## What is a quadratic equation?

A quadratic equation is an equation of the form:

where , and are coefficients. This is a polynomial of second degree. Solving the equation means to find all values of such that the left hand side equals some value which often is zero.

## Solving quadratic equations

Quadratic equations can be solved in different ways:

- Quadratic formula
- Factoring
- Completing the square

In the following we explain these different ways and demonstrate them with the following example which we want to solve:

### Quadratic formula

The quadratic formula can be used to solve any quadratic equation. The formula is as follows:

The sign denotes that there are two solutions and where one of them is the above equation with a plus and the other one with a minus.

In our example, our coefficients are:

(1)

Now we substitute them into the quadratic formula and solve it:

(2)

We found two solutions. Both and solve our example equation.

### Factoring

Some equations can be easily solved by factoring such as our example. Let us factor the equation (first we divide it by 2):

(3)

We can see now that the left hand side will be zero when either or . We can easily solve these equations to find the two solutions for which are for the left term and for the right one.

### Completing the square

For a quadratic equation where we can’t easily use factoring, we can *complete the square* so that it becomes a square trinomial. Let us look at our example and how we need to transform it to make the left hand side a square trinomial. We first bring to the right hand side so that we only have terms with on the left side:

(4)

To make the left hand side a square trinomial we must add to it. In this case =3, so we add to both sides and factor the left side:

(5)

Next, we solve the equation by taking the square root and solving for :

(6)

We found the same solutions for which are and as we found with the other ways. You can use any of the three presented ways to solve a quadratic equation. Of course you should know all of them in case your teacher asks you to use a specific way. But you might have a preferred way that you personally find the easiest and you can also use it to confirm your solution that you found with any of the other ways.