# How to use the discriminant to find out what type of solutions the equation has for x^2 = 0?

May 23, 2015

${x}^{2} = 0$ is of the form $a {x}^{2} + b x + c = 0$, with $a = 1$, $b = 0$ and $c = 0$.

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c = {0}^{2} - \left(4 \times 1 \times 0\right) = 0 - 0 = 0$

This means that ${x}^{2} = 0$ has one repeated real root and because $0 = {0}^{2}$ is a perfect square, that repeated root is rational.

The possible cases are:

$\Delta < 0$ No real roots. Two distinct complex roots.
$\Delta = 0$ One repeated, rational, real root.
$\Delta > 0$ Two distinct real roots. If $\Delta$ is also a perfect square then those roots are rational.

Severe overkill that it is, you can apply the standard quadratic solution formula to find the (repeated) solution as follows:

$x = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{- 0 \pm \sqrt{0}}{2 \times 1} = \frac{0}{2} = 0$