Task B-2.3. Without solving the quadratic equation $x^{2}+2 x+2=0$, calculate the value of the expression $\frac{x_{1}^{3}+x_{2}^{3}}{\left(x_{1}-x_{2}\right)^{2}}$, where $x_{1}, x_{2}$ are the solutions of the given equation.
## Solution.

$$
\begin{gathered}
x_{1}+x_{2}=-\frac{b}{a}=-2, \quad x_{1} \cdot x_{2}=\frac{c}{a}=2 \\
\frac{x_{1}^{3}+x_{2}^{3}}{\left(x_{1}-x_{2}\right)^{2}}=\frac{\left(x_{1}+x_{2}\right)^{3}-3 x_{1} x_{2}\left(x_{1}+x_{2}\right)}{\left(x_{1}+x_{2}\right)^{2}-4 x_{1} x_{2}}= \\
=\frac{(-2)^{3}-3 \cdot 2 \cdot(-2)}{(-2)^{2}-4 \cdot 2}=\frac{-8+12}{4-8}=-1
\end{gathered}
$$