2.78 For what values of a do the equations $x^{2}+a x+1=0$ and $x^{2}+x+a=0$ have a common root?
2.78 Subtracting the second equation from the first, we get

$$
a x - x + 1 - a = 0, \text{ or } (a-1)(x-1) = 0
$$

$$
x^{2} + x + 1 = 0
$$

such an equation has no real roots. If $a \neq 1$, then $x = 1$ and, therefore, $1 + a + 1 = 2$, i.e., $a = -2$.

Answer: when $a = -2$.