TRACT 28.

AND INFINITE SERIES*

97

63. Ex. 8. Again, from the equation x 3 — 12.r = — 8\Z2,

we have a = — 4, and b = — 4.^/2 ; hence c — >/(b 1 -\-a i ) =\/(82 — 64) =V~ 32 =4 V-2, s= V(J + c)==/(-4v'2 +4V-2)= v '2+^/-2, and d = y/ 2 - 2. So that

r = s+</=2^2 the middle root; and — s -~ ± —-3 =

— v / 2± v / — 2 . V- 3 = - V'2±v'6=;- V2 . (l=tV 3 ) thegreatest and least roots.

64. Ex. 9. But the equation x 3 — I5x = 22 gives a ——5,and 6 = 11; therefore c =^/(6 1 + a 3 ) =</(121 —125) =\/- 4, s = 3 /(b + c) =^(U + </- 4) =- l- a/- 4, andd = ~ 1 + \/ — 4. Consequently r = s-\-d = —2 the least

root; and — ± — S = l±v / —4 . V — 3 = 1± A /12

the two greater roots.

65. Ex. 10. Further, in the equation x 3 — 15.r = 20, wehave a — 5, and b = 10 ; consequently c =: a/( b 1 + a?) =V(100 — 125) =v/-25 = 5V-l, s :=£/(£+ c)=£/(10 +B\/— 1), and d =^/(10 — 5*/ — 1). Therefore r = .? +d =v/(10 + Bn/ — l) +^/(10 — 5^/ — 1) = the first root; and

+ S Z± aJ-% — Mio + V-i)+Mi°—W-i) +

2 2 ^ ” 2 —

^ _ 3 = the other two rO0ts .

*65. Ex. 11. Lastly, taking the equation x 3 — lx = 6. Herea = — |, and 6 = 3; therefore c = a/ (b z + a 3 ) — V(9 — 3 -~?) =V = V°^-3; i =^(6 + c) = $/(S + VV~») =i-l-Ay' -3 i and d —3; consequently, r=tf+d= 3;

and — ± — 3 = —1-±4. = — 1 and — 2, the two

less roots. So that all the three roots, in this example ofthe irreducible case, are rational.

66. Hence it appears, that Cardan’s rule, s + d, bringsout sometimes the greatest root, sometimes the middle root,and sometimes the least root.

VOL. II.

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