Condition for Commensurability of Roots of Quadratic Equation

Theorem
Consider the quadratic equation:
 * $(1): \quad a x - x^2 = \dfrac {b^2} 4$

Then $x$ and $a - x$ are commensurable $\sqrt{a^2 - b^2}$ and $a$ are commensurable.

Proof
We have that:

Let $a \frown b$ denote that $a$ is commensurable with $b$.

Necessary Condition
Let $\paren {a - x} \frown x$.

From Commensurability of Sum of Commensurable Magnitudes:
 * $a \frown x$

From Magnitudes with Rational Ratio are Commensurable:
 * $x \frown 2 x$

So:

Sufficient Condition
Let $a \frown \sqrt {a^2 - b^2}$.

Then:

From Magnitudes with Rational Ratio are Commensurable:
 * $x \frown 2 x$

From Commensurability is Transitive Relation:
 * $a \frown x$

From Commensurability of Sum of Commensurable Magnitudes:
 * $\paren {a - x} \frown x$