# How to use the discriminant to find out what type of solutions the equation has for 5n² + 6n + 7 = n² - 4n?

Jun 6, 2015

First subtract ${n}^{2} - 4 n$ from both sides to get:

$4 {n}^{2} + 10 n + 7 = 0$

This is of the form $a {n}^{2} + b n + c = 0$, with $a = 4$, $b = 10$ and $c = 7$

The discriminant is given by the formula:

$\Delta = {b}^{2} - 4 a c = {10}^{2} - \left(4 \times 4 \times 7\right) = 100 - 112 = - 12$

Since $\Delta < 0$ the quadratic has no real solutions. It has two distinct complex roots.

In general, the possible cases are:

$\Delta = 0$ : Means the quadratic has one repeated real root. If the coefficients of the quadratic are rational, that repeated root is also rational.

$\Delta > 0$ : Means that the quadratic has two distinct real roots. If $\Delta$ is also a perfect square and the coefficients of the quadratic are rational, then those roots are also rational.

$\Delta < 0$ : Means that the quadratic has no real roots. It has two distinct complex roots which are complex conjugates of one another.