Question

(i)If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$ ,express the roots of the equation $a^3{x}^2-a{b}^{2}x+b^{2}c=0$ in the terms of $\alpha$ and $\beta$.

Answer 1

Roots of $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$
$\alpha ,\beta =\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
Roots of $a^3{x}^2-a{b}^{2}x+b^{2}c=0$ is
$\begin{array}{c}x=\frac{-(-a{b}^2)\pm \sqrt{a^2{b}^4-4\left(a^3\right)\left(b^{2}c\right)}}{2a^3}\\ x=\frac{a{b}^2\pm ab\sqrt{b^2-4ac}}{\left(2a\right)a^2}\\ x=\frac{-ab(-b\pm \sqrt{b^2-4ac})}{\left(2a\right)a^2}\\ x=\frac{-ab}{a^2}\left(\alpha or\beta \right)\end{array}$
Roots of new equation are $\frac{-ab\alpha }{a^2}$ and $\frac{-ab\beta }{a^2}$