Quadratic formula

The quadratic formula is the formula for finding the solutions of quadratic equations. Students are sometimes taught a method known as factoring, however this is quickly tedious or impossible if possible solutions are nonreal or decimal. Using the quadratic equation, the values a, b, and c (considering the quadratic form ax2+bx+c) are substituted into the equation, then solved.

## Usage[edit]

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

Where a≠0 (if a were to equal zero, the equation would be linear). Then the coefficients a, b, and c can be substituted in the formula to find the solutions:

## Derivation[edit]

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

Now we need to get something of the form  that matches the first two terms. We have

So we need  to get a match.

Plugging that in, we get

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

## Application to Higher Degrees[edit]

The quadratic formula is most commonly used for solving quadratic equations, but can also be used for higher- or lower- degree powers in certain cases. This is achieved by replacing the x by another variable, for example x2.

For example, consider the equation:

Here, x is to the fourth and second powers. However, it is still possible to use the quadratic formula to solve this. If we suggest that:

Then we can substitute u into the equation, giving:

Note that x4 equals (x2)2. Then, using the quadratic formula, we have:

Keeping in mind that u is x squared, we have:

When simplified, we find:

This strategy can also be used for powers lower than 2, for instance in the equation:

In this case, we could consider u to equal x1/3. The solution would be found in the same way as the earlier example.