9. (15 points) Find all values of the parameter $a$ for which the equation

$$
3 x^{2}-4(3 a-2) x+a^{2}+2 a=0
$$

has roots $x_{1}$ and $x_{2}$, satisfying the condition $x_{1}<a<x_{2}$.
Answer: $(-\infty ; 0) \cup(1,25 ;+\infty)$

Solution. $y(x)=3 x^{2}-4(3 a-2) x+a^{2}+2 a-$ is a parabola with branches upwards, so the solution to the inequality $y<0$ is the interval $\left(x_{1} ; x_{2}\right)$. Then

$$
a \in\left(x_{1} ; x_{2}\right) \Longleftrightarrow y(a)<0
$$

Solving the inequality, we get the answer.