2. For what values of $q$ is one of the roots of the equation $x^{2}-12 x+q=0$ the square of the other?
Answer: -64 or 27.

Solution. Let the roots of the equation be $a$ and $a^{2}$. By Vieta's theorem, $a + a^{2} = 12$, $a \cdot a^{2} = q$. Therefore, $a = -4$ or $a = 3, q = a^{3}$.

Criteria. Only the answer - 1 point. In the case of a complete justification, the score is 7 points.