 
 (449, page 404.) 
 
 4. We have 3/)=— 6, or 7?=— 2; 
 
 2q=z 6, or q=z 3; 
 
 fY+F=^9-8=-l. 
 Whence by formula (^), 
 
 a;=(3 + l)* + (3-l)^=V4 + V/2=2.8473+,^«*. 
 
 6. Here, 3j9=9, or ;)=3; 
 
 22'=6, or 5' = 3; 
 
 ^7+?= *^^ + 27=f^36= ±6, 
 Whence by formula {A\ 
 
 a;=(3 + 6)' + (3--6)^=V9 + ^^=.63783+, ^n«. 
 
 6. To transform this equation into another deficient of its second 
 term, according to (437), put a;=y — 2, and we shall have for the 
 transformed equation, 
 
 y>_25y + 66 = 0. 
 
 Hence, in applying Cardan's rule, 
 
 3;)= -25, or p=-%^', 
 25-=— 66, or 5^=— 33; 
 f^«+P=V^1089— i-VV^==±22.689737 + . 
 (394-404) 
 
254 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 Whence by formula (^), 
 
 y=( — 33 + 22. 689737)^ + (-33-22. 589737)^ 
 = — 2.18-3.82=-6. 
 Therefore, x= — 6 — 2=z — 8. 
 
 Dividing the given equation by (a? + 8), we have for the depressed 
 equation, 
 
 ar' — 2a;H-3=0; 
 
 whence, a;=ldbV^— 2. 
 
 Hence, the three roots are 
 
 — 8, l+»^-2, and l-f^-2, Ans, 
 
 LIMITS OF REAL HOOTS. 
 (453, page 408.) 
 
 2. Here, n=2, and /*=25 ; hence, 
 
 VP + l=^^ + l=6yAn8. 
 
 3. Supplying the deficient term, we have 
 
 x* + 0x* — 5x^—9x-\-10 = 0. 
 Therefore, 71 = 2, and P — 9; hence, 
 
 VP + l=i^9-hl=iy Ans. 
 
 4. Here n=3, and P = 8; hence, 
 
 VP + l=V8 + l = d, Ans, 
 
 (454, page 408). 
 
 1. Changing the signs of the alternate terms, we have 
 a-' + 3z' + 5^—7 = 0. 
 Therefore, n=3, and Pr=7; and 
 
 VP + 1 = \/7 + 1 == 3, in whole numbers, Ans. 
 (404-408) 
 
HORNER'S METHOD. 255 
 
 2. Completing the terms, we have 
 
 a:*=bOx' — 15x'— 10i; + 24 = 0. 
 Changing the signs of the alternate terms, the equation becomes 
 
 The term having zero for a coefficient may always be regarded as 
 positive. Hence 
 
 w=2, and F = t5. 
 
 VP + l='^^ + l=5, Ans. 
 
 3. With the alternate signs changed, the equation is 
 
 x' + Sx' + 2x'—2lx'—4x^ + lz=0. 
 Hence, w=3, and P=2l. 
 
 VP + 1 = ^^27 + 1 = 4, ^w«. 
 
 HORNER'S METHOD OF APPROXIMATION. 
 (464, page 420.) 
 
 3. We have given 
 
 A''=a:' + 2x' — 23ar— 70. 
 
 The first derived polynomial is 
 
 X,~3x'-^4x—2S. 
 We multiply X by 3, and divide the result by X, . Thus, 
 
 3a;' + 6x'— 69j;— 210 
 3a;'4-4j;' — 23r 
 
 3rc' + 4a:— 23 
 
 X, +1 
 
 2a;»-46^-210 
 a;«_23.r— 105 
 3x'— 49-^ — 315, new prepared dividend, 
 3a;' + 4.r— 23 
 
 — 73^-292. Hence, B=x + i, 
 (408-420) 
 
256 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 We now divide X, by B, Thus, 
 
 3x' + 12r 
 
 3a:-8 
 
 — 8j:-23 
 
 — 8r— 32 
 
 ■f 9. Hence, i2, = — 9. 
 
 Therefore, the functions are as follows : 
 
 X =z* + 2f— 23x^10, 
 X,=3x* + Ax—2S, 
 R =ar + 4, 
 i2, = -9. 
 
 Substituting in these functions, x= — qo and x=z + oo successively, 
 we have the following results, in respect to signs : 
 
 For 
 
 ( j:= + oo, -f- 
 
 ~ + — — » 2 variations. 
 + — , 1 variation. 
 
 Hence the given equation has 1 real root. 
 
 If we substitute x=0 in the functions, the signs will be 
 
 - - + -, 
 
 giving 2 variations. Hence the real root must lie between and 
 + oo ; or, it is positive. To ascertain its position, make a;=l, aj=2, 
 ar=3, etc., successively. 
 
 For' 
 
 Hence, the initial figure is 5. The decimal part is found by the 
 following operation : 
 
 x=\, signs. 
 
 — 
 
 — 
 
 + 
 
 — , 
 
 2 variations. 
 
 ar=2, " 
 
 — 
 
 — 
 
 + 
 
 — , 
 
 2 
 
 (( 
 
 X=:3, « 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 (( 
 
 ar=4, « 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 i( 
 
 x=b, " 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 u 
 
 ir=6, « 
 
 + 
 
 + 
 
 + 
 
 — , 
 
 1 variation. 
 
 (420, Ex.3) 
 
HORNER'S METHOD. 
 
 257 
 
 + 2 
 
 _5 
 
 7 
 5 
 
 12 
 5 
 
 (^) 17.0 
 .1 
 
 17.1 
 .1 
 
 17.2 
 .1 
 
 (2) 17.30 
 3 
 
 17.33 
 3 
 
 17.36 
 3 
 
 (3> 17.390 
 4 
 
 17.394 
 4 
 
 17.398 
 4 
 
 <*> 17.402 
 <»> 17.4 
 
 {«) 1 
 
 — 23 
 35 
 12 
 60 
 
 0> 72.00 
 1.71 
 
 
 73.71 
 
 (2) 
 
 75.4300 
 .5199 
 
 
 75.9499 
 .5208 
 
 (8) 
 
 76.470700 
 69576 
 
 
 72.540276 
 69592 
 
 (4) 
 
 76.609868 
 8701 
 
 
 76.618569 
 8701 
 
 (5) 
 
 76.62727 
 122 
 
 
 76.62849 
 122 
 
 (d) 
 
 76.6297 
 
 1 
 
 
 76.6298 
 1 
 
 (7> 76.630 
 
 <^> 76.63 
 
 (9> 76.6 
 
 (10) 77 
 
 (420, Ex. 3 ) 
 
 -70 | 5.1345787253 
 60 
 
 <^>— 10.000 
 7.371 
 
 C2) — 2.629000 
 2.278497 
 
 (8) — .350503000 
 .306161104 
 
 (*>— 44341896 
 38309285 
 
 (5) _ 603 26 11 
 5363994 
 
 («> — 668617 
 613038 
 
 (7)-55579 
 53641 
 
 (8) — 1938 
 1533 
 
 (9)_405 
 383 
 
 (10)_22 
 23 
 
'Z58 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 4. AVe have given 
 
 X=ar'— x' + 70x-300. 
 The first derived polynomial is 
 
 Multiplying X by 3 to avoid fractions in dividing, we proceed as 
 follows : 
 
 3j:' — 3x' + 210ar— 900\3z^ — 2x-\-l0 
 
 32r' — 2:r'+ 70a; 
 
 k -1 
 
 — a;' + 140i:— 900 
 
 --3z' + 420-c — 2700, prepared dividend. 
 
 — 3a:'+- 2x~ 70 
 
 + 418X— 2630. Hence, i2=:—209j: + 1315. 
 To avoid fractions in the next division, multiply X, by 209. 
 
 627a:'- 418^+ 14630 
 627ar' — 3945.C 
 
 209a: + 1315 
 
 -3r, 
 
 3527 
 
 -f3527.r+ 14630 
 
 Multiplying by 209, +737 143a:— 3057670 
 
 + 737143j:-4038005 
 
 + 1580335. 
 
 
 i2, = -1580 
 
 ence, the functions are 
 
 
 
 X = a;' -a:' + 70a:- 300 
 
 1 
 
 
 X,= 3a;' -2a: + 70, 
 
 
 
 H =-209a: + 1315. 
 
 
 
 i2, = -1580335. 
 Let x=— cc\ we have — + + — , 
 
 2 
 
 variations ; 
 
 « a:= + 00, " + V ^ _ , 
 
 1 
 
 variation ; 
 
 " ^= 0, " _ + + _ , 
 
 2 
 
 variations ; 
 
 Hence, the given equation has but one real root, and this lies be- 
 tween and + oo ; it is therefore positive. 
 
 Leta:=l; we have — + + — , 
 
 « x=2 " _ + 4- — , 
 
 « x=3 " — + + _ , 
 
 " a:=4 " 4- + + _ , 
 
 (420, Ex. 4 ) 
 
 variations, 
 
HORNER'S METHOD. 
 
 259 
 
 Hence, the initial figure is 3. The decimal part is found by the 
 following operation : 
 
 1 —1 
 
 
 f 70 
 
 -300 |3.7387936878 
 
 3 
 
 
 
 (2) 
 (8) 
 
 (4) 
 (5) 
 (6) 
 
 6 
 
 76 
 15 
 
 228 
 
 2 
 3 
 
 0)- 72.000 
 67.963 
 
 5 
 3 
 
 ) 91.00 
 6.09 
 
 (2) — 4.037000 
 3.119217 
 
 0) 8.0 
 
 .7 
 
 97.09 
 6.58 
 
 (8) — .917783000 
 
 .834882272 
 
 8.7 
 
 .7 
 
 103.6700 
 .3039 
 
 (4) _ 82900728 
 73114357 
 
 9.4 
 
 7 
 
 103.9739 
 3048 
 
 (5) — 9786371 
 9401144 
 
 (2) 10.10 
 3 
 
 104.278700 
 81584 
 
 (6)-385227 
 313374 
 
 10.13 
 3 
 
 104.360284 
 81648 
 
 (7)-71853 
 62675 
 
 10.16 
 3 
 
 104.441932 
 7150 
 
 (^)-9l78 
 8357 
 
 (S) 10.190 
 
 8 
 
 104.449082 
 7150 
 
 (9) -831 
 732 
 
 10.198 
 8 
 
 104.45623 
 92 
 
 ao)-89 
 84 
 
 10.206 
 8 
 
 104.45715 
 92 
 
 —5 
 
 W 10.214 
 
 104.4581 
 
 
 (5) 10.2 
 
 (7) 
 
 104.458 
 
 
 C«) 1 
 
 (8) 
 
 104.46 
 
 
 
 (9) 
 
 104.5 
 
 
 
 (10) 
 
 105 
 
 (420, Ex.4) 
 
 
260 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREE3. 
 
 5. We have given 
 
 X=a:'+a;'-500. 
 The first derived polynomial is 
 
 Multiply X by 3 to avoid fractions in dividing. 
 
 3a;' + 3a;"-- 1500 
 Sx"4-2a; 
 
 Sx'-{-2x 
 
 X, +1 
 
 a;' — 1500 
 3jr'-4500 
 3a;' + 2a; 
 
 3ar'+ 2a; 
 3ar' + 6750a; 
 
 — 2a; -4500 
 a; + 2250 
 
 Hence, i2=a; + 2250 
 
 3a;— 6748 
 
 -6748a; 
 -6748jr- 15183000 
 
 + 15183000. Hence, i2, = — 15183000. 
 
 Thus the functions arc 
 
 X = 
 
 R = 
 
 Let x=. — oo ; we have 
 
 ** ar=: + O) ; " 
 
 0: 
 
 a:'+a;'— 500, 
 3x' + 2^-, 
 a; + 2250, 
 -15183000. 
 
 - + - 
 
 + + + 
 
 - ± + 
 
 2 variations. 
 
 1 variation. 
 
 2 variations. 
 
 Hence, there is but one real root ; and this is positive, because it 
 lies between and + QO. 
 
 .et 
 
 a;=l; 
 
 signs. 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 variations. 
 
 11 
 
 arz:r2; 
 
 <( 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 a 
 
 u 
 
 a;=3; 
 
 It 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 (( 
 
 u 
 
 a;=:4 
 
 u 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 u 
 
 u 
 
 a;=5 ; 
 
 u 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 (( 
 
 u 
 
 a;=6; 
 
 (( 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 u 
 
 t( 
 
 a;=7; 
 
 i( 
 
 — 
 
 + 
 
 + 
 
 — , 
 
 2 
 
 u 
 
 u 
 
 a;=8; 
 
 u 
 
 + 
 
 + 
 
 + 
 
 — , 
 
 1 
 
 variation. 
 
 (420, Ex. 5) 
 
HORNER'S METHOD. 
 
 261 
 
 Hence, the initial figure is 7 ; the decimal part is found as fol- 
 lows : 
 
 + 1 
 
 
 
 
 -500 17.6172797559 
 
 1 
 
 
 56 
 
 392 
 
 8 
 
 
 66 
 
 W— 108.000 
 
 7 
 
 
 105 
 
 104.736 
 
 15 
 
 (1) 
 
 161.00 
 
 (2) — 3.264000 
 
 7 
 
 
 13.56 
 
 1.887181 
 
 0) 22.0 
 
 
 174.56 
 
 (8)-1.376819000 
 
 .6 
 
 
 13.92 
 
 1.323862113 
 
 22.6 
 
 (2) 
 
 188.4800 
 
 W — .052956887 
 
 .6 
 
 
 .2381 
 
 37858967 
 
 23.2 
 
 
 188.7181 
 
 (5)- 15097920 
 
 .6 
 
 
 .2382 
 
 13251090 
 
 (2) 23.80 
 
 (8) 
 
 188.956300 
 
 (6)_i846830 
 
 .01 
 
 
 .166859 
 
 1703728 
 
 23.81 
 
 
 189.123159 
 
 W- 143102 
 
 .01 
 
 
 .166908 
 
 132512 
 
 23.82 
 
 (4) 
 
 189.290067 
 
 (S)-10590 
 
 .01 
 
 
 4770 
 
 9465 
 
 (8) 23.830 
 
 189.294837 
 
 (9)-1125 
 
 .007 
 
 
 4770 
 
 947 
 
 23.837 
 
 (5) 
 
 189.29961 
 
 (io)_n8 
 
 .007 
 
 
 167 
 
 170 
 
 23.844 
 
 
 189.30128 
 
 -8 
 
 .007 
 
 
 167 
 
 
 W 23.851 
 
 («) 
 
 189.3029 
 2 
 
 
 (5) 23.9 
 
 
 189.3031 
 2 
 189.303 
 
 
 («) 2 
 
 C) 
 
 
 
 (8) 
 
 189.30 
 
 
 
 (9) 
 
 189.3 
 
 
 
 (10) 
 
 189 
 (420, Ex. 6) 
 
 
262 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 6. We have given, 
 
 X=ar'— a:' — 40ar— 108. 
 The first derived polynomial is 
 
 X,=dx'-2x-40, 
 Multiply Xby 3, to avoid fractions in dividing. 
 
 3a;' — 32:'— 120j:4-324j3j:' — 2ar— 40 
 3jr' — 2ar'— 40.c Lr, —1 
 
 Multiply by 3, 
 
 — 242.1:4-932. Hence, i2=121ar— 46«, 
 Multiplying X, by 121, to avoid fractions in the next division, 
 
 - a:'- 
 
 80a: 4-324 
 
 -3a:'- 
 
 240a: 4- 9V2 
 
 -3a:' + 
 
 2a: 4- 40 
 
 363a:'— 242a:r- 4840 
 366.r'- 1398 a: 
 
 + 1156a:- 4840 
 Dividing by 4, 4- 289.i:— 1210 
 
 Multiplying by 121, 4- 34969a;— 146410 
 
 4- 34969a:— 134674 
 
 121a:— 466 
 
 3a:, 4-289 
 
 11736 =— J?. 
 
 Thus we find 
 
 X =a:'-a:*— 40a:4-108, 
 
 X,=3a:'— 2a:-40, 
 
 jR =121a:— 466, 
 
 i2, = + 11736. 
 Let a:= — oo ; we have -^ 4- — +,3 variations 
 " a:=4- oo; " + + + +, o 
 
 " ar= 0; « + _ _ +^ 2 « 
 
 Hence, there are three real roots, two between and 4- co, and 
 
 and one between and — oo ; that is, there arc two positive roots, 
 and one negative root. 
 
 To find the situation of the positive roots, let 
 
 a:=r:l ; signs, 4- — — +» 2 variations. 
 
 x=2; " + _ _ 4-, 2 
 
 x=d; « + _ _ 4-, 2 
 
 a:=4; " — ± + +, 1 variation. 
 
 x=5; " 4- + + 4-, 
 
 Hence the initial figures of the positive roots are 3 and 4. 
 To find the situation of the negative root, let 
 (420, Ex. 6) 
 
HORNER'S METHOD. 
 
 263 
 
 ar=-2; signs, 
 
 + 
 
 — 
 
 — 
 
 -f, 
 
 2 variations. 
 
 ar=-4; " 
 
 4- 
 
 4- 
 
 — 
 
 + , 
 
 2 " 
 
 x=-e; " 
 
 + 
 
 + 
 
 — 
 
 + 1 
 
 2 " 
 
 ar=-8; « 
 
 — 
 
 + 
 
 — 
 
 +, 
 
 3 " 
 
 Hence the negative root is situated between 6 and 8. 
 
 Let a: =—6; signs, + + — +> 2 variations. 
 
 " ar=-7; " _ 4- _ +, 3 
 
 Hence, the initial figure of the negative root is — 6, 
 The decimal part of the first root is obtained thus : 
 
 -1 
 
 -40 
 
 + 108 |3.3792053825 
 
 3 
 
 6 
 — 34 
 
 -102 
 
 2 
 
 0) 6.000 
 
 3 
 
 15 
 
 -4.953 
 
 6 
 
 (1)- 19.00 
 
 (2) 1.047000 
 
 3 
 
 2.49 
 -16.51 
 
 -.931147 
 
 0) 8.0 
 
 (8) .115853000 
 
 .3 
 
 2.58 
 
 -.113285061 
 
 8.3 
 
 (2) — 13.9300 
 
 (4) 2567939 
 
 .3 
 
 .6279 
 
 — 2500650 
 
 8.6 
 
 -13.3021 
 
 (5) 67289 
 
 8 
 
 <2) 8.90 
 
 .6328 
 (8)-12.669300 
 
 -62507 
 (6) 4782 
 
 7 
 8.9V 
 
 82071 
 -12.587229 
 
 -3750 
 (T) 1032 
 
 7 
 
 82152 
 
 -1000 
 
 9.04 
 
 C4)_12.505077 
 
 (8) 32 
 
 1 
 
 1827 
 
 -25 
 
 (8) 9.110 
 
 -12.503250 
 
 (9) 7 
 
 9 
 
 1827 
 
 -7 • 
 
 9.119 
 9 
 
 (5)- 12.50142 
 5 
 
 
 
 9.128 
 9 
 
 -12.50137 
 5 
 
 
 W 9.137 
 
 (6)- 12.501 
 
 
 (5) 9.1 
 
 (7)- 12.50 
 <S)-12.5 
 
 
 
 (9) -13. 
 
 (420, Ex. 
 
 6) 
 
264 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The decimal part of the second root is obtained thus : 
 
 —1 
 
 -40 
 
 + 108|4.5875359541 
 
 4 
 
 12 
 
 -112 
 
 3 
 
 -28 
 
 (1) -4.000 
 
 4 
 
 28 
 
 2.875 
 
 1 
 
 0) 0.00 
 
 W- 1.125000 
 
 4 
 
 5.75 
 
 1.020512 
 
 (1) 11.0 
 
 6.75 
 
 (3)— .104488000 
 
 .5 
 
 6.00 
 
 97009003 
 
 11.5 
 
 (2) 11.7500 
 
 (4)_ 7478997 
 
 .5 
 
 1.0064 
 
 6977044 
 
 12.0 
 
 12.7564 
 
 (5)-501953 
 
 .5 
 
 1.0128 
 
 418826 
 
 C2) 12.50 
 
 (8) 13.769200 
 
 (6)-83127 
 
 8 
 
 89229 
 
 69807 
 
 12.58 
 
 13.858429 
 
 (7)- 13320 
 
 8 
 
 89278 
 
 12565 
 
 12.66 
 
 W 13.947707 
 
 (3) -755 
 
 8 
 
 6381 
 
 698 
 
 (8) 12.740 
 
 13.954088 
 
 W-57 
 
 7 
 
 6381 
 
 56 
 
 12.747 
 
 (5) 13.96047 
 
 (10) _i 
 
 7 
 
 38 
 
 1 
 
 12.754 
 
 13.96085 
 
 
 
 7 
 
 38 
 
 
 W 12.761 
 
 («) 13.9612 
 
 1 
 
 13.9613 
 
 
 (») 12.8 
 
 W 13.96 
 
 
 1 
 
 (») 14.0 
 
 («) 1 
 
 (7) 13.961 
 
 (10) 14. 
 
 Note. — To obtain the trial figure for the second place in the root, proceed 
 according to Note 5, (464). Thus, we shall have 
 
 =/-f=/^:=^-- 
 
 By trial, we find that this must be diminished by .1, making .6 for the second 
 figure of the root. 
 
 (420, Ex. 6) 
 
Horner's method. 
 
 265 
 
 To obtain tlie negative root, we may change the signs of the 
 alternate terms ; the result obtained will then be positive, according 
 to Note 3, (464:). This is not necessary, however; for we niay 
 derive the negative root directly fiom the given coefficients, as will 
 be seen from the following operation : 
 
 W-21.880 
 
 — 6 
 
 — 21.886 
 — C 
 
 — 21.892 
 
 — 6 
 
 ")-2].898 
 
 ^5)^21.9 
 
 (6)_2 
 
 -40 
 
 + 108|— 6.9667413367 
 
 + 42 
 
 -12 
 
 2 
 
 0) 96.000 
 
 IS 
 
 -88.119 
 
 0) 80.00 
 
 
 17.91 
 
 
 97.91 
 
 
 18.72 
 
 (2) 
 
 116.6300 
 
 
 1.3056 
 
 
 117.9356 
 
 
 1.3092 
 
 (8) 
 
 119.244800 
 
 
 .131316 
 
 
 119.376116 
 
 
 .131352 
 
 (4) 
 
 119.507468 
 
 
 15329 
 
 
 119.522797 
 
 
 15329 
 
 (5) 
 
 119.53812 
 
 
 87 
 
 
 119.53899 
 
 
 87 
 
 («) 119.5399 
 (7) 119.540 
 W 119.54 
 <9) 119.5 
 (10) 120 
 
 — 7.076136 
 
 (8) .804864000 
 
 —.716256696 
 
 W 88607304 
 
 — 83665958 
 
 i^) 4941346 
 
 — 4781560 
 
 (6) 159786 
 
 4384 
 3586 
 
 (9) 
 
 798 
 -717 
 
 OO) 81 
 
 — 83 
 
 8 
 
 (420, Ek. 6) 
 
266 NUMERICAL EQUATIONS OF HIGHER DEGREES 
 
 7. We have given 
 
 X=a:'-4x' — 24j: + 48. 
 The first derived polynomial is 
 
 X,=z3x'-8x-2i. 
 Multiplying X by 3 to avoid fractions in dividing, 
 
 3.r' — 8a;— 24 
 
 3j:'-12a:'-72.r+144 
 3.c'— 8x' — 24.r 
 
 — 4j:'' — 48j;+144 
 Multiplying by 3, __ s^' — 36^+108 
 
 — 3r'+ 8x-\- 24 
 
 -442-+ 84. llence,ijj=lla:— 21. 
 Multiplying X, by 11, to avoid fractions, 
 
 33j;'— 882:— 264 
 
 llar-21 
 
 33j:'— Q3x 
 
 Sx, -25 
 
 — 25x— 264 
 
 Multiplying by 11, — 27oj;— 2904 
 
 — 215X+ 525 
 
 -3429. Hence, JR, =342 
 
 Hence, we have the functions, 
 
 X =x'-4jr'- 242^ + 48, 
 
 X, =3j:'-8a:-24, 
 
 E =ll2r-21, 
 
 i2,=3429. 
 
 Leta:=-Qo; we have — 4- — +,3 variations 
 
 " a:=+ 00; " 4- 4- + 4-, 
 
 " ar= 0; " 4- _ _ 
 
 + , 2 
 
 Hence, there are three real roots, — two situated between and 
 + 00 and therefore positive, and one situated between and — 00, 
 and therefore negative. 
 
 hetx= 
 
 1; 
 
 we have 
 
 + 
 
 — 
 
 — 
 
 + , 
 
 2 
 
 variations. 
 
 « x= 
 
 2; 
 
 (( 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 
 (( 
 
 " x= 
 
 3; 
 
 (( 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 
 u 
 
 " x= 
 
 4; 
 
 (i 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 
 u 
 
 « x= 
 
 5; 
 
 u 
 
 — 
 
 + 
 
 + 
 
 + , 
 
 
 u 
 
 « x= 
 
 6; 
 
 u 
 
 — 
 
 + 
 
 + 
 
 + , 
 
 
 u 
 
 " x= 
 
 7; 
 
 u 
 
 + 
 
 + 
 
 + 
 
 + , 
 
 
 
 u 
 
 « x=- 
 
 -1; 
 
 u 
 
 + 
 
 (420 
 
 , J^x 
 
 •7) 
 
 + , 
 
 2 
 
 u 
 
Horner's method. 
 
 267 
 
 Let ar=~2; we have 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 2 
 
 variations. 
 
 " ar= — 3; " 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 2 
 
 (t 
 
 " a:= — 4; " 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 2 
 
 (( 
 
 " x=-5\ 
 
 — 
 
 + 
 
 — 
 
 + , 
 
 3 
 
 (( 
 
 By inspecting the column of variations, we perceive that one root 
 is between 1 and 2, another between 6 and 7, and another between 
 — 4 and —5. Hence, the initial figures of the three roots are 
 
 1, 6, -4. 
 
 The first root is obtained as follows : 
 
 — 4 
 
 -24 
 
 + 48 |1.7191292611 
 
 1 
 — 3 
 
 - 3 
 
 -27 
 
 (l)~ 
 
 27 
 21.000 
 
 1 
 
 - 2 
 
 — 
 
 20.447 
 
 -2 
 
 0)-29.00 
 
 (2) 
 
 .553000 
 
 1 
 
 - .21 
 -29.21 
 
 (8; 
 
 — .289189 
 
 a)~i.o 
 
 > .263811000 
 
 .1 
 
 + .28 
 
 
 -.260077041 
 
 — .3 
 
 .1 
 
 (2) -28.9300 
 + 111 
 
 
 W 3733969 
 
 — 2888700 
 
 + A 
 
 ,1 
 
 — 28.9189 
 -f 112 
 
 
 (5) 845259 
 
 -577737 
 
 (2) 1.10 
 
 (3)-28.907700 
 
 (6) 267522 
 
 1 
 
 4- 10251 
 
 
 -259981 
 
 1.11 
 
 — 28.897449 
 
 
 (7) 7541 
 
 1 
 
 + 10332 
 
 
 -5777 ' 
 
 1.12 
 
 W-28.887117 
 
 
 (8) 1764 
 
 1 
 
 + 116 
 
 
 -1733 
 
 (8) 1.130 
 
 -28.887001 
 
 
 (9) 31 
 
 9 
 
 + 116 
 
 
 -29 
 
 1.139 
 
 (S)-28.88688 
 
 
 (10) 2 
 
 9 
 
 1.148 
 
 9 
 
 + 2 
 — 28.88686 
 4- 2 
 
 (S) 
 
 • -3 
 
 -28.89 
 
 <4) 1.157 
 
 (6)-28.8868 
 
 (9) 
 
 -28.9 
 
 (5) 1.2 
 
 (7) -28.887 
 
 (420, Ex.7) 
 
 (10) 
 
 -29. 
 
268 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The decimal part of the second root is found thus : 
 
 + 48 | 6.546145'7261 
 -72 
 
 1 
 
 -4 
 
 
 6 
 
 
 ■1 « 
 
 
 2 
 
 
 6 
 
 
 8 
 
 
 6 
 
 (1) 
 
 14.0 
 
 
 .5 
 
 
 14.6 
 
 
 .5 
 
 
 15.0 
 
 
 .5 
 
 (2) 
 
 15.50 
 
 
 4 
 
 
 15.54 
 
 
 4 
 
 
 15.58 
 
 
 4 
 
 (8) 
 
 15.620 
 
 
 6 
 
 
 15.626 
 
 
 6 
 
 
 15.632 
 
 
 6 
 
 (4) 
 
 15.638 
 
 (5) 
 
 15.6 
 
 (6) 
 
 2 
 
 -24 
 
 12 
 48 
 
 Oy 36.00 
 7.25 
 
 43.25 
 7.50 
 
 <«> 60.7500 
 .6216 
 
 51.3716 
 .6232 
 
 <3> 51.994800 
 93756 
 
 52.088556 
 93792 
 
 <*> 52.182348 
 1564 
 
 52.183912 
 1564 
 
 (5) 52.18548 
 
 62 
 
 52.18610 
 62 
 
 (6> 52.1867 
 1 
 
 52.1868 
 1 
 
 (^ 52.187 
 (5> 52.19 
 <9> 52.2 
 (10) 52 
 
 (420, Ex. 
 
 (i>-24.000 
 21.625 
 
 C2) — 2.375000 
 2.054864 
 
 (8> -.3201 36000 
 .312531336 
 
 (4) _ 7604664 
 
 5218391 
 
 (5) — 2386273 
 
 2087444 
 
 c«>- 298829 
 260934 
 
 (7) -37895 
 36531 
 
 (8) -1364 
 1044 
 
 t9) — 320 
 313 
 
 (10) _ 7 
 
 5 
 
 — 2 
 
 n 
 
HORNERS METHOD. 
 
 269 
 
 . And the decimal part of the third root is found, without chang- 
 ing the signs of the alternate terms of the equation, as follows : 
 
 1 -4 
 
 -24 
 
 + 48 -4.2652749871 
 
 -4 
 
 + 32 
 
 -32 
 
 -8 
 
 8 
 
 0) 16.000 
 
 -4 
 
 48 
 0) 56.00 
 
 — 11.848 
 
 — 12 
 
 (2) 4.152000 
 
 — 4 
 
 3.24 
 
 — 3.811176 
 
 (l)-16.0 
 
 59.24 
 
 <3) .340824000 
 
 — .2 
 
 3.28 
 
 — .323033625 
 
 -16.2 
 
 (2) 62.5200 
 
 W 17790375 
 
 — .2 
 
 9996 
 
 -12938807 
 
 -16.4 
 
 63.5196 
 
 (5) 4851568 
 
 — .2 
 
 1.0032 
 
 -4528900 
 
 (2)- 16.60 
 
 (3) 64.522800 
 
 (6) 322668 
 
 - 6 
 
 83925 
 64.606725 
 
 -258799 
 
 -16.66 
 
 0) 63869 
 
 - 6 
 
 83950 
 
 —58230 
 
 — 16.72 
 
 W 64.690675 
 
 (8) 5639 
 
 - 6 
 
 3359 
 64.694034 
 
 -5176 
 
 W- 16.780 
 
 (9) 463 
 
 - 5 
 
 3359 
 
 -453 
 
 -16.785 
 
 (5) 64.69739 
 
 (10) 10 
 
 — 5 
 
 118 
 
 -1 
 
 — 16.790 
 
 64.69857 
 
 3 
 
 — 5 
 
 118 
 
 
 (*)- 16.795 
 
 (6) 64.6997 
 1 
 
 
 (»J-16.8 
 
 64.6998 
 1 
 (7) 64.700 
 
 
 («)— 2 
 
 
 
 (8) 64.70 
 
 
 
 (9) 64.7 
 
 
 
 00) 65. 
 
 
 
 (420, Ex.7) 
 
 
*270 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 4a:* + 4j:'4-4j;'— 4^—2000 4a:' + 3;c' + 2ar— 1 
 
 8, Given, 
 Whence, 
 
 X, +1 
 
 or, 
 
 a:' + 2a:'- 82:— 2000 
 4a;' + 8j:' — 12a; -8000 
 4x' + 3a;"+ 2a;— 1 
 
 5a;''-14a;-7999. i?=-5a;« + 14a; + 7999. 
 
 or. 
 
 20a;' + 15a:' + 10a:— 
 
 20a:'— 56a;'— 31996a; 
 
 71a;' + 32006a-— 
 855a;' + 160030a; — 
 
 — 5a;' + 14a;-f-7999 
 
 -4a;, -71 
 
 5 
 25 
 
 355a;'— 994a;— 567929 
 
 161024a; + 567904 
 
 Whence, 
 
 Multiplying R by 5032, 
 
 — 25160a;' + 70448a- + 
 
 — 25160a;'— 88735a; 
 
 i2.=—5032j:— 17747, 
 
 40250968 
 
 -5032a;-l7747 
 
 5a;, —159183 
 
 or. 
 
 159183a;+ 40250968 
 
 5032 •159183a; + 202542870976 
 6032 -1591833;+ 2825020701 
 
 + 199717850275 =— i2j 
 
 In these functions, let 
 
 a;= — 00 ; we have 
 a;= + 00 ; " 
 x= 0; « 
 
 + — — + — , 3 variations. 
 + + — — — , 1 variation. 
 — — + — — , 2 variations. 
 
 Hence, the given equation has one real root between and 00, 
 which must be positive ; and one real root between and — 00, 
 which must be negative. By proper substitutions, we shall find the 
 initial figures to be 4 and —4. 
 
 (420, Ex. 8) 
 
HORNER'S METHOD. 271 
 
 The decimal part of the positive root is found as follows : 
 
 +1 
 
 4 
 
 + 1 
 20 
 
 21 
 36 
 
 57 
 52 
 
 0) 109.00 
 6.96 
 
 — 1 
 
 84 
 
 83 
 
 228 
 
 — 500 14.4604168201 
 332 
 
 5 
 
 4 
 
 (1) — 168.0000 
 142.9536 
 
 9 
 4 
 
 (1)311.000 
 46.384 
 
 (2)— 25.04640000 
 24.87028656 
 
 13 
 4 
 
 357.384 
 49.232 
 
 (8)-. 17611344 
 .16900578 
 
 (1)17.0 
 .4 
 
 115.96 
 7.12 
 
 (2)406.616000 
 7.888776 
 
 (*)- 710766 
 422569 
 
 17.4 
 .4 
 
 123.08 
 
 7.28 
 
 414.504776 
 7.956168 
 
 (5) — 288197 
 253543 
 
 17.8 
 .4 
 
 (2)130.3600 
 1.1196 
 
 (•^) 422.460944 
 53495 
 
 (6)— 34654 
 33806 
 
 18.2 
 .4 
 
 131.4796 
 1.1232 
 
 132.6028 
 1.1268 
 
 422.514439 
 53495 
 
 (■)-848 
 845 
 
 (2)18.60 
 .06 
 
 ('t) 422.5679 
 13 
 
 (8) -3 
 
 4 
 
 18.66 
 .06 
 
 (8)133.7296 
 75 
 
 422.5692 
 13 
 
 
 18.72 
 .06 
 
 133.7271 
 75 
 
 (•"') 422.571 
 (6)422.57 
 
 
 18.78 
 .06 
 
 133.7446 
 75 
 
 (7) 422.6 
 
 (8) 423 
 
 
 (8)18.84 (4)134. 
 
 Note. — The first contracted terms in the operation, marked (4), occur in con- 
 nection with the cipher in the root ; and the pupil will observe that they are 
 therefore contracted twice the usual number of places. 
 
 (420, Ex. 8) 
 
272 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 To obtain the decimal part of the negative root, change the signs 
 of the alternate terms, and proceed as in the following operation.: 
 
 -1 
 
 + 1 
 
 + 1 
 
 -500 14.9296646474 
 
 4 
 
 12 
 13 
 
 52 
 63 
 
 212 
 
 3 
 
 (l)-288.0000 
 
 4 
 
 28 
 
 164 
 
 275.7411 
 
 7 
 
 41 
 
 0) 217.000 
 
 (2)-12.25890000 
 
 4 
 
 44 
 
 89.379 
 
 8.23961296 
 
 11 
 
 0) 85.00 
 
 306.379 
 
 (3) -4.0 1928704 
 
 4 
 
 14.31 
 
 102.987 
 
 3.74208814 
 
 0) 15.0 
 
 99.31 
 
 (2) 409.366000 
 
 (4)-.27919890 
 
 .9 
 
 15.12 
 
 2.614648 
 
 .25023185 
 
 15.9 
 
 114.43 
 
 411.980648 
 
 (5)-2696705 
 
 .9 
 
 15.93 
 
 2.622104 
 
 2502841 
 
 16.8 
 
 («) 130.3600 
 
 (3) 414.602752 
 
 (6) -193864 
 
 .9 
 
 .3724 
 
 1.184819 
 
 166859 
 
 17.Y 
 
 130.7324 
 
 415.787571 
 
 (^)- 27005 
 
 .9 
 
 .3728 
 
 1.186331 
 
 25029 
 
 (2)18.60 
 
 131.1052 
 
 W 416.97390 
 
 (8) -1976 
 
 2 
 
 .3732 
 
 7919 
 
 1669 
 
 18.62 
 
 (^) 131.4784 
 
 417.05309 
 
 (^-307 
 
 2 
 
 .1681 
 
 7920 
 
 292 
 
 18.64 
 
 131.6465 
 
 (5) 417.1323 
 
 (io)_iv 
 
 2 
 
 1681 
 
 79 
 
 17 
 
 18.66 
 
 131.8146 
 
 417.1402 
 
 
 
 2 
 
 .1681 
 
 79 
 
 
 CS)18.68 
 
 (4) 131.98 
 
 (<5) 417.148 
 
 
 W2 
 
 1 
 
 (') 417.15 
 
 
 
 131.99 
 
 (3) 417.2 
 
 
 
 1 
 
 (9) 417. Hence, 
 
 , -4.9296646474, root. 
 
 
 132.00 
 1 
 (5) 132. 
 
 (10) 42 
 
 
 
 
 
 
 (6)1 
 
 
 
 (420, Ex. 8) 
 
<^^'^^ 
 
 HORNER S MET] 
 
 9. We have given 
 
 TY, 
 
 k..-9x.-n..-2o>^££fOR^ 
 
 The first derived polynomial is 
 Multiplying X by 4, to avoid fractions in dividing, 
 
 4a;*-362:'- 44ar'- 
 4x*-2lx'— 22^'- 
 
 8Qx-\- 16 
 202r 
 
 4^' — 27a?'— 22jr— 20 
 
 9 
 
 — f)x'— 22r'— 60x+ 16 
 or -36x'— 88a;' — 240jr+ 64 
 
 — 36a:' + 243a:' + 198a: + 180 
 
 -331a;'-438a:- 
 
 -116. 
 
 tiplying X, by 331, 
 
 
 1324a:'— 893Ya:'- 
 
 7282a;- 
 
 1324a:' -f- 1752a:' + 
 
 464a-- 
 
 i2=331a:' + 438a; + 116. 
 
 — 10689a:'- 
 
 6620 
 
 7746a;— 6620 
 
 331a:' + 438a: + 116 
 
 4a:,— 10689 
 
 or — 331* 10689t'— 2563926a;— 2191220, prepared dividend, 
 — 331-10689.r'— 468l782a:-1239924 
 
 + 21l7866a:- 951296 
 
 Dividing by 32, and changing signs, 
 
 i?, = — 66183ar + 29728. 
 Multiplying i2by 66183, 
 
 66183- 331a:' + 28988154a;4- 
 66183-331a:'— 9839968a; + 
 
 7677228 
 
 -66183.r 4-29728 
 -331a-,-19414061 
 
 + 38828122a:+ 7677228 
 
 or, +19414061a;+ 3838614 
 
 or, +66183 -19414061a: + 254050990362, prepared dividend, 
 
 + 66183 -1941406^—577141205408 
 
 + 831192195770=-/?,. 
 (420, Ex. 9) 
 
274 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 Thus we have, for the several functions, 
 
 X = x*—9x*—Ux^ — 20x-\-4, 
 X, = 4x' — 2lx'-22x-20, 
 R = 331a:' + 438a;+116, 
 i2,=— 66183ar + 29728, 
 7?3 = -831192195770. 
 
 To find the number of real roots, let 
 
 x=— OO; signs, 
 
 + 
 
 N 
 
 + 
 
 + 
 
 — , 3 variations. 
 
 ar=4- Oo; " 
 
 + 
 
 + 
 
 4- 
 
 — 
 
 — , 1 variation. 
 
 x= ; " 
 
 + 
 
 — 
 
 + 
 
 + 
 
 — , 3 variations. 
 
 Hence, there are two real roets between and oo. To find their 
 situation, let 
 
 x= 
 
 signs. 
 
 + 
 
 — 
 
 + 
 
 + 
 
 — 
 
 , 3 variations. 
 
 Xz= 1 
 
 u 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 , 2 
 
 u 
 
 x= 2 
 
 t( 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 , 2 
 
 u 
 
 x= 3 
 
 (( 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 ► 2 
 
 u 
 
 «= 4 
 
 ti 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 2 
 
 a 
 
 ar= 5; 
 
 u 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — , 
 
 2 
 
 (i 
 
 x= 6 
 
 u 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 . 2 
 
 a 
 
 x= 7 
 
 t( 
 
 — 
 
 — 
 
 + 
 
 — 
 
 — 
 
 2 
 
 (( 
 
 x= 8; 
 
 u 
 
 — 
 
 + 
 
 + 
 
 — 
 
 — 
 
 2 
 
 (( 
 
 «= 9 
 
 u 
 
 — 
 
 + 
 
 + 
 
 — 
 
 — 
 
 2 
 
 (( 
 
 ar=10; 
 
 u 
 
 — 
 
 4- 
 
 + 
 
 — 
 
 — , 
 
 2 
 
 (( 
 
 ar=ll 
 
 u 
 
 + 
 
 + 
 
 + 
 
 — 
 
 — , 
 
 1 variation. 
 
 Hence the lesser root is situated between and 1, and the greater 
 between 10 and 11. 
 
 The initial figures of the greater are 10. To find the initial figure 
 of the lesser, let 
 
 x=,l ; signs, 
 x=.2: « 
 
 + - 
 
 3 variations. 
 
 2 " 
 
 Hence, the initial figure is .1. The decimal part of this root is 
 as follows : 
 
 (420, Ex. 9) 
 
HORNER'S METHOD. 
 
 275 
 
 1 —9.0 
 
 -11. 
 
 -20. 
 
 + 4 1. 17968402504 
 
 .1 
 
 — .89 
 
 — 11.89 
 
 - 1.189 
 
 — 21.189 ( 
 
 -2.1189 
 
 — 8.9 
 
 ■^) 1.88110000 
 
 .1 
 
 - .88 
 -12.77 
 
 - 1.277 
 
 — 1.64238179 
 
 — 8.8 
 
 (1) -22.466000 
 
 (2) .238718210000 
 
 .1 
 
 - .87 
 
 — .996597 
 -23.462597 
 
 -.221760635319 
 
 -8.7 
 
 0)_ 13.6400 
 
 (8) 16957574681 
 
 .1 
 
 - .5971 
 -14.2371 
 
 — 1.038051 
 
 — 14873731847 
 
 a)-8.60 
 
 (2)-24.500648000 
 
 W 2083842834 
 
 7 
 
 — .5922 
 
 — 14.8293 
 
 — .139422591 
 
 -1984015679 
 
 — 8.53 
 
 -24.640070591 
 
 (5) 99827155 
 
 7 
 
 — .5873 
 
 - .140095053 
 
 -99206044 
 
 — 8.46 
 
 (2)-15.416600 
 
 (3)-24.780165644 
 
 (6) 621111 
 
 7 
 
 — . 74799 
 
 - 9387434 
 
 -496031 
 
 — 8.39 
 
 — 16.491399 
 
 — 24.789553078 
 
 (^) 125080 
 
 7 
 
 - . 74718 
 
 - 9390416 
 
 -124008 
 
 (2)_8.320 
 
 — 15.566117 
 
 W— 24.79894349 
 
 (8) 1072 
 
 9 
 
 - . 74637 
 
 - 125250 
 
 -992 
 
 -8.311 
 
 (8)-15.640754 
 
 -24.80019599 
 
 80 
 
 9 
 
 - . 4970 
 
 - 125255 
 
 
 — 8.302 
 
 -15.645724 
 
 (5)-24.8014485 
 
 
 9 
 
 - . 4970 
 
 — 15.650694 
 
 - 626 
 
 
 —8.293 
 
 -24.8016111 
 
 
 9 
 
 - . 4970 
 
 - 626 
 
 
 (8) — 8.284 
 
 W- 15.6557 
 
 (6)-24.801574 
 
 
 <*>— 8. 
 
 — 6 
 
 (7)-24.8016 
 
 
 
 — 15.6563 
 
 — 6 
 
 (8) -24.802 
 
 
 
 -15.6569 
 
 
 
 - 6 
 
 
 
 
 (5)-15.66 
 
 
 Note — The term marked (6), in the second column from the right, is contracted two 
 places, to obtain the next term (7), This is on account of the cipher which occurs in 
 the corresponding part of the root. 
 
 (420, Ex. 9) 
 
276 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The decimal part of the greater root is found as follows 
 
 -9 
 
 -11 
 
 — 20 
 
 + 4 |10.2586086356 
 
 10 
 
 10 
 
 -1 
 110 
 
 109 
 210 
 
 -10 
 
 — 30 ' 
 1090 
 
 — 300 
 
 1 
 10 
 
 0)-296.0000 
 225.0096 
 
 11 
 10 
 
 0) 1060.000 
 65.048 
 
 (2)- 70.99040000 
 60.41618125 
 
 21 
 10 
 
 0)319.00 
 6.24 
 
 1125.048 
 66.304 
 
 («)- 10.57421875 
 9.82494438 
 
 0)31.0 
 ,2 
 
 325.24 
 6.28 
 
 (2)1191.352000 
 16.971625 
 
 (4)-.74927437 
 .73864151 
 
 81.2 
 .2 
 
 331.52 
 6.32 
 
 1208.323625 
 17.051375 
 
 (5) -1063286 
 985022 
 
 31.4 
 .2 
 
 (2)337.8400 
 1.5925 
 
 (5)1225.375000 
 2.743048 
 
 (6)-78264 
 73877 
 
 31.6 
 .2 
 
 339.4325 
 1.5950 
 
 1228.118048 
 2.745096 
 
 (^-4387 
 3694 
 
 W 31.80 
 5 
 
 341.0275 
 1.5975 
 
 (*) 1230.86314 
 .20605 
 
 (8) -693 
 616 
 
 31.85 
 5 
 
 (8)342.6250 
 .2560 
 
 1231.06919 
 .20606 
 
 (»)-77 
 74 
 
 31.90 
 6 
 
 342.8810 
 .2560 
 
 (5)1231.2753 
 27 
 
 -3 
 
 31.95 
 5 
 
 343.1370 
 .2560 
 
 1231.2780 
 
 27 
 
 
 (3) 32.00 
 
 (4)343.39 
 
 2 
 
 343.41 
 
 2 
 
 343.43 
 
 2 
 
 (5) 343 
 
 (6)1231.28 
 
 
 W3 
 
 (7)1231.3 
 
 
 
 (8)1231 
 
 
 
 (»)123 
 (420, Kx. 9) 
 
 
HORNERS METHOD. 
 
 277 
 
 10. We readily find the functions to be 
 
 X =:c*-12x' + 12j-3, 
 
 R =2i;'-3jr+l, 
 
 3 variations. 
 " 
 
 Let ar=— oo; we have 4- — + — +, 
 " «=+ oo; " + + + 4- +, 
 
 Hence the roots are all real. And since the signs of the given 
 equation give three variations and one permanance, three of the 
 roots are positive, and one is negative, ( 447 ). 
 
 To find the situation of the roots, 
 
 Let xz=z ; 
 
 we have 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 variations. 
 
 ** «= 1; 
 
 (4 
 
 — 
 
 — 
 
 ± 
 
 + 
 
 + , 
 
 1 variation. 
 
 « «= 2; 
 
 (( 
 
 — 
 
 — 
 
 + 
 
 + 
 
 + , 
 
 1 
 
 " x= 3; 
 
 t( 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + , 
 
 " 
 
 " ar=~l; 
 
 (i 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 variations. 
 
 « ar=-2; 
 
 i( 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 
 
 " ar=— 3; 
 
 (( 
 
 — 
 
 — 
 
 + 
 
 — 
 
 + , 
 
 3 " 
 
 " ar=-4; 
 
 (( 
 
 + 
 
 — 
 
 + 
 
 — 
 
 + , 
 
 4 " 
 
 Hence, there are two roots situated between and 1, one between 
 2 and 3, and one between —3 and —4. 
 
 To find more definitely the situation of the roots between and 1, 
 
 Let x=.l ; 
 
 we have 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 variations. 
 
 " x=.2; 
 
 a 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 
 
 (( 
 
 " x=.Z\ 
 
 u 
 
 — 
 
 4- 
 
 + 
 
 — 
 
 + , 
 
 3 
 
 t( 
 
 " x=.4; 
 
 a 
 
 — 
 
 + 
 
 + 
 
 — 
 
 + , 
 
 3 
 
 u 
 
 « «=.5; 
 
 u 
 
 + 
 
 + 
 
 ± 
 
 — 
 
 + , 
 
 2 
 
 u 
 
 " ^=.6; 
 
 u 
 
 + 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 2 
 
 i( 
 
 « x=.T; 
 
 u 
 
 — 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 1 variation. 
 
 Hence, there is one root between .4 and .5, and one between .6 
 and .7. 
 
 The initial figures of the four roots, taken in the order of their 
 values, are as follows : 
 
 2, .6, .4, -3. 
 (420. Ex. 10) 
 
278 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The decimal part of the first root is found as follows : 
 
 1+0 
 2 
 
 — 12 
 
 4 
 
 — 8 
 
 8 
 
 
 12 
 
 + 12 
 -16 
 
 - 4 
 
 
 
 — 8 12.858083082 
 —8 
 
 2 
 
 2 
 
 0)-11.0000 
 8.9856 
 
 4 
 
 2 
 
 15.232 
 
 (2) _ 2.0 1440000 
 1.71940625 
 
 6 
 2 
 
 0) + 12.00 
 7.04 
 
 19.04 
 7.68 
 
 + 11.232 
 21.376 
 
 (3)-.29499375 
 .29192888 
 
 0) 8.0 
 .8 
 
 (2) 32.608000 
 1.780125 
 
 (4)_306487 
 294315 
 
 8.8 
 .8 
 
 26.72 
 8.32 
 
 34.388125 
 1.808375 
 
 W 36.196500 
 .294610 
 
 (5)- 121 72 
 11038 
 
 0.6 
 
 .8 
 
 («) 35.0400 
 .5625 
 
 («)-1134 
 1104 
 
 10.4 
 .8 
 
 35.6025 
 .5650 
 
 36.491110 
 .295340 
 
 (7)-30 
 29 
 
 (2)11.20 
 5 
 
 36.1675 
 .5675 
 
 W 36.7350 
 912 
 
 (*) 36.78645 
 296 
 
 (S)_l 
 
 1 
 
 11.25 
 5 
 
 36.78941 
 296 
 
 
 
 11.30 
 5 
 
 36.8262 
 912 
 
 (5) 36.792 
 
 (6) 36.79 
 
 • 
 
 11.35 
 5 
 
 36.9174 
 912 
 
 (7) 36.8 
 
 (8) 4 
 
 
 (3)11.40 
 (*)1 
 
 W 37.01 
 
 
 (420, Ex. 10) 
 
Horner's method. 
 The second root is found as follows : 
 
 1 
 
 279 
 
 + 0. 
 .6 
 
 -12. 
 + .36 
 
 -11.68 
 + .72 
 
 -10.92 
 + 1.08 
 
 + 12 
 
 - 6.984 
 
 + 5.016 
 -6.552 
 
 -3 1.6060183069 
 3.0096 
 
 .6 
 .6 
 
 <!) + 9600000000 
 — 9569720304 
 
 1.2 
 .6 
 
 a)- 1.536000000 
 - 58953384 
 
 (2) + 30279696 
 — 16539179 
 
 1.8 
 6 
 
 a) -9.840000 
 -f 14436 
 
 — 1.594953384 
 
 - 58866552 
 
 (3)4-13740517 
 — 13232754 
 
 (1) 2.400 
 6 
 
 -9.825564 
 + 14472 
 
 (2) — 1.653819936 
 - 97996 
 
 W + 507 763 
 -496253 
 
 2.406 
 6 
 
 -9.811092 
 + 14508 
 
 (2) -9.796584 
 + 24 
 
 -1.653917902 
 - 97965 
 
 (5) + 11510 
 — 9925 
 
 2.412 
 6 
 
 (S)- 1.6540059 
 
 - 784 
 
 («) + 1585 
 -1489 
 
 2.418 
 6 
 
 — 9.796560 
 + 24 
 
 -1.6540943 
 
 - 784 
 
 + 96 
 
 («) 2.424 
 
 — 9.796536 
 24 
 
 W-1.654173 
 - 3 
 
 
 
 <8)— 9.80 
 
 • 
 W-10 
 
 -1.654176 
 3 
 
 
 
 (5)-1.65418 
 <fi) 1.654 
 
 
 (420, Ex. 10) 
 
280 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The third root is found as follows 
 
 + 
 
 -12. 
 
 + 12 
 
 -3 |.443276939ft 
 
 A 
 
 .16 
 
 - 4.736 
 
 2.9056 
 
 A 
 
 -11.84 
 
 7.264 0) 
 
 -.09440000 
 
 A 
 
 .32 
 
 -4.308 
 
 8864096 
 
 .8 
 
 -11.52 
 
 (^) 2.656000 
 
 (2)-57l9040000 
 
 .4 
 
 .48 
 
 — .438976 
 2.217024 
 
 5244710001 
 
 1.2 
 
 0)— 11.0400 
 
 (8)_474329999 
 
 .4 
 
 656 
 — 10.9744 
 
 — .436288 
 
 342717760 
 
 0) 1.60 
 
 (2) 1.780736000 
 
 W-131612239 
 
 4 
 1.64 
 
 672 
 — 10.9072 
 
 — 32499333 
 1.748236667 
 
 119746687 
 (5)-11865552 
 
 4 
 
 688 
 
 — 32483439 
 
 10259068 
 
 1.68 
 
 (2)- 10.838400 
 
 (8) 1.715753228 
 
 (6)-1606484 
 
 4 
 
 5289 
 
 -2164430 
 
 1538793 
 
 1.72 
 
 -10.833111 
 
 1.713588798 
 
 <7)-67691 
 
 4 
 
 5298 
 
 — 2164360 
 
 51293 
 
 («) 1.760 
 
 -10.827813 
 
 W 1.71142444 
 
 (8)- 16398 
 
 3 
 
 5307 
 
 -75749 
 
 15388 
 
 1.763 
 
 W- 10.822506 
 
 1.71076695 
 
 W-IOIO 
 
 3 
 
 354 
 
 -75748 
 
 1026 
 
 1.766 
 
 -10.822152 
 
 (5) 1.7099095 
 
 
 3 
 
 354 
 
 -649 
 
 
 1.769 
 
 -10.821798 
 
 1.7098446 
 
 
 3 
 
 354 
 
 -649 
 
 
 (3) 1.772 
 
 W— 10.8214 
 
 («) 1.709780 • 
 
 
 
 1 
 
 -10 
 
 
 (4) 2 
 
 — 10.8213 
 
 1.709770 
 
 
 
 1 
 
 -10 
 
 
 
 -10.8212 
 1 
 W- 10.82 
 
 (7) 1.60976 
 
 
 
 (S) 1.7098 
 
 
 
 («)-ll 
 
 (9) 1.710 
 
 
 
 (430, Ex. 10) 
 
 
HORNER'S METHOD. 
 
 281 
 
 The decimal part of the fourth root is found as follows, the alter- 
 nate signs in the given equation being changed : 
 
 -0 
 
 -12 
 
 — 12 
 
 — 3 
 
 [3.9073785547 
 
 3 
 
 9 
 — 3 
 
 — 9 
 
 -21 
 
 -63 
 
 
 3 
 
 )-66.0000 
 
 3 
 
 18 
 15 
 
 45 
 
 65.0241 
 
 6 
 
 (1) 24.000 
 
 (2)_, 
 
 .97590000 
 
 3 
 
 27 
 
 48.249 
 
 (3)_ 
 
 .92562109 
 
 9 
 
 <^> 42.00 
 
 72.249 
 
 -5027891 
 
 3 
 
 11.61 
 
 59.427 
 
 
 3984354 
 
 ('> 12.0 
 
 53.61 
 
 (2) 131.676000 
 
 (4). 
 
 -1043537 
 
 .9 
 
 12.42 
 
 .555584 
 
 (-•5^ 
 
 929889 
 
 12.9 
 
 66.03 
 
 132.231584 
 
 •-113648 
 
 .9 
 
 13.23 
 
 .556349 
 
 
 106278 
 
 13.8 
 
 <2) 79.2600 
 
 <•''> 132.78793 
 
 <6>-7370 
 
 .9 
 
 .1092 
 
 2388 
 
 
 6643 
 
 14.7 
 
 79.3692 
 
 132.81181 
 
 
 (7) -727 
 
 .9 
 
 .1092 
 
 2388 
 
 
 665 
 
 <2> 15.60 
 
 79.4784 
 
 (*> 132.8357 
 
 (S)-62 
 
 
 .1092 
 
 56 
 
 
 53 
 
 (8) 2 
 
 (»> 79.59 
 
 132.8413 
 
 C9)— 9 
 
 
 
 56 
 
 
 9 
 
 
 <*> 80. 
 
 <-'^> 132.847 
 <6> 132.85 
 (7> 132.9 
 (5) 133. 
 
 
 
 
 
 
 <5> 13. —3. 
 
 9073785547, root 
 
 (420, Ex. 10) 
 
282 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 11. We have given, 
 
 The first derived polynomial is 
 
 X,=5x* — 30x^-\-Q. 
 Multiplying X by 5, to avoid fractions, 
 
 5x*— 50j:'4-30a; + 5 
 5a;* — 30i:*+ Qx 
 
 5x* — 30x'-{-6 
 
 . — 20j;' + 24a; + 5. 
 
 Hence, 
 Multiplying X, by 4, to avoid fractions, 
 
 7?=20a:* — 24ar— 5. 
 
 20jr* — 120j' + 24 
 20x*— 24.r' — 5.C 
 
 20j:» — 24.r— 5 
 
 — 96i:' + 5j: + 24 
 llencc, 
 Multiplying i? by 24, 
 
 480x'— 5lex — 120 
 480x'— 2 ox'— 120.r 
 
 :96x' — oar— 24, 
 
 96^:'- 5j:— 24 
 
 5x, +25 
 
 25j:'— 456j: — 120 ; multiply by 96 
 2400j:' — 43776^ -11520 
 2400^'— 1252: — 600 
 
 i2,==43651a; + 10920. 
 
 — 43651jr— 10920 
 Hence, 
 Multiplying i?, by 43651, 
 
 4190496x'— 218255.r— 1047624|43651.r + 10920 
 
 4190496ar* + 
 
 1048320jr 
 
 '96.r, —1266575 
 
 — 1266575^:- 1047624; multiply by 43651 
 
 — 55287265325a;-45729835224 
 
 — 55287265325a; — 13830999000 
 
 _ 
 
 -31898836224=— i?,. 
 
 Therefore, the functions are 
 
 
 
 X =x^~l0x'-\-6x—l, 
 
 
 X,=52;*-302;' + 6, 
 
 R =20x»-24i;-5, 
 
 
 i2,=96x'-52;-24. 
 
 iJj =4365 1.C + 10920, 
 
 
 i23=31898836224. 
 
 Let a;=: - oo ; we have — 
 
 + 
 
 — + — + , 5 variations. 
 
 " x=+ oo; " 4- 
 
 + 
 
 + + + +, 
 
 (420, 
 
 Ex. 11) 
 
HORNER S METHOD. 
 
 283 
 
 Hence, there are 5 real roots. And since the signs of the given 
 equation present two variations, two of the roots are positive, 
 {4L4LT ) ; consequently, three are negative. 
 
 To ascertai 
 
 n the 
 
 situation of the roots, let 
 
 
 
 
 x= 
 
 signs, + 
 
 + 
 
 — 
 
 — 
 
 + 
 
 + , 
 
 2 
 
 variations. 
 
 x= 1 
 
 i( 
 
 — 
 
 — 
 
 — 
 
 + 
 
 + 
 
 + , 
 
 1 
 
 variation. 
 
 x= 2 
 
 u 
 
 — 
 
 — 
 
 + 
 
 4- 
 
 + 
 
 + , 
 
 1 
 
 u 
 
 x= 3 
 
 u 
 
 — 
 
 + 
 
 4- 
 
 + 
 
 + 
 
 + , 
 
 1 
 
 u 
 
 x=z 4; 
 
 u 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + 
 
 + , 
 
 
 
 n 
 
 «=-! 
 
 u 
 
 + 
 
 — 
 
 — 
 
 + 
 
 — 
 
 + , 
 
 4 
 
 variations. 
 
 x=-2 
 
 u 
 
 + 
 
 — 
 
 — 
 
 + 
 
 — 
 
 +, 
 
 4 
 
 a 
 
 ar=-3 
 
 u 
 
 + 
 
 + 
 
 — 
 
 + 
 
 ■— 
 
 + , 
 
 4 
 
 u 
 
 x= — 4' 
 
 (( 
 
 — 
 
 + 
 
 — 
 
 + 
 
 — 
 
 + , 
 
 5 
 
 u 
 
 Hence we 
 
 have 
 
 
 
 
 
 
 
 
 
 
 1 
 
 positive 
 
 root between 
 
 . 
 
 and 
 
 1, 
 
 
 
 1 
 
 (( 
 
 u 
 
 
 u 
 
 3 and 
 
 4, 
 
 
 
 2 
 
 negative roots 
 
 a 
 
 and - 
 
 -1, 
 
 
 1 " root " —3 and -4. 
 
 In order to limit still further the roots situated between and 1, 
 and and —1, we may employ JT alone, according to ( 453, 3 ). 
 
 As x=zl reduces X nearly to 0, the positive root between and 1 
 must be nearly 1. AVc therefore commence with x=ly and substi- 
 tute the descending scale of tenths, till we come to the initial figure. 
 Thus, 
 
 Substitute x=z 1, .9, .8, 
 
 The signs of X are — — + 
 
 Hence, this root is situated between .8 and .9. 
 
 For the roots situated between and —1, we proceed thus: 
 Substitute x= 0, —.1, —.2, —.3, —.4, —.5, —.6, —.7, 
 
 The signs of X are ++ — — — — — + 
 
 Hence, one root is between —.1 and —.2; and the other between 
 — .6 and —.7. 
 
 Thus, we find the initial ficfures of the several roots, taken in the 
 order of their algebraic values, to be 
 
 -3, -.6, -.1, +.8, 4-3. 
 
 Note. — In extracting the negative roots in this example, wo may change the 
 signs of the alternate terms, remembering always to supply the de&cieat terms, 
 with for coefficients. 
 
 (420, Ex. 11) 
 
284 
 
 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The operation for the first root is 
 
 as follows : 
 
 
 1-0 
 
 -10 
 
 — 
 
 + 6 
 
 - i;3.0653157913 
 
 3 
 
 9 
 -1 
 
 — 3 
 -3 
 
 -9 
 
 -3 (^> 
 
 - 9 
 
 3 
 
 -10.0000000000 
 
 3 
 
 18 
 
 51 
 
 144 
 
 9.1254751776 
 
 6 
 
 17 
 
 48 
 
 141.00000000 
 
 (2)-.8745248224 
 
 3 
 
 27 
 
 132 
 
 11.09125296 
 
 .8222637421 
 
 9 
 
 44 
 
 0) 180.000000 
 
 152.09125296 
 
 (3)- 522610803 
 
 3 
 
 36 
 
 4.854216 
 
 11.38577184 
 
 496468108 
 
 12 
 
 0) 80.0000 
 
 184.854216 
 
 (2)163.47702480 
 
 (*)-26142695 
 
 3 
 
 .9036 
 
 4.908648 
 
 .97572362 
 
 16555014 
 
 (1) 15.00 
 
 80.9036 
 
 189.762864 
 
 164.45274842 
 
 (5)-9587681 
 
 6 
 
 .9072 
 
 4.963296 
 
 .97781834 
 
 8277654 
 
 15.06 
 
 81.8108 
 
 (2)194.726160 
 
 (3)165.4305668 
 
 (6)- 1310027 
 
 6 
 
 .9108 
 
 .418563 
 
 588025 
 
 1158879 
 
 16.12 
 
 82.7216 
 
 195.144723 
 
 165.4893093 
 
 (7)-151148 
 
 6 
 
 .9144 
 
 .418945 
 
 588100 
 
 148999 
 
 15.18 
 
 (2)83.6360 
 
 195.563668 
 
 (4)165.548179 
 
 (S)-2149 
 
 6 
 
 765 
 
 .419328 
 
 1961 
 
 1656 
 
 15.24 
 
 83.7125 
 
 (8>195.9830 
 
 165.550140 
 
 (»)-493 
 
 6 
 
 765 
 
 252 
 
 1961 
 
 497 
 
 (2) 15.30 
 
 83.7890 
 
 196.0082 
 
 (5)165.55210 
 
 + 4 
 
 
 765 
 
 252 
 
 98 
 
 
 
 83.8655 
 
 196.0334 
 
 165.55308 
 
 
 
 765 
 
 252 
 
 98 
 
 
 
 C8>83.9 
 
 (« 196.06 
 (5)196. 
 
 (6)2 
 
 (6)165.5541 
 1 
 165.5542 
 (7)165.554 
 (8)165.55 
 
 
 
 
 
 (»)165.6 —3.0653157913, root 
 
 
 
 (420, 
 
 Ex. 11) 
 
 
HORNER'S METHOD. 
 
 285 
 
 The c 
 1-0 
 .6 
 
 >peration for 
 -10 
 .36 
 
 -9.64 
 .72 
 
 -8.92 
 1.08 
 
 -7.84 
 1.44 
 
 the second root 
 
 - 
 
 - 5.784 
 
 is as follows : 
 + 6 
 -3.4704 
 
 — 1|.6915752805 
 1.51776 
 
 .6 
 .6 
 
 - 5.784 
 
 — 5.352 
 
 + 2.5296 
 -6.6816 
 
 ci) + . 51 77600000 
 — .5064468651 
 
 1.2 
 .6 
 
 -11.136 
 - 4.704 
 
 (1)- 4.15200000 
 -1.47518739 
 
 (2) 113131349 
 -71670641 
 
 1.8 
 .6 
 
 (^) — 15.840000 
 — .550971 
 
 — 5.62718739 
 
 — 1.52245656 
 
 (3) 41460708 
 — 35906031 
 
 2.4 
 .6 
 
 (i)_ 6.4000 
 .2.781 
 
 — 16.390971 
 
 — .525213 
 
 <2)-7.14964395 
 - 1742015 
 
 (4) 5494677 
 — 5042202 
 
 0) 3.00 
 9 
 
 -6.1219 
 
 .2862 
 
 — 16.916184 
 
 - .498726 
 
 -7.16706410 
 — 1742538 
 
 (5) 452475 
 — 432268 
 
 3.09 
 9 
 
 -5.8357 
 2943 
 
 (2) — 17.414910 
 — 5236 
 
 (") — 7.1844895 
 - 87166 
 
 (6) 20207 
 -14409 
 
 3.18 
 9 
 
 -5.5414 
 .3024 
 
 -17.420146 
 - 5232 
 
 -7.1932061 
 - 87179 
 
 C7) 5798 
 -5764 
 
 3.27 
 9 
 
 (2)-5.2390 
 35 
 
 -5.2355 
 35 
 
 -17.425378 
 - 5229 
 
 (4)-7.201924 
 - 1221 
 
 (S) 34 
 -36 
 
 3.36 
 9 
 
 (2)- 17.4306 
 - 26 
 
 — 7.203145 
 
 - 1221 
 
 -2 
 
 (2) 3.45 
 
 -6.2320 
 35 
 
 -17.4332 
 
 - 26 
 
 - 17.4358 
 
 - 26 
 
 (5)- 7.20437 
 - 10 
 
 
 
 -5.2285 
 35 
 
 -7.20447 
 - 10 
 
 
 
 —6.2 
 
 (*>- 17.44 
 (»)-l7. 
 
 («)-7.2046 
 (7)-7.205 
 (8)-7.21 -. 
 
 ,6915762805, root. 
 
 (420, Ex.11) 
 
286 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The operation for the third root is as follows : 
 
 1- 
 .1 
 
 .1 
 .1 
 
 .2 
 .1 
 
 .3 
 .1 
 
 .4 
 
 .1 
 
 0) .50 
 
 1 
 
 .67 
 1 
 
 .64 
 
 1 
 
 .71 
 
 7 
 
 .78 
 
 7 
 
 (2) .85 
 
 10 
 
 .01 
 
 -9.99 
 2 
 
 — 9.97 
 
 3 
 
 — 9.94 
 
 4 
 
 (i)_9.9000 
 399 
 
 — 9.8601 
 
 448 
 
 -0 
 
 — .999 
 
 — .999 
 
 — .997 
 
 — 1.996 
 
 — .994 
 
 0) — 2.990000 
 
 — .600207 
 
 + 6. 
 
 - .0999 
 
 6.9001 
 
 — .1996 
 
 — 1. |.1756747993 
 .59001 
 
 0> -.4099900000 
 .3810019857 
 
 c^) 5.70050000 (2) — 289880143 
 — .25761449 255583952 
 
 5.44288551 
 — .30570946 
 
 — 3.680207 (2) 5.13717605 
 
 — .687071 — 2549702 
 
 (8) -34296191 
 30496909 
 
 (*)- 3799282 
 3555530 
 
 — 4.367278 
 
 — .683592 
 
 5.11167903 
 2573958 
 
 <« — 243752 
 203158 
 
 — 9.8153 (2)-5.050870 W 6.0859394 
 497 — 48534 — 31213 
 
 <6)_40594 
 36562 
 
 -9.7656 -5.099404 5.0828181 
 546 — 48512 — 31248 
 
 (7) -5042 
 4571 
 
 (2)-9.7ll0 
 
 5.147916 (*> 5.079693 
 
 43 — 48491 
 
 363 
 
 (8)-47l 
 467 
 
 9.7067 (8)_5.i964 
 
 43 
 
 58 
 
 — 9.7024 
 43 
 
 —9.6981 
 43 
 
 5.2022 
 58 
 
 5.2080 
 
 58 
 
 5.079328 
 
 — 365 
 
 (•'^) 5.07896 
 
 — 2 
 
 (»)-14 
 15 
 
 + 1 
 
 5.07894 
 2 
 
 (8) -9.7 
 
 W-5.21 
 
 (6) 5.0789 
 
 (5)— 5. 
 
 (7) 5.079 
 
 
 (8) 6.08 
 
 
 (9) 5.1 
 
 (420, 
 
 Ex. 11) 
 
 -.1766747093, root 
 

 
 HORNER'S 
 
 METHOD. 
 
 2«7 
 
 The operation for the fourth root is as follows : 
 
 
 1 +0 
 .8 
 
 — 10. 
 .64 
 
 + 
 
 -7.488 
 
 + 6 +1 1-8795087084 
 -5.9904 -f .00768 
 
 .8 
 .8 
 
 -9.36 
 1.28 
 
 -7.488 
 -6.464 
 
 + .0096 0)1.0076800000 
 -11.1616 -.8742890793 
 
 1.6 
 
 .8 
 
 — 8.08 
 1.92 
 
 -13.952 
 — 4.928 
 
 0) — 11.15200000 
 — 1.38784399 
 
 (2). 1333909207 
 -.1261650637 
 
 2.4 
 8 
 
 -6.16 
 2.56 
 
 0)- 18.880000 
 - .232057 
 
 — 12.48984399 
 
 - 1.35266796 
 
 (S) 72258570 
 -71020747 
 
 3.2 
 
 .8 
 
 0)- 3.6000 
 .2849 
 
 -19.112057 
 - .211771 
 
 (2) — 13.84251195 
 — .17582846 
 
 W 1237823 
 -1137128 
 
 0)4.00 
 
 7 
 
 — 3.3151 
 
 .2898 
 
 -19.323828 
 - .191142 
 
 -14.01834041 
 — .17601901 
 
 (5) 100695 
 -99500 
 
 4.07 
 
 7 
 
 -3.0253 
 .2947 
 
 (2)-19.514970 
 — 21525 
 
 (8)— 14.1943594 
 
 — 97899 
 
 -14.2041493 
 
 - 97905 
 
 (6) 1195 
 -1137 
 
 4.14 
 
 7 
 
 -2.7306 
 .2996 
 
 -19.536495 
 - 21173 
 
 (7) 58 
 -5^ 
 
 4.21 
 
 7 
 
 (2)-2.4310 
 392 
 
 -19.557668 
 - 20821 
 
 W-14.213940 
 - 157 
 
 1 
 
 4.28 
 
 7 
 
 -2.3918 
 392 
 
 (^)- 19.5 785 
 - 12 
 
 -14.214097 
 - 157 
 
 
 <2) 4.35 
 
 -2.3526 
 392 
 
 -19.5797 
 - 12 
 
 (5)— 14.2143 
 
 (6)-14.214 
 
 (7)-14.2 
 
 
 
 -2.3134 
 392 
 
 -19.5809 
 — 12 
 
 
 
 W-2.3 
 
 (*)- 19.58 
 
 
 
 
 (420, Ex.11) 
 
 
288 NUMERICAL EQUATIONS OF HIGHER DEGREES. 
 
 The operation for the fifth root is as follows ; 
 
 1+0 
 3 
 
 3 
 
 -10 
 9 
 
 — 1 
 
 + 
 
 — 3 
 
 — 3 
 
 + 6 +1 |3.0535816265 
 
 -9 -9 
 
 — 3 0)- 8.0000000000 
 
 3 
 
 18 
 17 
 
 51 
 
 48 
 
 144 
 
 7.5100940625 
 
 6 
 
 0) 141,00000000 
 
 (2) -^.4899059375 
 
 3 
 
 27 
 44 
 
 132 
 
 9.20188125 
 
 .4805548729 
 
 9 
 
 0)180.000000 
 
 150.20188125 
 
 (3)- 93510646 
 
 3 
 
 36 
 
 4.037625 
 
 9.40565000 
 
 80386391 
 
 12 
 
 0) 80.0000 
 
 184.037025 
 
 (2)159.60753125 
 
 (4) — 13124255 
 
 3 
 
 .7525 
 
 4.075375 
 
 .57742639 
 
 12862718 
 
 (1)15.00 
 
 80.7525 
 
 188.113000 
 
 160.18495764 
 
 (5)-261537 
 
 5 
 
 .7550 
 
 4.113250 
 
 .57817444 
 
 160786 
 
 15.05 
 
 81.5075 
 
 (2) 192.226250 C^) 160.7631321 
 
 (G)_ 100751 
 
 5 
 
 .7575 
 
 .249212 
 
 96489 
 
 96472 
 
 16.10 
 
 82.2650 
 
 192.475462 
 
 160.7727810 
 
 (^-4279 
 
 5 
 
 .7600 
 
 .249350 
 
 96491 
 
 3216 
 
 15.15 
 
 (3) 83.0250 
 
 192.724812 W 160.78243 
 
 (8) -1063 
 
 5 
 
 458 
 
 .249487 
 
 154 
 
 965 
 
 15.20 
 
 83.0708 
 
 (8)192.9743 
 
 160.78397 
 
 («)-98 
 
 5 
 
 458 
 
 42 
 
 154 
 
 97 
 
 (2) 15.25 
 
 83.1166 
 
 192.9785 
 
 (^) 160.7855 
 
 ^1 
 
 
 .458 
 
 42 
 
 (6) 160.786 
 (7)160.79 
 
 
 
 83.1624 
 
 192.9827 
 
 
 
 458 
 
 42 
 
 (8)160.8 
 
 
 (3) 83.2 
 
 (4) 193. 
 
 (9) 161. 
 
 (420, Ex. 11) 
 
AN \N1T1AU FINE °l^^^^o return 
 
 ^^^ U ASSESSED !,°« ''^Ce THE PENALTY 
 TH^S BOOK ON THE OATE °U ^^^ ^„„„„ 
 
 ^«,LU.NCREASET0 50CEN^^^ ^^^^^^„ „^V 
 
 DAY AND TO S'"" 
 OVERDUE. 
 
 SFP 9 1933 
 
 DEC 12 1938 
 
 MAY 10 1940 
 
 LD 21-^ 
 
;> -ji