complete the square

A Simple Explanation

Completing the Square is a factoring technique that we can sometimes use to help solve quadratic equations. The basic idea behind completing the square is to transform a quadratic polynomial (a polynomial whose highest power is 2) that cannot be factored "nicely", into the sum of a perfect square and a constant.

What is a Perfect Square?

For the purposes of completing the square, a perfect square is a polynomial such that if we square it (multiply it by itself) we get our original quadratic polynomial back.

For example consider the following polynomial:

example polynomial 1

Notice this polynomial factors nicely into the product (x+1)*(x+1)

quadratic polynomial 2

Notice that the polynomial (x+1) times itself equals our original quadratic polynomial. This is what it means to be a perfect square.

The Three Steps to Completing the Square

Step 0

Before we can attempt to complete the square, we must have our polynomial in the form of


such that a, b, and c are arbitrary constants, and a = 1. If the coefficient for x2 does not equal one you must factor it out before we can begin the completing the square process (remember, 1*x2=x2 so we don't usually write the 1 in front of the x2 when 1 is the coefficient).

Step 1

Look at the coefficient for the x term in your polynomial (the number in front of x), and take half of it. So, step one says take half of the number represented by b in the above polynomial.

Step 2

Take the number we found in step one and square it

Step 3

Add zero. Add zero? yes. What I mean by this is to take the number we found in step 2, and then add it and subtract it to our polynomial, effectively adding zero and not changing the value of our polynomial

Video Example


Alright now that we know the steps, lets look at a solid example. Consider the following polynomial:

example polynomial 3

Notice that this polynomial does not factor nicely into anything useful. If we were attempting to solve an equation of this polynomial set equal to zero, we would have to use the quadratic formula if we did not know how to complete the square.

However, by following the steps above completing the square becomes an easy task. First we need to take half of the x coefficient. In this case the x coefficient is 6, so we will take half and get 3. The next step says to square that number, so we will square 3 and get 9. The final step says to add zero (and then simplify). After doing these three things we will wind up with the following polynomial:

example polynomial 4

Now all that is left to do is simplify! If we focus on the first three terms in the polynomial above, we should notice that these three terms make a perfect square, and the remaining terms are both constants that can be combined. 

completing the square

And that is how you complete the square. This is an extremely useful technique that comes up in several areas of mathematics and its applications beyond algebra class, so I hope this article helps you master the skill of completing the square.

Challenge Yourself!

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