Video: Finding the Center and Radius of a Circle by Completing the Square

Video Transcript

By completing the square, find the center and radius of the circle 𝑥 squared minus four 𝑥 plus 𝑦 squared minus four 𝑦 minus eight equals zero.

The equation of a circle is 𝑥 minus ℎ squared plus 𝑦 minus 𝑘 squared equals 𝑟 squared. ℎ, 𝑘 is the center point, and 𝑟 is the radius, the length. So we will be completing the square because that’s the standard equation that the formula is in.

So we want to somehow go from 𝑥 squared minus four 𝑥 to something squared in terms of 𝑥, and the same thing for 𝑦, and then have a constant on the right-hand side of the equation that will be our radius. So to complete the square, we will take our 𝑏 term which is the negative four — it’s the coefficient in front of the 𝑥 term — and we will take 𝑏, divide by two, and square it.

So when we take negative four divided by two, that’s negative two, and when we square it, that’s positive four. So let’s do the exact same steps for the 𝑦s. Now notice our 𝑏 term that we’ll divide by two and then square is actually the exact same as for the 𝑥; they’re both negative four, so we can add the exact same number for the 𝑦.

And then to move the negative eight over to the right, we add eight to both sides, so it’s equal to eight. So again, the whole point of completing the square is to end up with something squared, 𝑥 squared minus four 𝑥 plus four, that is, 𝑥 minus two squared, which is the same thing as 𝑥 minus two times 𝑥 minus two.

We could always FOIL that back and you would still get 𝑥 squared minus four 𝑥 plus four. And since it’s the same expression for 𝑦, it’s the same thing, 𝑦 minus two squared. Now, however, the right-hand side of the equation is not eight.

Notice how we added two fours to the left-hand side of the equation. You have to keep equations balanced. So if you added them to one side of the equation, you’re going to have to add them to the other side of the equation, so this will actually be equal to not eight, but 16.

So this will be the equation of our circle. Both of these equations are equal; they’re just in different forms. And the equation of a circle is in this form because it’s useful when you’re trying to find the center. So, for example, this center of this equation would be two, two because two is ℎ and two is the same as 𝑘, in our equation. And then it’s equal to 𝑟 squared.

So if we just wanted the radius, we would have to square-root 𝑟 squared to find 𝑟, so the square root of 16 would mean the radius is four, but again, our equation would be 𝑥 minus two squared plus 𝑦 minus two squared equals 16.