Find $p$ and $q$ in $x^2-px+q=0$

Question

If $p$ and $q$ are the roots of the equation $x^2-px+q=0,\ \{x,p,q\}\in\mathbb{R} $, then find $p$ and $q$.

I tried sum and product of the roots formula and got ,

$$p+q=p \\pq=q$$

I found $q=0$ ,but I am confused on how to find $p$ since I cannot divide $q$ when $q=0$.

I look for a short and simple way.

I have studied maths up to $12$th grade.

Answer 1

Go back to the initial equation: $x^2 - px + 0 = x(x-p) = 0$ has roots $p$ and $0$ for all $p \in \mathbb{R}$. So it works for any $p$.