96

ON CUBIC EQUATIONS

TRACT 28.

J+d , S-rf . „ 3/1S — 3/12

T ± Tv _ j= -a—

y' _ 3 the other

two roots.

59. Ex. 4. In the equation .r 3 — 1 5x =4, we have a — — 5,

5 = 2; hence c ■ V(b'- + a 3 ) — ^/(4 — 125) = \/ — 121 =1 V-l, 5 = l/ib + c) = V(2 + 1V-1) = 2 + A /-l,and d = £/(5 — c) = 3 /(2 — 11 a/ — 1) —■ 2 — \/ — I. There-fore r = i + d = 4 the first root; and — + s — V — 3 =

— 2 ± a/ —1 • V — 3 = — 2±^/3 the other two roots,which are also real.

60. Ex. 5. The equation .r 3 — 6.r = 4 gives a = — 2, and

5= 2; therefore c : ^7(6* + a 3 ) = ^/(4 —8) = \/—A =.

2V-1, s =l/(b+ c) = 1/(2 + 2^- 1) = - 1 + V-l,and d - l/(b — c) = s /(2 — 2^/ —l) = — 1 — —1. And

hence r — s + d = — 2 the first root; and — ± ^4

v'-3=l± v /-l. V — 3 = 1 ± v /3, which are the twoextremes, or the greatest and least roots. So that in tinsexample, Cardan’s rule gives the middle root.

61. Ex. 6. Let the equation be .r 3 — 9x — — 10. Thena — — 3 and b = — 5 ; so that c = V (b 1, 4- a 3 ) = V(25 — 21) =V -2, s = £/(i +£•)==/{- 5+ V — 2) = 1 +a/-2, andd = £/(5 — £.’) = 3 /(— 5 — a/ — 2) = 1 — a/ — 2. Hence

7’ = ^ 4- d — 2 the middle root; and — a/ — 3 =

« *

— 1±a/ — 2. v' — 3 = — 1±a/ 6 the greatest and least

roots.

62. Ex. 1. Take the equation x 3 — 12jt = 9. Here a =

n Q1

— 4, and b = '-; therefore c = V(5 I + a 3 ) = V (— — 64) =

= 7 > * = + 0 =$4+ !✓-'*) =

+ and ^ - c ) = Mj -1 ^ - 7 ) =

a/ — 7. Hence r = s + d = —3 the middle root;

3=tv'2i

2 1 23_ _ 1_" 2 2

, s+d s —d ,

and --pi— a/~ 3 =the greatest and least roots.

[- a/ “ 7 • v/ ~ 3 =