Problem 9.1. Find all values of $a$ such that the equation

$$
\left(a^{2}-a-9\right) x^{2}-6 x-a=0
$$

has two distinct positive roots.

Ivan Landjev
9.1. The equation has two distinct positive roots if and only if

$$
\left\lvert\, \begin{aligned}
& D=a^{3}-a^{2}-9 a+9>0 \\
& x_{1}+x_{2}=\frac{6}{a^{2}-a-9}>0 \\
& x_{1} x_{2}=\frac{-a}{a^{2}-a-9}>0
\end{aligned}\right.
$$

The first inequality is satisfied for $a \in(-3,1) \cup(3,+\infty)$, the second one for $a \in((1-\sqrt{37}) / 2,(1+\sqrt{37}) / 2)$, and the third - for every $a \in(-\infty, 0)$. Therefore the required values of $a$ are $a \in(-3,(1-\sqrt{37}) / 2)$.