x^2+y^2-6x+8y=-9

I am assuming that you are converting the equation of the circle standard form (possibly for graphing purposes).

First, I am going to arrange the x terms together and the y terms together:

x^{2} - 6x + y^{2} + 8y = -9

Next, I am going to group the x's terms in one parenthesis and the y terms in another parenthesis:

(x^{2} - 6x) + (y^{2} + 8y) = -9

We are going to complete the square in each of the parenthesis. To complete the square, we need to divde the coefficient of the first power term by 2 and square it.

For the x's, we need to divide -6 by 2 and square the result. (-6/2)^{2} = (-3)^{2} = 9. We need to add 9 in the first parenthesis.

For the y's, we need to divide 8 by 2 and squre the result. (8/2)^{2} = 4^{2} = 16. We need to add 16 in the second parenthesis.

Remember that the same numbers that are added on the left side of the equal sign must also be added on the right side of the equal sign.

(x^{2} - 6x + 9) + (y^{2} + 8y + 16) = -9 + 9 + 16

We can factor the parenthesis as perfect squares, if we completed the square correctly. The factors will be the square root of the first term, sign of the middle term, square root of the last term. The whole factor will be squared.

(x - 3)^{2} + (y + 4)^{2} = -9 + 9 + 16

Simplifying the right side, we get the standard form of a circle

(x -3)^{2} + (y + 4)^{2} = 16

The standard form of a circle is: (x - h)^{2} + (y - k)^{2} = r^{2}, where (h, k) is the center and r is the radius.

The center of your circle is (3, -4), since 3 is subtracted from x and -4 is subtracted from y.

The radius of the circle is 4, since 4 is the square root of 16.