Question

If one root of the equation $a{x}^2+bx+c=0$ be the square of the other, then the relation of $a,b$ and $c$ is given by

Answer 1

Let the roots be $\alpha ,{\alpha }^2$
$\therefore \alpha +{\alpha }^2=-\frac{b}{a}$ and $\therefore \alpha .{\alpha }^2=\frac{c}{a}$
$\Rightarrow {(-\frac{b}{a})}^3={(\alpha +{\beta }^2)}^3$
$={\alpha }^3+{\alpha }^6+3.\alpha .{\alpha }^2(\alpha +{\alpha }^2)$
which gives $b^3+ac(a+c)3abc$