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
- Completing the square
In the following we explain these different ways and demonstrate them with the following example which we want to solve:
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:
Now we substitute them into the quadratic formula and solve it:
We found two solutions. Both and solve our example equation.
Some equations can be easily solved by factoring such as our example. Let us factor the equation (first we divide it by 2):
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:
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:
Next, we solve the equation by taking the square root and solving for :
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.