Complete the square and write the equation in standard form. Then give the center and radius of each circle and graph the equation. x² + y²+6x+2y+6 = 0

Verified step by step guidance

1. Start with the given equation: \(x^{2} + y^{2} + 6x + 2y + 6 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^{2} + 6x) + (y^{2} + 2y) = -6\) (move the constant to the right side).
3. Complete the square for the \(x\) terms: take half of 6, which is 3, then square it to get 9. Add 9 inside the \(x\) group.
4. Complete the square for the \(y\) terms: take half of 2, which is 1, then square it to get 1. Add 1 inside the \(y\) group.
5. Since you added \$9 + 1 = 10\( to the left side, add 10 to the right side as well to keep the equation balanced. Then rewrite the equation as \)(x + 3)^{2} + (y + 1)^{2} = ext{(new right side)}$.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x + p)² = q. It involves adding and subtracting a constant to create a perfect square trinomial, which simplifies solving or rewriting equations, especially for conic sections like circles.

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. Converting the general form to this form helps identify the circle's key features and makes graphing straightforward.

Identifying the Center and Radius

Once the equation is in standard form, the center of the circle is given by the coordinates (h, k), and the radius is the square root of r². These values are essential for graphing the circle accurately and understanding its position and size.