# How do you find the value of the discriminant and determine the nature of the roots 4x² – 8x = 3 ?

Feb 9, 2018

$\Delta = 112 > 0$ is not a perfect square, so this quadratic equation has two distinct real but irrational roots.

#### Explanation:

Given:

$4 {x}^{2} - 8 x = 3$

Subtract $3$ from both sides to get:

$4 {x}^{2} - 8 x - 3 = 0$

This is in the standard form $a {x}^{2} + b x + c = 0$, with $a = 4$, $b = - 8$ and $c = - 3$.

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

$\Delta = {b}^{2} - 4 a c = {\left(- 8\right)}^{2} - 4 \left(4\right) \left(- 3\right) = 64 + 48 = 112$

Since $\Delta > 0$ this quadratic has two distinct real roots.

Note however that $\Delta = 112$ is not a perfect square. Hence we can deduce that the roots are irrational.

In general, we find:

• If $\Delta > 0$ is a perfect square, then the quadratic equation has two distinct rational roots.

• If $\Delta > 0$ is not a perfect square, then the quadratic equation has two distinct real, but irrational roots.

• If $\Delta = 0$ then the quadratic equation has one repeated rational real root.

• If $\Delta < 0$ then the quadratic equation has no real roots. It has a complex conjugate pair of non-real roots. If $- \Delta$ is a perfect square then the imaginary coefficient is rational.