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:

(\frac{c}{a})^{\frac{1}{3}} + (\frac{c}{a})^{\frac{2}{3}} = -\frac{b}{a}.

The above is the desired solution.

🙂 🙂 🙂

Nalin Pithwa

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