# How do you determine if a solution to a quadratic equation is rational or irrational by using the discriminant?

Oct 30, 2014

Consider Quadratic Equation $a {x}^{2} + b x + c = 0$

the solutions for above quadratic equation are as below
$x = \frac{- b \pm \sqrt{D}}{2 a}$

Here, $D = {b}^{2} - 4 a c$

so, if $D > 0$, $\sqrt{D}$ is real and we have two real solutions viz., $x = \frac{- b + \sqrt{D}}{2 a}$ and $x = \frac{- b - \sqrt{D}}{2 a}$

If $D = 0$, $\sqrt{D} = 0$ and we have one real solution viz., $x = \frac{- b}{2 a}$

If $D < 0$, $\sqrt{D}$ is imaginary and we have two imaginary solutions viz., $x = \frac{- b + i \cdot \sqrt{|} D |}{2 a}$and $x = \frac{- b - i \cdot \sqrt{|} D |}{2 a}$