Question

If $x_1$ and$x_2$ are the roots of $a{x}^2+bx+c=0$, find the value of
(1) ${(a{x}_1+b)}^{-2}+{(a{x}_2+b)}^{-2}$
(2) ${(a{x}_1+b)}^{-3}+{(a{x}_2+b)}^{-3}$

Answer 1

Given that $x_1,x_2$ are roots of equation $a{x}^2+bx+c=0$

We have $x_1+x_2=-\frac{b}{a}$ and $x_1{x}_2=\frac{c}{a}$

1. ${(a{x}_1+b)}^{-2}+{(a{x}_2+b)}^{-2}={(-a{x}_2)}^{-2}+{(-a{x}_1)}^{-2}=\frac{1}{a^2}\left(\frac{x_1^2+x_2^2}{{\left(x_1{x}_2\right)}^2}\right)=\frac{b^2-2ac}{a^2{c}^2}$

2. ${(a{x}_1+b)}^{-3}+{(a{x}_2+b)}^{-3}={(-a{x}_2)}^{-3}+{(-a{x}_1)}^{-3}=-\frac{1}{a^3}\left(\frac{x_1^3+x_2^3}{{\left(x_1{x}_2\right)}^3}\right)=-\frac{1}{a^3}\left(\frac{b(3ac-b^2)}{c^3}\right)$