Find the coefficients of the equation $x^2 + px + q = 0$ such that its roots are equal to $p$ and $q$.
Given quadratic equation: $\;$ $x^2 + px + q = 0$
Roots of the given quadratic equation are $\;$ $p, \; q$.
Sum of roots $= p + q = -p$ $\;\;\; \cdots \; (1)$
Product of roots $= p \cdot q = q$ $\;\;\; \cdots \; (2)$
From equation $(2)$, when $q \neq 0$, $\;$ $p = 1$
Substituting $\;$ $p = 1$ $\;$ in equation $(1)$ gives
$1 + q = -1$ $\implies$ $q = -2$
When $\;$ $q = 0$, $\;$ we have from $(1)$
$p + 0 = -p$
i.e. $\;$ $2p = 0$ $\implies$ $p = 0$
$\therefore \;$ Values of $p$ and $q$ are
$p = 0, \; q = 0$ $\;\;$ OR $\;\;$ $p = 1, \; q = -2$