 So let's take a detailed look at finding the equation of a circle. So let's try to find the center and radius of the circle with this equation, and then let's try and graph it. So the easiest way to find the center and radius of a circle is to have the equation of the circle in a standard form. And the easiest standard form to work with is the following. The circle with center hk and radius r has the equation x minus h squared plus y minus k squared equals r squared. And the thing to notice here is that all of our x terms are going to be part of a perfect square and all of our y terms are going to be part of a perfect square. So if we want to find the equation of a circle in the easiest form, we'll want to complete the square on both x and y. So if you look at the standard equation, you see that all of our x terms are together, all of our y terms are together, and all of the constants are over on the right-hand side of the equation. So let's rearrange the given equation so that we have all of the x terms together, all of the y terms together, and we'll shove this constant onto the other side of the equation. So now we want to make sure all of our x terms are part of a perfect square, and so that means we want to find c so that x squared plus 8x plus c squared is a perfect square. So let's complete the square on x. So our first term, x squared, well, that's the square of x. Our last term, c squared, well, that's the square of c, and we want our middle term, 8x, to be 2 times the square roots x and c. So let's compare. 8x is supposed to be equal to 2xc. Well, I can rearrange these factors over on the right so I can move the 2 and the c together, and I see that 8 times x is supposed to be the same as 2c times x. So that means 8 must be equal to 2c. So we'll solve for c, and we find that c must be equal to 4, and this means that we want to add c squared for squared. Since this is an equation, we can add 4 squared to one side as long as we pay for it by adding 4 squared to the other side. So now our x terms plus that extra constant make up a perfect square. We also want to do the same thing for our y terms. So we want to find c so that y squared minus 4y plus c squared is a perfect square. So completing the square on y, our first term, y squared, is the square of y. Our last term, c squared, is the square of c, and our middle term, minus 4y, we need that to be 2 times the square roots y and c. So comparing, we want minus 4y to be equal to 2 times y times c. We'll rearrange things a little bit. And if they're equal, we have to have minus 4 the same as 2c. So our equation becomes minus 4 must be equal to 2c, which we can solve, and that gives us c is equal to minus 2, and we need to add c squared minus 2 squared. Again, since we have an equation, we can add minus 2 squared as long as we pay for it by adding minus 2 squared to the other side as well. Now remember, we went through all this trouble so that we could make our x terms part of a perfect square and our y terms part of a perfect square. So let's go ahead and do that factoring x squared plus 8x plus 4 squared. Well, that's the square of x plus 4. And y squared minus 4y plus minus 2 squared, that factors as y minus 2 squared. And while we're at it, let's clean up some of this arithmetic. 4 squared is 16, and minus 2 squared is 4. And equals means replaceable. So everywhere we see x squared plus 8x plus 4 squared, how about right here? We can replace it with x plus 4 squared. Everywhere we see a y squared minus 4y plus negative 2 squared, we can replace it with a y minus 2 squared. And we can replace 4 squared with 16 and minus 2 squared with 4. Bringing back our equation of a circle, we actually want this in the form x minus h squared plus y minus k squared equals r squared. So we want to rewrite this x plus 4 squared as x minus something, x minus a negative 4 squared, and then this mass of arithmetic, 44 plus 16 plus 4, well that's equal to 64, which is equal to 8 squared. And again, equals means replaceable. So instead of x plus 4 squared, I can write x minus negative 4 squared, and instead of 44 plus 16 plus 4, I can write 8 squared. And instead of y minus 2 squared, I can write, well algebra is generalized arithmetic, and arithmetic is bookkeeping, and part of that is if it's there, it's still there. We still have y minus 2 squared. And now our equation of the circle is in the same form as the standard equation, so we can see that the center is at minus 4, 2, and the radius is r equal to 8. And finally, we can graph the circle. We know where the center is, so we'll graph that center point. And remember, if it's not written down, it didn't happen. We should label the coordinates of our point. Remember, the radius is the distance of every point on the circle from the center. So from the center, we'll go out 8 units in any direction at all. Again, if it's not written down, it didn't happen, so let's label our radius with its length, and we'll draw a circle where that length is our radius. And again, if it's not written down, it didn't happen. We should label our graph with the equation.