[ Quadratic equations. Vieta's theorem ]  [ Methods for solving problems with parameters ]

For what value of the parameter $m$ is the sum of the squares of the roots of the equation $x^{2}-(m+1) x+m-1=0$ the smallest?
The sum of the squares of the roots $\left(x_{1}+x_{2}\right)^{2}-2 x_{1} x_{2}=(m+1)^{2}-2(m-1)=m^{2}+3$ is minimal when $m=0$. Note that in this case, the equation

$x^{2}-x-1=0$ has roots.

## Answer

When $m=0$.