10.2. Find all pairs of real values $a$ and $b$, for which both equations $x^{2} + a x + b^{2} = 0$ and $x^{2} + b x + a^{2} = 0$ have at least one common root.
Answer. $a=b=0$.

Solution. For the equations to have a common root, it is necessary that each of them has roots, that is, their discriminants must be non-negative. Therefore, $a^{2} \geq 4 b^{2}, b^{2} \geq 4 a^{2}$, from which $a^{2} \geq 16 a^{2}, b^{2} \geq 16 b^{2}$, which is possible only when $a=b=0$. In this case, both equations have a common root of 0.

Grading criteria. The statement about the necessity of the inequalities $a^{2} \geq 4 b^{2}, b^{2} \geq 4 a^{2}$ being satisfied: 3 points. The explicit check that when $a=b=0$ the equations indeed have a common root equal to 0 is missing: deduct 1 point.