Quadratic Irrational is Root of Quadratic Equation

Theorem
Let $$x$$ be a quadratic irrational.

Then $$x$$ is a solution to a Quadratic Equation with rational coefficients.

Proof
Let $$x = r + s \sqrt n$$.

From Quadratic Equation, the solutions of $$a x^2 + b x + c$$ are:
 * $$x = \frac {-b \pm \sqrt {b^2 - 4 a c}} {2a}$$

given the appropriate condition on the discriminant.

So if $$x = r + s \sqrt n$$ is a solution, then so is $$x = r - s \sqrt n$$.

Hence we have:

$$ $$ $$

As $$r$$ and $$s$$ are rational and $$n$$ is an integer, it follows that $$-2 r$$ and $$r^2 - s^2 n$$ are also rational from the fact that rational numbers form a field.

Hence the result.