What is a quadratic equation?

A quadratic equation is an equation of the form:

    equation

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

    equation

Quadratic formula

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

    equation

The equation sign denotes that there are two solutions equation and equation 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)   equation

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

(2)   equation

We found two solutions. Both equation and equation 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)   equation

We can see now that the left hand side will be zero when either equation or equation. We can easily solve these equations to find the two solutions for equation which are equation for the left term and equation 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 equation to the right hand side so that we only have terms with equation on the left side:

(4)   equation

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

(5)   equation

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

(6)   equation

We found the same solutions for equation which are equation and equation 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.