# How to use the discriminant to find out what type of solutions the equation has for 3x^2 - x + 2 = 0?

Mar 26, 2018

Zero roots

#### Explanation:

Quadratic formula is $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$
or
$x = - \frac{b}{2 a} \pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a}$

We can see that the only part that matters is $\pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a}$
as if this is zero then it says that only the vertex $- \frac{b}{2 a}$ lies on the x-axis

We also know that $\sqrt{- 1}$ is undefined as it doesn't exist so when ${b}^{2} - 4 a c = - v e$ then the function is undefined at that point showing no roots

Whilst if $\pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a}$ does exist then we know it is being plussed and minused from the vertex showing their are two roots

Summary:
${b}^{2} - 4 a c = - v e$ then no real roots
${b}^{2} - 4 a c = 0$ one real root
${b}^{2} - 4 a c = + v e$ two real roots

So
${\left(- 1\right)}^{2} - 4 \cdot 3 \cdot 2 = 1 - 24 = - 23$ so it has zero roots