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.

### General Solution For A Quadratic by Completing the Square $$ax^{2}+bx+c=0$$

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

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.

### Variation

In some situations, it is preferable to use this variation of the quadratic formula: $$\frac{2c}{-b\pm\sqrt{b^2-4ac}}$$