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 x1 and x2:

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


Video lesson

Solve the given equation using the quadratic formula

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