Video Transcript
By completing the square, find the center and radius of the circle ๐ฅ squared minus four ๐ฅ plus ๐ฆ squared minus four ๐ฆ minus eight equals zero.
The equation of a circle is ๐ฅ minus โ squared plus ๐ฆ minus ๐ squared equals ๐ squared. โ, ๐ is the center point, and ๐ is the radius, the length. So we will be completing the square because thatโs the standard equation that the formula is in.
So we want to somehow go from ๐ฅ squared minus four ๐ฅ to something squared in terms of ๐ฅ, and the same thing for ๐ฆ, 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 ๐ term which is the negative four โ itโs the coefficient in front of the ๐ฅ term โ and we will take ๐, divide by two, and square it.
So when we take negative four divided by two, thatโs negative two, and when we square it, thatโs positive four. So letโs do the exact same steps for the ๐ฆs. Now notice our ๐ term that weโll divide by two and then square is actually the exact same as for the ๐ฅ; theyโre both negative four, so we can add the exact same number for the ๐ฆ.
And then to move the negative eight over to the right, we add eight to both sides, so itโs equal to eight. So again, the whole point of completing the square is to end up with something squared, ๐ฅ squared minus four ๐ฅ plus four, that is, ๐ฅ minus two squared, which is the same thing as ๐ฅ minus two times ๐ฅ minus two.
We could always FOIL that back and you would still get ๐ฅ squared minus four ๐ฅ plus four. And since itโs the same expression for ๐ฆ, itโs the same thing, ๐ฆ 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, youโre 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; theyโre just in different forms. And the equation of a circle is in this form because itโs useful when youโre trying to find the center. So, for example, this center of this equation would be two, two because two is โ and two is the same as ๐, in our equation. And then itโs equal to ๐ squared.
So if we just wanted the radius, we would have to square-root ๐ squared to find ๐, so the square root of 16 would mean the radius is four, but again, our equation would be ๐ฅ minus two squared plus ๐ฆ minus two squared equals 16.