In Exercises 53–64, 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 by rearranging the equation to group the x and y terms: \(x^2 + 6x + y^2 + 2y = -6\).
2. Complete the square for the x terms: Take half of the coefficient of x (which is 6), square it, and add it to both sides. This gives \((x^2 + 6x + 9)\).
3. Complete the square for the y terms: Take half of the coefficient of y (which is 2), square it, and add it to both sides. This gives \((y^2 + 2y + 1)\).
4. Rewrite the equation with the completed squares: \((x + 3)^2 + (y + 1)^2 = 4\).
5. Identify the center and radius of the circle: The center is \((-3, -1)\) and the radius is \(2\).

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique involves rearranging the equation and adding or subtracting constants to create a binomial squared. It is essential for converting equations into standard form, particularly for circles and parabolas, allowing for easier identification of key features such as the center and radius.

Standard Form of a Circle

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. This format makes it straightforward to identify the circle's center and radius directly from the equation. Understanding this form is crucial for graphing circles and solving related problems in coordinate geometry.

Graphing Circles

Graphing circles involves plotting points that satisfy the circle's equation on a coordinate plane. The center of the circle is marked at (h, k), and the radius r determines the distance from the center to any point on the circle. Recognizing the relationship between the equation and its graphical representation is vital for visualizing geometric concepts and solving problems involving circles.