Question

If alpha and beta are the roots of ax2+bx+c=0,then (alpha ÷a beta+b)^3-(beta÷a alpha +b)^3

Answer 1

ax² + bx + c =0

alpha + beta = -b/a
a.alpha + a.beta = -b
a.alpha + b = -a.beta ------(1)
in the same way ,
a.beta + b = -a.alpha -----(2)
alpha.beta = c/a
a.alpha.beta = c ----------(3)

={alpha /( a.beta + b)}³ -{ beta /( a.beta +b)}³
put equation (1)and (2) here

={ alpha/( -a.beta)}³ -{ beta/( -a.alpha)}³

=-alpha/a³.beta³ + beta /a³alpha³

=(-alpha⁴ + beta⁴)/(a.alpha.beta)³

=(beta² -alpha²)(beta²+alpha²)/(a.alpha.beta)³

=(beta -alpha)(alpah-beta)(b²/a² -2c/a)/(c)³

beta -alpha =±√(b²/a² -4c/a)

use this ,

=(±√(b²/a² -4c/a))(-b/a)(b²/a² -2c/a)/c³