9.

11.

x4-10x2+1=0.

10. x4-10x3- 19x2+480x-1392=0.

x2-6x3+18x2 - 26x+21=0. 12. x8-16x6 +88x4+ 192x2+144=0. 13. One positive, one negative, two imaginary. [Compare Art. 554.]

15. One positive, one negative, at least four imaginary. [Compare Art. 554.]

5. 16axh (x6+7xah2+7x2ha +ho) +2bh (5x2+10x2h2+h1)+2ch.

22.2±√√6,

25. 2±√3.

2. 10, -57-3.

5.

1 2+/-3

9. 4, -1,

7

3. 4, -2±5/- 3.

6. 11, 11, 7.

11. ±1, -4±√6.

12. 1, 2, -2,

14. 1, 3, 2/5.

3 -3±3/5

18. q3+8r2=0;'
2'

2, 2±√2. 23. 83y4+q8 (1-8)2 y2+r(1 − s)3y+(1-8)=0.

28. x1- 8x3+21x2 - 20x+5=(x2 – 5x+5) (x2 - 3x+1); on putting x=4-y, the expressions x2-5x+5 and 2-3x+1 become y2-3y+1 and y2-5y+5 respectively, so that we merely reproduce the original equation.

MISCELLANEOUS EXAMPLES. PAGES 490-524.

2. 6, 8.

4. (1) 1±5; 1±2√5.

3. Eight.

(2) x=1, y=3, z=-5; or x=-1, y=-3; z=5.

24. Wages 15s.; loaf 6d.

26. (1) 1,

c(a - b)
a(b-c)'

25. 6, 10, 14, 18.

(2)

29. x=3k, y=4k, z=5k; where k3=1, so that k=1, w, or w2. 31. Either 33 half-crowns, 19 shillings, 8 fourpenny pieces; or 37 half-crowns, 6 shillings, 17 fourpenny pieces.

33. 40 minutes.

1±√/21 ̧ [x* -x−5 (x2+x+1)=0.]

40. The first term.

0,

0;

2

48.

150 persons changed their mind; at first the minority was 250, the

[Put (a–c) (b − d) = {(x − c ) − (x − a)} { (x − d) – (x − b)}; then square.]

58. (1) 1. (2) ±4/2 [putting x2-16=y1, we find y4-16-4y (y2 — 4)=0.]

[(a + b)3 − a3 − b3 =3ab (a+b), and (a − b)3 − a3+b3— — 3ab (a − b).]

80.

81.

a=3, b=1.

с

[Put x-au and y-b=v.] 82. x=3. 84. 126. 85. Sums invested were £7700 and £3500: the fortune of each was £1400.

91. 25 miles from London.

(2) x=y=z=1.

1+ 4x
-x- 2x2
nth term = {2"+(−1)n} xn−1.
a2+b-c2- d.

100. Generating function is

(1) x=a, y=b, z=c.

107.

109.

108. 12 persons, £14. 18s. (2) x=3, or 1; y=1, or 3.

117. (1) x=α, y=b; x=a, y=2a; x=2b, y=b. (2) x=3 or 1, y=2, z=1 or 3;

x=

113. £12. 158.

120. (1) 1

(n+1)2

138. £3. 2s. at the first sale and £2. 128. at the second sale.

(2) x, y, z may have the permutations of the values 3, 5, 7.

142. y3+qy2-q2y — q3 — 8r=0.

(3) 2n+1+1⁄2n (n+7) — 2.

145. -2, -2, -2, 2.

146.

A walks in successive days 1, 3, 5, 7, 9, | 11, 13, 15, 17, 19, 21, 23, | miles,
B walks
12, 13, 14, 15, 16, 17, 18, 19, 20,

so that B overtakes A in 2 days and passes him on the third day; A
subsequently gains on B and overtakes him on B's 9th day.

162. (1) x2 y2=

92 1 1
= +
5

585

-7±√√217

[(12x-1) (12x-2) (12x − 3) (12x-4)=120.]

9

2

13

161. 44 hours.

; x=±1, ±2; y = 2, 1; x=-y= ±√3

(2) x=k(b1+c1 − a2b2 — a2c2), &c., where 2k2 (a+b¤+c® − 3a2b2c2)=1. [It is easy to shew that a2x + b2y+c2z=0, and

a2y+b2z + c2x=x3 + y3 + z3 - 3xyz=a2z+b2x + c2y.]

-

163. 2(a+b+c) x = ( (bc+ca+ab)± √(bc+ca+ab)2 - 4abc (a+b+c). [Equation reduces to (a+b+c) x2 - (bc+ca+ab) x+abc=0.]

4. (1) ̧n (n + 1) (n + 2) (3n+13). (2) 2e - 5.

12