230. Given two equations

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

Find all values of $a$ for which these equations have at least one common root.
Instruction. From the second equation, we find $a$ and substitute it into the first:

$$
a=-(x^{2}+x), \quad x^{2}-(x^{2}+x) x+1=0, \quad x^{3}-1=0
$$

or

$$
(x-1)\left(x^{2}+x+1\right)=0
$$

Therefore,

$$
x_{1}=1 \text { and } x_{2,3}=\frac{-1 \pm i \sqrt{3}}{2}
$$

But

$$
a=-(x^{2}+x)
$$

thus,

$$
a_{1}=1, \quad a_{2}=-2 *
$$