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How to Complete the Square

  • 1:51 Standard Form
  • 2:32 Vertex Form
  • 3:01 Perfect Square Binomial
  • 3:28 Finding and Completing…
  • 5:20 Example
  • 8:07 Lesson Summary
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Lesson Transcript
Instructor: Luke Winspur

Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.

Completing the square can help you learn where the maximum or minimum of a parabola is. If you're running a business and trying to make some money, it might be a good idea to know how to do this! Find out what I'm talking about here.

Mathematics, Economics and Yogurt

I decided pretty early on in my freshman year of college to become a math major, but a little ways in I also discovered that I was really into economics as well. I thought about getting an econ minor but was told that the minor was only two classes less than the major. Well, if I was going to take the time to go through 9 econ classes, why not 11? So I went for it - double major, here I come! And I'm really glad I did, too. It ended up being a breeze because there was enough math in my economics classes to make it hard for everybody else but pretty easy for me.

Anyways, I bring this up because I wanted to share with you a math problem that many of my econ classmates might have gotten stuck on. Here's the scenario: I've decided to hop on the bandwagon and open up a new frozen yogurt chain - you know, the kind where they just let you put whatever you want on it, and then you weigh it at the end?

Well, I know that people are going to want to be able to choose from a lot of different toppings, so having more is probably better. But how many should I have? If I went overboard and bought like 200 different types of toppings, some of them would probably never get eaten and I'd end up losing money.

So this is what economists would do. Using data and theory, they would set up a mathematical model that predicts how much money they'd make with different numbers of toppings. Let's say that, based on how previous yogurt shops had done and how the economy is doing right now, they came up with this: y = -x^2 + 36x - 224, where y is the amount of profit I'll make (in thousands of dollars) and x is the number of toppings I'm going to offer.

Because the biggest exponent on the x is a 2, what we've got here is a quadratic equation, which means our graph will be a parabola. This quadratic is in standard form (y = ax^2 + bx + c), which means we can make a few observations.

The c value will tell us our y-intercept, which means if I were to offer no toppings, or substitute in x = 0, I would end up losing quite a lot of money. Next, the negative a value tells me that this will be a concave down parabola, so I'm guessing that our graph will look something like this. So, this makes sense to me - as I offer more and more toppings, my profit will slowly go up. But eventually I'll reach a point where I'm offering too many toppings and they're not getting eaten, so I start to lose money.

Graph of the concave down parabola
Yogurt Profit Parabola

The question is, where is the vertex of this parabola? Well, too bad this function wasn't given to us in vertex form instead of standard form. If it was, I'd know exactly what the profit-maximizing number of toppings was. So how can we change it into the form that we want?

Well, that's exactly what completing the square is: taking a standard form equation like this one (y = ax^2 + bx + c) and changing it into vertex form, like this: y = a(x - h)^2 + k. The main part of the problem comes down to this little guy: (x - h)^2. We call this a perfect square binomial and need to remember that it's not equal to x^2 - h^2, but instead equal to (x - h)(x - h). So if we're trying to change our standard form equation into vertex form, what we need to do is factor the standard form equation into something like this, a perfect square binomial. As is often the case in math, it's all going to come down to pattern.

Finding and Completing the Square

Let's step away from our yogurt problem real quick to a slightly different one with a little bit nicer numbers to make finding the pattern easier - say, (x + 5)^2. Well, that's just (x + 5)(x + 5), which we can FOIL out to be x^2 + 10x + 25.

So what relationship do the numbers in the standard form expression have with each other? Well, it might be a little bit tricky to see, but it turns out that our b value, 10, divided by 2 (which is 5) and then squared turns into our c value, 25. Since we know that this trinomial can be factored into a perfect square binomial, (x + 5)^2, it will turn out that any trinomial where the middle value divided by 2 and then squared is our final value, the c value, will be able to be factored as a perfect square binomial. If we want to say that same thing in math, we're looking for a trinomial with a c value that is equal to (b/2)^2. If that's the case, it can be factored as a perfect square binomial.

So, in order to do this, we need our c value to be a specific number. But when we've got an equation that we can, say, add 20 to both sides of, we can make that c value whatever we want. This allows us to force the c value into being the exact number we need it to be in order to become a perfect square binomial. This process of changing the c value into the exact right number that makes it fit the pattern and become a perfect square binomial is what mathematicians call completing the square.

Completing the square involves turning a quadratic equation in standard form into one in vertex form
Completing the Square Equations

Yogurt Profit

Let's see how we would do this with our yogurt problem in order to determine the right number of toppings to offer. We were here - y = -x^2 + 36x - 224 - but now we know that our c value needs to be something specific in order for it to work as a perfect square binomial. What it needs to be is the middle number, 36, divided by 2 (18) and squared (324). Right now our c value is -224, so it's not quite the right number. Our trinomial is not ready to be factored as a perfect square binomial... yet.

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