In math, we have a process called completing the square where you take your quadratic equation and rewrite it to make it easier to solve. Watch this video lesson to learn how you can do this process easily yourself.
A Quadratic Equation
Many things in math start off with a quadratic equation. And the process of completing the square is no exception. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are numbers and x is your variable.
The process known as completing the square changes the quadratic equation to the form a(x + h)^2 + k = 0, where a, h, and k are numbers and x is the variable. A good way to remember this process is to think of the square. You want to rewrite the equation so that instead of having your variable in two places, you only have your variable once inside the parentheses that are squared.
Why Complete the Square?
Why would you want to complete the square? It's much easier to solve a quadratic when your variable is in one spot only, instead of two. Look at the form of the equation after we have completed the square: a(x + h)^2 + k = 0. It's much easier to solve this equation for x than it is if we left it in our quadratic equation form.
We can easily solve our completed square equation by moving the k over and then dividing by a, followed by taking the square root and then moving the h to get x by itself. To solve the quadratic equation, we'd have to use the quadratic formula, x = (-b +/- square root of (b^2 - 4ac))/(2a), which is not as easy.
Finding the Square
So, what is this process of completing the square? The first step is to find our square. This is not as hard as it sounds. Let's go through this process by completing the square for the quadratic equation 2x^2 + 4x + 6 = 0.
First, we move the 6 over so that all our variables are on one side and our constants on the other. Because our 6 is being added, we will subtract the 6 from both sides to move it.
2x^2 + 4x + 6 - 6 = 0 - 6
2x^2 + 4x = -6
Now we are going to divide the whole equation by 2, meaning we are going to divide each term by 2 so that we have our x^2 by itself.
2x^2/2 + 4x/2 = -6/2
x^2 + 2x = -3
This is the part where we find the square. We want the left side of our equation to look like x plus something squared: (x + something)^2. So, the question we need to ask ourselves is what number do we need to add to the left side so that when we factor the left side we get (x + something)^2?
To find this number, we divide the number next to the x by 2 and then square it. The number we have next to the x is a 2, so we will divide this by 2 and then square it. If our sign in front of our 2 was a minus, we would be working with a -2. So, keep an eye on your sign. Doing this, we get 2/2 = 1 and 1^2 = 1. So the number we need to add is a 1. We have found our square.
Completing the Square
Now, to complete our square, I add this 1 to both sides.
x^2 + 2x + 1 = -3 + 1
I can now factor the left side of this equation to get (x + 1)^2. The number I use inside the parentheses is the number I get when I divide my number next to the x by 2. Remember to keep the sign in front of the 2 when you do this.
(x + 1)^2 = -3 + 1
(x + 1)^2 = -2
To get it into the form I want, my last and final step is to move the number on the right side to the left. My number on the right is a -2, so I add a 2 to both sides.
(x + 1)^2 + 2 = -2 + 2
(x + 1)^2 + 2 = 0
And I have my answer of (x + 1)^2 + 2 = 0, and I am done! I went from my quadratic equation of 2x^2 + 4x + 6 = 0 to my completed square equation of (x + 1)^2 + 2 = 0.
Give yourself a pat on the back for a job well done! Now you can test yourself with quiz questions to see if you've really understood this process. But before we do that, let's do a quick review.
We've learned that a quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are numbers and x is your variable, and the process known as completing the square changes the quadratic equation to the form a(x + h)^2 + k = 0, where a, h, and k are numbers and x is the variable.
We perform the process of completing the square because it gives us an equation that is easier to solve. The process itself involves finding the number that will allow us to rewrite our equation so that our variable is only written once inside a pair of parentheses squared. To do this, we first move our constant c over to the right. We then divide all our terms by our number a. Then we divide our number b by 2 and then square it to find the number to complete the square. We then add this number to both sides, factor the left side, and move the number on the right back to the left to get our final completed square equation.
Once you've completed this lesson you should be able to:
- Identify a quadratic equation
- Apply the completing the square process to more easily solve a quadratic equation