Roots of Quadratic with Rational Coefficients of form r plus s Root 2

Theorem
Consider the quadratic equation:
 * $(1): \quad a^2 x + b x + c = 0$

where $a, b, c$ are rational.

Let $\alpha = r + s \sqrt 2$ be one of the roots of $(1)$.

Then $\beta = r - s \sqrt 2$ is the other root of $(1)$.

Proof
We have that:

Because $a$, $b$, $c$, $r$ and $s$ are rational, it must be that $\paren {2 a + b} s = 0$.

Hence:

and so $\beta$ is also a root of $(1)$.