# How to use the discriminant to find out what type of solutions the equation has for 4/3x^2 - 2x + 3/4 = 0?

May 26, 2015

$\frac{4}{3} {x}^{2} - 2 x + \frac{3}{4} = 0$ is of the form $a {x}^{2} + b x + c = 0$ with $a = \frac{4}{3}$, $b = - 2$ and $c = \frac{3}{4}$.

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(- 2\right)}^{2} - \left(4 \times \left(\frac{4}{3}\right) \times \left(\frac{3}{4}\right)\right)$

$= 4 - 4 = 0$

Since $\Delta = 0$, the quadratic has one repeated rational root.

The possible cases are:

$\Delta < 0$ The quadratic has no real roots. It has two complex roots that are conjugates of one another.

$\Delta = 0$ The quadratic has one repeated root. If the coefficients of the quadratic are rational then that repeated root is rational too.

$\Delta > 0$ The quadratic has two distinct real roots. If $\Delta$ is a perfect square and the coefficients of the quadratic are rational then those roots are rational too.