A quadratic equation question for pRMO or preRMO

Question:

Find the necessary and sufficient condition that the quadratic equation $ax^{2}+bx+c=0$ where $a \neq 0$ has one root which is the square of the other.

Solution:

Let the two roots of the given quadratic equation $ax^{2}+bx+c=0$ , with $a \neq 0$ be $\alpha$ and $\beta$ such that $\beta = \alpha^{2}$ .

Then, we know $\alpha+\beta=-\frac{b}{a}$ and $\alpha\beta=\frac{c}{a}$ so that $\alpha+\alpha^{2}=-\frac{b}{a}$ and $\alpha^{3}=\frac{c}{a}$ . From the latter relation, we get that $\alpha = (\frac{c}{a})^{\frac{1}{3}}$ . Substituting this in the first relation of sum of roots, we get the following necessary and sufficient condition: