2.38 Without solving the equation $a x^{2}+b x+c=0$, find $x_{1}^{-2}+x_{2}^{-2}$, where $x_{1}$ and $x_{2}$ are the roots of this equation.
2.38 After performing the transformations and using Vieta's theorem, we get

$$
\begin{aligned}
& \frac{1}{x_{1}^{2}}+\frac{1}{x_{2}^{2}}=\frac{x_{1}^{2}+x_{2}^{2}}{\left(x_{1} x_{2}\right)^{2}}=\frac{\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}}{\left(x_{1} x_{2}\right)^{2}}= \\
& =\frac{\frac{b^{2}}{a^{2}}-\frac{2 c}{a}}{\frac{c^{2}}{a^{2}}}=\frac{b^{2}-2 a c}{c^{2}}
\end{aligned}
$$

Result: $\frac{b^{2}-2 a c}{c^{2}}$.