Definition:Quadratic Irrational/Reduced

Definition
An irrational root $\alpha$ of a quadratic equation with integer coefficients is a reduced quadratic irrational
 * $(1): \quad \alpha > 1$
 * $(1): \quad$ 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.

Also see

 * Definition:Quadratic Equation
 * Quadratic Irrational is Root of Quadratic Equation