Definition:Quadratic Irrational

Definition
A quadratic irrational is an irrational number of the form:
 * $r + s \sqrt n$

where $r, s$ are rational and $n$ is a positive integer which is not a square.

Reduced Form
An irrational root $\alpha$ of a quadratic equation with integer coefficients is called reduced if $\alpha > 1$ and its conjugate $\tilde{\alpha}$ satisfies $-1 < \tilde{\alpha} < 0$.

Solutions of such quadratics can be written as:
 * $\alpha = \dfrac{\sqrt D + P} Q$

where $D, P, Q \in \Z$ and $D, Q > 0$.

It is also possible (though not required) to ensure that $Q$ divides $D - P^2$.

This is actually a necessary assumption for some proofs and warrants its own definition.

As an example for some enlightenment, notice $\alpha = \dfrac{2 + \sqrt 7} 4$ is reduced but $4$ does not divide $7 - 2^2$.

However, if we write this as $\dfrac{8 + \sqrt{112}}{16}$, we have our desired condition.

Association
We say a reduced quadratic irrational $\alpha$ is associated to $D$ if we can write $\alpha = \dfrac {P + \sqrt D} Q$ and $Q$ divides $D - P^2$.

Also see

 * Quadratic Equation
 * Quadratic Irrational is Root of Quadratic Equation