## Task A-2.2. (4 points)

Given are two quadratic equations

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

Determine all values of the parameter $a$ for which these equations have at least one common solution.
## Solution.

If we subtract the second equation from the first, we get

$$
(a-1) x+1-a=0 \quad, \text { i.e. } \quad(a-1)(x-1)=0
$$

In the case when $a=1$, the condition of the problem is obviously satisfied since the first and second equations coincide.

If $a \neq 1$, it must be $x=1$, and by substituting into the original equations, we find that $a=-2$ also satisfies the conditions of the problem.

(2 points)

Thus, the equations from the problem have at least one common solution for $a=-2$ and $a=1$.

Note. If the student writes one or both solutions without a valid explanation that there are no other solutions, they should receive 1 point for each solution.