6.124. Determine the coefficients of the quadratic equation

$x^{2}+p x+q=0$

so that its roots are equal to $p$ and $q$.
## Solution.

By Vieta's theorem $\left\{\begin{array}{l}p+q=-p, \\ p q=q\end{array} \Leftrightarrow\left\{\begin{array}{l}2 p+q=0, \\ q(p-1)=0 .\end{array}\right.\right.$

From the second equation of the system, we have $q=0$ or $p-1=0$. Then $q_{1}=0, p_{1}=0 ; p_{2}=1, q_{2}=-2 p_{2}=-2$.

Answer: $p_{1}=q_{1}=0 ; p_{2}=1, q_{2}=-2$.