How To “Complete the Square” Visually

and the Trouble with Memorization

Brett Berry
Jun 7, 2017 · 6 min read

There’s this notion that gets thrown around that the gifted can see past the numbers. That things hidden to the average person pop out to the math genius. We are convinced that only the prodigies have the mathematical intuition to make sense out of numbers and symbols.

Today I want to challenge this belief we’ve bought into. The belief that it takes some natural brilliance to see past the numbers.

Instead, consider this:

Perhaps many never see past the numbers because they are never shown?

Image for post
Image for post
insta

Learning to Complete the Square

I remember learning how to “complete the square” as a teenager and I did what most students do, I memorized the steps. I was pretty good at memorization, so I got by.

I understood that completing the square was a method for solving a quadratic, but it wasn’t until years later that I realized I hadn’t really understood what I was doing at all. I was just going through the motions.

And that’s the way it often is, isn’t it? Our teacher scribbles a series of steps and formulas on the whiteboard and we do our best to replicate it.

But is memorization and regurgitation really what math is about?

Instead of developing understanding that we can carry with us for years, we memorize steps that we’ll forget next week. I believe there is always a better way of seeing math that will help you understand what the gibberish is all about.

But first, pick your poison!

Before we jump into the math, I’m going to give you two options 😉

Option 1: you can keep reading below for an easy, breezy breakdown of how to complete the square visually, which is pretty awesome, or

Option 2: you can watch my video tutorial where I go a little bit more in-depth on how to complete the square using both the visual geometric method and the traditional algebraic method. I also cover how to complete the square when you have a leading coefficient that isn’t 1 (e.g. y=2x²+8x+11) in case you’re interested in that sort of thing 🙃

Check out Math Hacks on YouTube for more video tutorials!

Quadratics: A Quick Recap

I haven’t talked about quadratics much in this series, but I really wanted to cover this topic so I’m going to assume you know quadratics are algebraic equations where the term with the highest power is 2.

Image for post
Image for post

Remember that when we graph a quadratic we get a parabola, or U-shaped graph (or an upside down U).

Sometimes the U-shaped graph crosses the horizontal x-axis, and when it does we like to find the x-values where it crosses (and even when it doesn’t, we still like to solve for these guys, we just call them complex roots instead). These x-values have lots of names. Some people call them roots, others call them zeroes, and some simply call them solutions.

Now there are lots of techniques for finding roots, and depending on the equation some methods are easier than others. We can solve by graphing, factoring, the quadratic formula or by completing the square.

Typical Instructions for Completing the Square

I could show you an example of how to complete the square by following these instructions, but honestly how much of these steps will you remember?

  • Subtract the constant to the other side.
  • Take half of the x terms coefficient, square it and add to both sides.
  • Factor the trinomial into a binomial squared.
  • Take the square root of both sides (including a plus or minus sign).
  • Finish solving for X and simplify the answer, if needed.

A Different Approach

Instead let’s use an area model to visualize how to complete the square of the following equation:

Image for post
Image for post

Our area model is fairly straightforward. Begin by representing x² as a square whose sides are both x units in length making an area of x².

Image for post
Image for post

Next we’ll add 2x to the area model. For 2x to represent an area we’ll need to either add a block that is 2 units by x units, or we can add two blocks each 1 unit by x units. Because we want to make a square, we’ll choose to add two 1-by-x blocks to our model.

Image for post
Image for post

Now as you can see we are trying to make a square but there is a chunk missing in the bottom right corner. What is the area of this missing piece?

Image for post
Image for post

Since the blue blocks adjacent to our missing piece are both 1 unit wide, we can deduce that the missing block has an area of 1•1=1.

Also note that by adding together the outermost units of the square, the area of the square becomes our desired binomial squared → (x+1)²

Image for post
Image for post

Now all that’s left to do is literally complete the square and adjust for the extra units. To do this first we can fill in the area of the purple square, which we know to be 1. Since our original equation had a constant of 12, we’ll subtract 1 from 12 to account for the 1 we add to the square.

Image for post
Image for post

And like magic, we have completed the square! Our square is (x+1)² with 11 leftover. The extra 11 can simply be added to the end of our binomial squared.

Image for post
Image for post

Now if we wanted to find the roots we can set the equation equal to zero and solve with basic algebra.


I bet if you look back at the algebraic instructions for completing the square you’ll understand why you take half of the middle term’s coefficient and square it. You’ll see how that helps you factor a binomial squared out of the equation, and how the constants get adjusted along the way.

Of course it isn’t efficient to draw an area model every time you complete the square. You’ll need to know how to do the process strictly using algebra, but if you ever forget the steps having a solid understanding of how completing the square works can help you derive the steps in a pinch.

And look there, you just saw past the algebra in quest of the true meaning of completing the square! It wasn’t that bad, was it?

Thanks for reading!

Math Hacks

Tutorials with a fresh perspective.

Welcome to a place where words matter. On Medium, smart voices and original ideas take center stage - with no ads in sight. Watch
Follow all the topics you care about, and we’ll deliver the best stories for you to your homepage and inbox. Explore
Get unlimited access to the best stories on Medium — and support writers while you’re at it. Just $5/month. Upgrade

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store