11.41 For what value of a is the sum of the squares of the roots of the equation $x^{2}+a x+a-2=0$ the smallest?
11.41 Let $x_{1}$ and $x_{2}$ be the roots of the equation. Then $x_{1} x_{2}=a-2$, $x_{1}+x_{2}=-a$. From this, we can express the sum of the squares of the roots:

$$
x_{1}^{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}=a^{2}-2 a+4=f(a)
$$

Since $f(a)$ is a quadratic function with a positive leading coefficient, $f(a)$ attains its minimum value. This minimum value is achieved at $a=1$.

Answer: at $a=1$.