1. For what values of $a$ do the equations

$$
x^{2}+a x+1=0 \quad \text { and } \quad x^{2}+x+a=0
$$

have at least one common root?
Solution. Let $x_{0}$ be the common root of the equations. Substitute into the equations, equate the expressions for $x_{0}^{2}$.

$$
\begin{aligned}
& \left\{\begin{array}{l}
x_{0}^{2} + a x_{0} + 1 = 0, \\
x_{0}^{2} + x_{0} + a = 0
\end{array}\right. \\
& (a-1) x_{0} = a-1 \quad a x_{0} + 1 = x_{0} + a \\
&
\end{aligned}
$$

We get two cases. When $a=1$, the equations coincide, $x^{2} + x + 1 = 0$ has no roots. When $x_{0}=1$, by substitution we get $a=-2$.

Answer. When $a=-2$.