Answer:
Explanation
To rewrite the equation in this form, we need to complete the square for both the x and y terms. Let's break it down step by step:
1. Move the constant term (15) to the right side of the equation:
x^2 + 12x + y^2 + 4y = -15
2. Complete the square for the x terms:
To complete the square for x, we take half of the coefficient of x (12/2 = 6) and square it (6^2 = 36). Add this value to both sides of the equation:
x^2 + 12x + 36 + y^2 + 4y = -15 + 36
3. Complete the square for the y terms:
To complete the square for y, we take half of the coefficient of y (4/2 = 2) and square it (2^2 = 4). Add this value to both sides of the equation:
x^2 + 12x + 36 + y^2 + 4y + 4 = -15 + 36 + 4
4. Simplify:
(x^2 + 12x + 36) + (y^2 + 4y + 4) = 25
5. Factor the squared terms:
(x + 6)^2 + (y + 2)^2 = 25
Comparing this equation to the standard form, we can see that the center of the circle is (-6, -2) and the radius is sqrt(25) = 5.
Therefore, the center of the circle is (-6, -2) and the radius is 5 units.