# The Quadratic formula

Instead of solving a quadratic equation by completing the squares (shown in algebra 1) we could solve any quadratic equation by using the quadratic formula.

$$\\ If\; ax^{2}+bx+c=0\; and\; a\neq 0\; then\\ \\ x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$$

A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.

**Example**

Solve the following equation using the quadratic formula:

$$x^{2}+x-2=0$$

First we identify our a, b and c:

$$\left\{\begin{matrix} a=1\\ b=1\\ c=-2 \end{matrix}\right.$$

Then we plug our values into the quadratic formula to determine our x:

$$x=\frac{-1\pm \sqrt{1^{2}-(4\cdot 1\cdot -2)}}{2\cdot 1}=$$

$$=\frac{-1\pm \sqrt{1+8}}{2}=$$

$$=\frac{-1\pm 3}{2}$$

From here we can determine our \(x_1 \) and \(x_2\):

$$\\ x_{1}=\frac{-1+3}{2}=1\\ \\ x_{2}=\frac{-1-3}{2}=-2\\$$

**Video lesson**

Solve the given equation using the quadratic formula

$$x^{2}+2x-8=0$$