2.1. Find the smallest value of $a$, for which the sum of the squares of the roots of the equation $x^{2}-3 a x+a^{2}=0$ is $0.28$.
Answer: $-0.2$ Solution. By Vieta's theorem $x_{1}^{2}+x_{2}^{2}=\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}=(3 a)^{2}-2 a^{2}=7 a^{2}$. According to the condition $7 a^{2}=\frac{28}{100}$, hence $a^{2}=\frac{1}{25}$ and $a= \pm 0.2$. It is important that for $a= \pm 0.2$ the discriminant $D=9 a^{2}-4 a^{2}=5 a^{2}>0$.