Quadratic formula

The quadratic formula is the formula for finding the solutions of quadratic equations. Students are sometimes taught a method known as factoring, but that's really just looking for a lucky guess.

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>

The formula is derived in the following way, which is known as completing the square:

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

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

Now we need to get something of the form <math>(x+Q)^2\!</math> that matches the first two terms. We have

<math>(x+Q)^2 = x^2 + 2Qx + Q^2\!</math>

So we need <math>Q = \frac{b}{2a}\!</math> to get a match.

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

Plugging that in, we get

<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>

You can check that the formula is correct by substituting the formula (with either sign for the square root) 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.