TRACT 28.

AND INFINITE SERIES.

95

v — w — s/{r'- — —); which being added to, and taken fromv + w = — r, and dividing by 2 , we have ” j- = — fr ±

iVi 1 '"" — j) = - jr x [ 1 ± \/{l - ^)], the same with one

of the formulae above given; and then by substitution theothers will be obtained.

56. To illustrate now the rules x — s + d, or— -j- ± ■</ ~ 3 > by some examples; suppose the given

equation to be .r 3 36.r = 91. Here p — — 36, q ~ 91,

a = j-p = — 12, b — —; then c = ^/{]? + a 3 ) =

✓(T - 1728 ) = S™ = ?>'=W + c >=Mt + t) =

1/6 4 = 4 , and d = 3 /[b -c)= ^ - j -) = 3 /27 = 3. Con-sequently, r = 5 + cl = 4 + 3= 7 the first root; and

— Lti? + »/ _ 3 — ~ —? th e other two roots, which

2 2 v 2 5

are imaginary.

57. Ex. 2. Let the equation be x % + 3CXr 117.Here a = ^p = 10, b — \q — ^; then c = ^(/j* + a 3 ) =

\/(—*— + 1000) = -v/— = —, 5 = ftb + c) =

+-qf) = = ^125 = 5, and d = %/(b - c) =

133 ,

250

-^) = v' —= $/- 8 = - 2 . Consequently, ?’ =i + rf=5 — 2 = 3 the first root; and — s -~ ± s -^- */ — 3 =——— the other two roots, which are imaginary.

58. Ex. 3. If the equation be .r 3 + 1 Sjp = 6, we shall havea — 6, and b — 3 ; then c — + a 3 ) = */{9 + 216) =

1/ 225 = 15, s = %/(b + c) = 1/(3 + 15) =$/18, and d =y/(b — c) = £/(3 — 15) = l/— 12 = — v/12. Therefore= s + d = ^18 — v/12 = '331313 the first root; and