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

Let $\alpha$ be the common root. 
\begin{align}
   \alpha^2 + p \alpha + q &= 0 \\
   \alpha^2 + q \alpha + p &= 0 \\
\hline
   (p-q)\alpha + (q-p) &= 0 \\
   (p-q)(\alpha-1) &= 0
\end{align}
$\alpha = 1$ or $p=q$.
If $\alpha = 1$, then $\alpha^2 + p \alpha + q = 0$ becomes
$1+p+q = 0$
