The problem asks us to analyze the equation $x^2 + y^2 + 6x - 2y + 6 = 0$. We need to determine what the equation represents. We can accomplish this by completing the square for both $x$ and $y$.

1. Problem Description

The problem asks us to analyze the equation $x^2 + y^2 + 6x - 2y + 6 = 0$. We need to determine what the equation represents. We can accomplish this by completing the square for both $x$ and $y$.

2. Solution Steps

First, we group the $x$ terms and the $y$ terms together:

$(x^2 + 6x) + (y^2 - 2y) + 6 = 0$

To complete the square for the $x$ terms, we take half of the coefficient of $x$, which is $\frac{6}{2} = 3$, and square it, which is $3^2 = 9$. So we add and subtract

9. To complete the square for the $y$ terms, we take half of the coefficient of $y$, which is $\frac{-2}{2} = -1$, and square it, which is $(-1)^2 = 1$. So we add and subtract

1.

$(x^2 + 6x + 9 - 9) + (y^2 - 2y + 1 - 1) + 6 = 0$

$(x^2 + 6x + 9) - 9 + (y^2 - 2y + 1) - 1 + 6 = 0$

Now we rewrite the expressions in parentheses as squared terms:

$(x + 3)^2 + (y - 1)^2 - 9 - 1 + 6 = 0$

$(x + 3)^2 + (y - 1)^2 - 4 = 0$

Add 4 to both sides:

$(x + 3)^2 + (y - 1)^2 = 4$

This is the equation of a circle with center $(-3, 1)$ and radius $\sqrt{4} = 2$.

3. Final Answer

The equation $x^2 + y^2 + 6x - 2y + 6 = 0$ represents a circle with center $(-3, 1)$ and radius

2.