Question

If one root is square of the other root of the equation $x^2+px+q=0$, then find the relation between $p$ and $q$.

Answer 1

Let $\alpha$ and ${\alpha }^2$ be the roots of the given equation $x^2+px+q=0$
$\Rightarrow \alpha +{\alpha }^2=-p=\alpha (1+\alpha )$ and $\alpha .{\alpha }^2=q={\alpha }^3$
Now $\left(\alpha \right(1+\alpha ){)}^3=(-p{)}^3$
$\Rightarrow {\alpha }^3[1+{\alpha }^3+3\alpha (1+\alpha \left)\right]=-p^3$
$\Rightarrow q[1+q-3p]=-p^3$
$\Rightarrow p^3-(3p-1)q+q^2=0$,
which is the required relations between $p$ and $q$.