'S'RACT 28.

And infinite series.

9 *

63. Ex. 8. Again, from the equation x 3 — 12r =— 8v"2,we have «rr-4, and b = — 4^2; hence c = v '(6 l + « 3 ) =

V (32 — 64) =^/~ 32=4 V r -2,^ = V(* + t) = v'(~ 4fi ' /2+4V-2)=v'Hv'" 2 i and d — <J2—*/—2. So that

r =: s +d=2V2 the middle root; and —± —3 =

— \Z2±^/ — 2 . V- 3 = - v / 2±v' 6=-V2.(lq:v , 3) thegreatest and least roots.

64. Ex. 9. But the equation .y 3 — I5x = 22 gives a— — 5,and b — II; therefore c = \/{b z + a 3 ) =*/(i21 — 125) =*/ — 4, s = 3 /{b + c) = ^/( 11 + ^/ — 4) = — 1 — y' — 4, andd = — 1 h \/ — 4. Consequently r = s + d = — 2 the least

root; and - ± — 3 = l±^/ — 4 . „/ — 3 = 1± V / T2

the two greater roots.

65. Ex. 10. Further, in the equation x 3 — 15x = 20, wehave a =— 5, and 6 = 10 ; consequently c — */(b z + a 3 ) —V(i00 — 125) = */ — 25 = 5«/ — 1, s = 3 /{b + c)-t/{\0 +5\/ —1), and d =^/(l0— 5</ — l). Therefore r = s + d —-v/(10 -f- 5«/ — l) +%/(l0 — 5a/— 1) = the first root; and

_ i+i Mio + V-'O+Vdo-V-i) j_

a — 2 v o — 3 —

---—- - ^/ — 3 — the other two roots.

*65. Ex. 11. Lastly, taking the equation x 3 —lx = 6. Here« = -|, and b — 3 ; therefore c = s/(b 1 + a 3 } = v'i'9 — ^) =

V - V° T ° = v° ^ - 3 ; « =; ^ + c) = tfS + VV -3) =

f+-|V , ~ 3 > and d =-J—|V — 3; consequently, r=s-{-d=z3;

an( l ~~~T~ — ~Z~V~ 3 = “t — r = — 1 and — 2, the two

less roots. So that all the three roots, in this exaipple 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,