4* % Dr . Hutton on Cubic Equations 
given equation to be x l — 36a: = 91. Here p — —36, 
q = 91, a = Jp ~ 1 2, b - then c = \/ b L + <2 3 = 
- 1728 = V 7 ^ - 12, j ■■= v'iTt - ^21 + 22 = 
4 7 4 2 7 .22 
^64 = 4> and d - \/ b - c - v 7 ^- — — — v 7 27 =• 3. 
Confequently, r = r + c/ = 4 + 3 = 7 the fir ft root; and 
_ i±i;dt J — v 7 — 3= — — ' / ~ 3 the other two roots, which 
are imaginary. 
57. Ex. 2. Let the equation be a; 3 + 30a? = 117. 
Here a-\p- 10, b = -t? = ^ 7 ; then c = ^ b 2 + a z 
= v/!25i9 + 1QOO - yiz««9 _ 03 J = ^ c + ill 
4 4 2 2 2. 
= ^ = ^125 = 5,and^ = &b - c = ^ = d/-L s 
= = -2. Confequently, r^r + r/ = 5- 2 = 3 the 
firft root; and - ^ ± - ~ 3 the. other 
two roots, which are imaginary. 
58. Ex. 3. If the equation be at 3 + 18 x = 6, w r e fhall 
have a = 6, and b — 3; then c — vV + « 3 = + 216 
= ^225 = 15, j = v 7 ^ + c — v 7 3 + 15 = \/i 8 , and 
d-s/b-c-s/^- 15 = v 7 - 1 2 = - \/i 2. Therefore 
r - s + d- -y 18 - \/i 2 = *331313 the firft root; and 
J + t ± v/_ ~ = _ fit -# 1 * + ^.8+^12 ./_ 
22^ 2 2 ** 
the other two roots. 
59. Ex. 4. In the equation a: 3 - 15^ = 4, we have 
a - - 5, b =: 2 ; hence r s= */ b 2 + a? - ^4 - 125 = 
■V 7 "* 1 21 
