# How to use the discriminant to find out what type of solutions the equation has for 2m^2 - m - 6 = 0?

May 25, 2015

$2 {m}^{2} - m - 6$ is of the form $a {m}^{2} + b m + c$, with $a = 2$, $b = - 1$ and $c = - 6$.

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c$

$= {\left(- 1\right)}^{2} - \left(4 \times 2 \times - 6\right) = 1 + 48 = 49 = {7}^{2}$

$\Delta > 0$ so the equation $2 {m}^{2} - m - 6 = 0$ has two distinct real roots.

In addition, it is a perfect square, so those roots are rational.

The roots are given by the formula:

$m = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{1 \pm 7}{4}$

That is $m = - \frac{3}{2}$ and $m = 2$