# How do you describe the roots of 4x^2 - 3x + 2 = 0?

Jan 30, 2016

A conjugate pair of Complex roots.

#### Explanation:

$4 {x}^{2} - 3 x + 2$ is of the form $a {x}^{2} + b x + c$ with $a = 4$, $b = - 3$ and $c = 2$.

This has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(- 3\right)}^{2} - \left(4 \cdot 4 \cdot 2\right) = 9 - 32 = - 23$

Since $\Delta$ is negative, the quadratic equation has no Real roots. It has a pair of Complex roots, conjugate to one another, given by the quadratic formula:

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

$= \frac{- b \pm \sqrt{\Delta}}{2 a}$

$= \frac{3 \pm \sqrt{- 23}}{2 \cdot 4}$

$= \frac{3}{8} \pm \frac{\sqrt{23}}{8} i$

where $i$ is the imaginary unit.