 In this video, we will look at determining the center and radius of a circle and we'll consider the example 2x squared plus 2y squared plus 24x plus 12y plus 40 equals 0. Notice that this circle is given to us in general form and if we can convert to standard form, we can determine the center and radius. Recall that our standard form is x minus h squared plus y minus k squared equals r squared where hk is our center and r is the radius of the circle. And so now again, we want to write in this form, we know we want to end up with x plus or minus some numbers squared. And so if we think about multiplying that out, we'd have x plus n times x plus n which is x squared plus 2nx plus n squared. And so we want to get the x terms of our circle in this form x squared plus 2 times some number times x plus that same number squared. And so let's look at this. To start with, notice that we have a 2 in front of our x squared. Likewise, we have a 2 in front of our y squared. We want to remove that to get into this goal form. And so we're going to divide all terms by 2. When we divide by 2, we end up with x squared plus y squared plus 12x plus 6y plus 20. And we're going to rearrange this so that our x is together so we have x squared plus 12x. We rearrange so we have our y's together, y squared plus 6y. And we move the 20 to the other side which makes it negative 20. The next piece of the form we want to have is that 2nx. Notice that the number in front of x is 12. And so for us, we want that to be set equal to 2n. That's twice our number that we're looking for. And for our y value, we're looking at the number in front of y which is 6 which is our 2n value for y. And in our form, we have again x squared plus 2nx plus n squared. We need to find out what n squared is. Well, if 2n is 12 for x, we know that n is 6. And n squared would be 36. For y, we have that 2n is 6. And so n being half of that would be 3. And n squared 3 squared is 9. We want to add 36 to both sides of the equation to get our perfect square trinomial for x's. And we want to add the 9 to both sides of the equation for the y's. We now are going to factor. We have x squared plus 12x plus 36 and y squared plus 6y plus 9. And we're going to factor each of those independently. And then we add the constants on the right side of the equation. So we'd add the negative 20 to 36 to the 9. When we factor x squared plus 12x plus 36, we get x plus 6 times x plus 6 or x plus 6 squared. And for y, we'd have y plus 3 squared. And we add our three terms to get 25. And so now our center is going to be negative 6, negative 3. We're pulling out the h and k values from r. And the radius of our circle, square root of 25, is 5.