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

It is known that the equation $x^{2}+5 b x+c=0$ has roots $x_{1}$ and $x_{2}, x_{1} \neq x_{2}$, and some number is a root of the equation $y^{2}+2 x_{1} y+2 x_{2}=0$ and a root of the equation $z^{2}+2 x_{2} z+2 x_{1}=0$. Find $b$.
The common root of the equations $y^{2}+2 x_{1} y+2 x_{2}=0$ and $y^{2}+2 x_{2} y+2 x_{1}=0$ is also a root of their difference equation $2\left(x_{1}-x_{2}\right) y+2\left(x_{2}-x_{1}\right)=0$. Since $x_{1}-x_{2} \neq 0$, then $y-1=0$, that is, $y=1$. Substituting into the first equation, we get $1+2\left(x_{1}+x_{2}\right)=0$. Hence, $5 b=-\left(x_{1}+x_{2}\right)=1 / 2$.

## Answer

$b=1 / 10$.