9.2. For what values of $q$ does the quadratic equation $x^{2}-12 x+q=0$ have two distinct roots, one of which is 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, $a + a^{2} = 12$ and $a \cdot a^{2} = q$. Solving the quadratic equation for $a$, we find that $a = -4$ or $a = 3$. Since $q = a^{3}$, then $q = -64$ or $q = 27$.

## Criteria.

7 points. Correct solution with the correct answer.

2 points. Vieta's theorem is correctly applied and a system of two equations is formed.

1 point. Only the correct answer.