13.189. The sum of the squares of the roots of the equation $x^{2}-3 a x+a^{2}=0$ is 1.75. Find the value of $a$.
Solution.

Solve the equation with respect to $x: x^{2}-3 a x+a^{2}=0$;

$D=9 a^{2}-4 a^{2}=5 a^{2} . x_{1}=\frac{3 a-a \sqrt{5}}{2} ; x_{2}=\frac{3 a+a \sqrt{5}}{2}$. According to the condition

$\frac{a^{2}}{4}(3-\sqrt{5})^{2}+\frac{a^{2}}{4}(3+\sqrt{5})^{2}=1.75$, from which $a^{2}=\frac{1}{4}, a= \pm 0.5$.

Answer: $\pm 0.5$.