Solving Quadratic Equations by Completing the Square

Instructor: Yuanxin (Amy) Yang Alcocer

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

After reading this how to lesson, you'll be well on your way to using the completing the square method of solving quadratic equations. Learn what numbers you need to focus on to find your missing square.

The Steps

Say you are trying to find the solutions to this problem:


completing the square


You would set your y equal to 0 and then proceed to find the solutions. If you aren't able to easily factor this quadratic equation, then you can use the method that is called completing the square. In this method, you manipulate your equation so you end up with one squared part that equals a number. This way, you can easily find your two solutions.

Here are the steps to solve a quadratic by completing the square.

Step 1: Set your equation to 0.

Whenever a problem asks you to find the solutions or x-intercepts, it means that you need to set your equation equal to 0 (i.e. set y = 0).


completing the square


Step 2: Move your single constant to the other side.

You want just your variables on the left and your numbers on the right. In our example, this means we move the 8 over to the other side. We can do so by adding it to both sides since it is being subtracted. Remember, when moving terms from one side to the other, you always perform the opposite or inverse operation.


completing the square


Step 3: Divide by the coefficient of the squared term if there is one.

You want your squared term to be just that, your variable squared, with no other constants multiplying with it. In our example, our squared term is being multiplied by a 3, so we need to divide both sides by 3.


completing the square


Step 4: Take the coefficient of your single x-term, half it including its sign, and then add the square of this number to both sides.

This step is a little bit tricky. You're going to take the coefficient of the x-term, then you're going to divide it by 2. Then you're going to square this number and add it to both sides. So, for our example, the x-term's coefficient is 4 / 3. Dividing it by 2, we get 4 / 6 or 2 / 3. Then we'll square the 2 / 3 and add that to both sides. This is what we get:


completing the square


Step 5: Convert to squared form and combine like terms.

Now that you've figured out the square of the coefficient of the x-term, you can now convert your equation into squared form. You'll use what you found to be half of the x-term's coefficient. You'll also add your like terms together on the right side of the equation. For our problem, this is what we get:


completing the square


Step 6: Take the square root of both sides.

The next step in solving your equation is to take the square root of both sides. Doing this will cancel out your square. This is what we get for our problem:


completing the square


Remember that when you take the square root of a number, you'll have both a positive and a negative component.

Step 7: Solve for your variable.

Now that you've canceled out your square, you can now to ahead and solve for your variable. Since you have a positive and a negative part, you'll have two equations to solve for. For our problem, you have these two equations you need to solve, one for the positive part and one for the negative part:


completing the square


To find your solutions, solve for your variable by isolating it. For our problem, we'll need to subtract our 2 / 3 from both sides to find our solutions.

The Solution

After isolating our variable, this is what we get for our answers:


completing the square


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