2.42 Find all values of $a$ for which the sum of the roots of the equation $x^{2}-2 a(x-1)-1=0$ is equal to the sum of the squares of the roots.
2.42 We have $x^{2}-2 a x+(2 a-1)=0$ and $x_{1}+x_{2}=x_{1}^{2}+x_{2}^{2}$, i.e., $x_{1}+x_{2}=\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}$. According to Vieta's theorem, $2 a=4 a^{2}-4 a+2$, or $2 a^{2}-3 a+1=0$, from which $a_{1}=1$, $a_{2}=\frac{1}{2}$.

Answer: $a_{1}=1 ; a_{2}=0.5$.