If $r$ is a nonzero solution $ x^2 + ax + b$, prove that $r | b$

Question

I know that if $r$ is a solution, then there exist two factors of $b$ that when multiplied equal $b$ and that $r$ is one of them. So clearly $r$ divides $b$, but I don't know if there is any other way to formally prove it?

Answer 1

Assuming that $a$, $b$ and $r$ are integers, it is true. $r$ is a nonzero solution of $x^2 + ax + b$, that is, $r^2 + ar + b = 0$ or $b = -r(r+a)$.

But if either of $b$ or $r$ are not integers, "$r|b$" is nonsense, and if $a$ is not an integer it may or may not be true. For example, $x^2 + 2.5x + 1$ has a solution $r=-2$, and $-2$ does not divide $b=1$.