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a certain circle can be represented by the follow equation x^2+y^2+12x+4y+15=0. What is the center of the circle and what is the radius of the circle

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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.

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