[Math] If the quadratic equation $x^2+px+q=0$ and $x^2+qx+p=0$ have a common root

Question

If quadratic equations $x^2+px+q=0$ and $x^2+qx+p=0$ have a common root, prove that: either $p=q$ or $p+q+1=0$.

My attempt:

Let $\alpha $ be the common root of these equations. Since one root is common, we know:
$$(q-p)(p^2-q^2)=(q-p)^2.$$
How do I get to the proof from here?

Answer 1

Let $x$ be the common root.

Thus, $$px+q=qx+p$$ or $$(x-1)(p-q)=0,$$ which gives $p=q$ or $x=1$ and from this we obtain $p+q+1=0$.