The equation of a circle is x^2-6x+y^2+2y=15

what is the center and the radius?

factor it as perfect squares

x^2-6x + 9 + y^2+2y + 1= 15+9+1
(x-3)^2 + (y+1)^2=25
so center is 3,1 and radius is 5

oops, center 3,-1

To find the center and radius of a circle, we need to rewrite the given equation in the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

Let's complete the square for the given equation:

x^2 - 6x + y^2 + 2y = 15
To complete the square for the x terms, we need to take half of the coefficient of x (-6/2) and square it ((-6/2)^2 = 9).

(x^2 - 6x + 9) + y^2 + 2y = 15 + 9
(x^2 - 6x + 9) + (y^2 + 2y + 1) = 24

Simplifying the equation, we have:

(x - 3)^2 + (y + 1)^2 = 24

Comparing this to the standard form, we can see that the center of the circle is at the point (3, -1), and the radius is sqrt(24), which simplifies to 2√6.

Therefore, the center of the circle is (3, -1) and the radius is 2√6.

To find the center and radius of a circle given its equation, we need to write the equation in a specific form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius.

Let's start by completing the square for both the x and y terms in the equation.

1. x^2 - 6x + y^2 + 2y = 15
To complete the square for the x-terms, we need to find the value to add and subtract inside the parentheses. Take half of the coefficient of x (-6), square it, and add it to both sides of the equation.
(x^2 - 6x + __) + y^2 + 2y = 15 + __ + __

Since we added (-6/2)^2 = 9 to the left side of the equation, we also need to add it on the right side.
(x^2 - 6x + 9) + y^2 + 2y = 15 + 9

Do the same for the y-terms.
(x^2 - 6x + 9) + (y^2 + 2y + 1) = 15 + 9 + 1
(x - 3)^2 + (y + 1)^2 = 25

Now that we have the equation in the correct form, we can identify the center and radius of the circle.

The center of the circle is given by the values inside the parentheses, (h, k). Comparing to the equation we obtained, we have h = 3 and k = -1.

Therefore, the center of the circle is at (3, -1).

The radius of the circle, r, can be determined by taking the square root of the value on the right side of the equation. In this case, r = √25 = 5.

Therefore, the radius of the circle is 5.