6.250. For what values of $a$ do the equations $x^{2}+a x+1=0$ and $x^{2}+x+a=0$ have a common root?
Solution.

If for some $a$ and $x$ the left parts of the equations are equal to 0, then they are equal to each other:

$$
\left\{\begin{array}{l}
x^{2}+a x+1=0 \\
x^{2}+x+a=0
\end{array}\right.
$$

Subtracting the second equation of the system from the first, we get

$a x-x+1-a=0,(a-1) x-(a-1)=0,(a-1)(x-1)=0$, from which:

1) if $a-1=0$, then $a=1$ and each equation will take the form $a^{2}+a+1=0$, i.e., it has no roots;
2) if $a-1 \neq 0$, then $x=1$, from which $1+a+1=0$, i.e., $a=-2$.

Answer: $a=-2$.