1. If $x_{1}$ and $x_{2}$ are solutions to the equation $x^{2}+k x+1=0$, determine those values of $k$ for which the inequality

$$
\left(\frac{x_{1}}{x_{2}}\right)^{2}+\left(\frac{x_{2}}{x_{1}}\right)^{2}>2
$$

holds.
Solution. From Vieta's formulas, it follows that $x_{1}+x_{2}=-k, x_{1} x_{2}=1$. Further:

$$
\begin{aligned}
\left(\frac{x_{1}}{x_{2}}\right)^{2}+\left(\frac{x_{2}}{x_{1}}\right)^{2} & =\frac{x_{1}^{4}+x_{2}^{4}}{\left(x_{1} x_{2}\right)^{2}}=\left(\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}\right)^{2}-2 x_{1}^{2} x_{2}^{2} \\
& =\left(k^{2}-2\right)^{2}-2=k^{2}\left(k^{2}-4\right)+2
\end{aligned}
$$

From the previous equalities, it follows that inequality (1) holds if and only if $k^{2}>4$, i.e., if and only if $k \in (-\infty,-2) \cup(2,+\infty)$.