Quadratic formula

The quadratic formula can be used when simpler methods of solving a quadratic equation do not work.

First, the quadratic equation must be reduced to this format:

<math>ax^2+bx+c=0\!</math>

Then the coefficients a, b, and c can be substituted in the formula to find the solutions:

<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

You can prove the formula the following way:

<math>ax^2+bx+c=0\!</math>

<math>x^2+\frac{b}{a}x+\frac{c}{a}=0\!</math>

<math>(x+\frac{b}{2a})^2-(\frac{b}{2a})^2+\frac{c}{a}=0\!</math>

<math>(x+\frac{b}{2a})^2=(\frac{b}{2a})^2-\frac{c}{a}\!</math>

<math>(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\!</math>

<math>x+\frac{b}{2a}=\frac{\pm \sqrt {b^2-4ac}}{2a}\!</math>

<math>x = \frac{-b \pm \sqrt {b^2-4ac}}{2a}</math>

This method of deriving the formula is done via completing the square.

You can assert that the formula is correct by substituting the formula in place of x in <math>ax^2+bx+c=0\!</math> and then gradually simplifying the rather complicated formula that results, step by step. Eventually, if all the steps are done correctly, it will simplify to 0.