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Completing the Square in Math

Examples & Formula for completing the square

Objective The goal of this web page is to explain how to complete the square, how the formula works and provide lots of practice problems.
On a different page, we have a completing the square calculator which does all the work for this topic. You just enter the quadratic.

What does "completing the square" do for us?

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Answer
It gives us a way to find the last term of a perfect square trinomial .

Okay, what is a "perfect square trinomial"??

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So, what does it really do for us?

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Before we look answer that, let's first examine the three equations below. How would you solve each one?
  • Equation 1) $$ x^2 = 25 $$
  • Equation 2) $$ (x + 1 ) ^2 = 36 $$
  • Equation 3) $$ x^2 +10x = 24 $$
Formula for Completing the Square
To best understand the formula and logic behind completing the square, look at each example below and you should see the pattern that occurs whenever you square a binomial to produce a perfect square trinomial.


loading completing square animation
Practice Problems


Directions Find the missing value to complete the square

Problem 1) $$x^2 + \color{red}{16}x + \blacksquare $$
Step 1


Problem 2) $$x^2 + \color{red}{20}x + \blacksquare $$
Step 1


Problem 3) $$x^2 + \color{red}{18}x + \blacksquare $$
Step 1


Problem 4) $$x^2 + \color{red}{5}x + \blacksquare $$
Step 1


Problem 5) $$x^2 + \color{red}{7}x + \blacksquare $$
Step 1
II. Completing Square when a is not 1

What about $$\color{blue}{3}x^2 + 18x$$, what then ?

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loading completing square animation
Practice Problems


Problem 6) $$ 2x^2 + 12x + \blacksquare $$
Step 1


Problem 7) $$ 3x^2 + 36x + \blacksquare $$
Step 1


Problem 8) $$ 5x^2 + 20x + \blacksquare $$
Step 1

challenge problems The next problems are quite challenging, good luck!


Problem 9) $$ 2ax^2 + 12ax + \blacksquare $$
Step 1


Problem 10) $$ 3x^3 + 18x^2 + \blacksquare $$
Step 1


Problem 11) $$ 4x^3 + 2x^2 + \blacksquare $$
Step 1
Further Reading