Quadratic formula

The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.

Statement

For any quadratic equation $ax^2+bx+c=0$ , the following equation holds. $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

We start with

$ax^{2}+bx+c=0$

Dividing by $a$ , we get

$x^{2}+\frac{b}{a}x+\frac{c}{a}=0$

Add $\frac{b^{2}}{4a^{2}}$ to both sides in order to complete the square:

$\left(x^{2}+\frac{b}{a}x+\frac{b^{2}}{4a^{2}}\right)+\frac{c}{a}=\frac{b^{2}}{4a^{2}}$

Complete the square:

$\left(x+\frac{b}{2a}\right)^{2}+\frac{c}{a}=\frac{b^{2}}{4a^{2}}$

Move $\frac{c}{a}$ to the other side:

$\left(x+\frac{b}{2a}\right)^{2}=\frac{b^{2}}{4a^{2}}-\frac{c}{a}=\frac{ab^{2}-4a^{2}c}{4a^{3}}=\frac{b^{2}-4ac}{4a^{2}}$

Take the square root of both sides:

$x+\frac{b}{2a}=\pm\sqrt{\frac{b^{2}-4ac}{4a^{2}}}=\frac{\pm\sqrt{b^{2}-4ac}}{2a}$

Finally, move the $\frac{b}{2a}$ to the other side:

$x=-\frac{b}{2a}+\frac{\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

This is the quadratic formula, and we are done.

In some situations, it is preferable to use this variation of the quadratic formula:

$\frac{2c}{-b\mp\sqrt{b^2-4ac}}$