Question

If $x_1,x_2$ are the roots of the quadratic equation $a{x}^2+bx+c=0$ and $c\ne 0$, find the value of $(a{x}_1+b{)}^{-2}+(a{x}_2+b{)}^{-2}$ in term of $a,b,c$.

Answer 1

Given $x_1,x_2$ are the roots of $a{x}^2+bx+c=0⟹ax+b=-\frac{c}{x}$

$a{x}_1+b=-\frac{c}{x_1},a{x}_2+b=-\frac{c}{x_2}$

$⟹x_1+x_2=-\frac{b}{a},x_1{x}_2=\frac{c}{a}$

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

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