Write the Standard Form of an Equation by Completing the Square

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  • 0:03 A Quadratic Equation
  • 0:52 Why Complete the Square?
  • 1:44 Finding the Square
  • 3:37 Completing the Square
  • 4:52 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

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

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