Give the center and radius of the circle represented by each equation. See Examples 3 and 4. x^2+y^2+6x+8y+9=0

Verified step by step guidance

1. Rewrite the equation in the standard form of a circle by completing the square for both the x and y terms.
2. Group the x terms together and the y terms together: \( (x^2 + 6x) + (y^2 + 8y) = -9 \).
3. Complete the square for the x terms: Take half of the coefficient of x (which is 6), square it, and add it inside the parentheses: \( (x^2 + 6x + 9) \).
4. Complete the square for the y terms: Take half of the coefficient of y (which is 8), square it, and add it inside the parentheses: \( (y^2 + 8y + 16) \).
5. Adjust the equation to maintain equality: Add 9 and 16 to the right side of the equation as well, resulting in \( (x+3)^2 + (y+4)^2 = 16 \).

Key Concepts

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 form allows for easy identification of the circle's center and radius by comparing it to the general equation.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial. This technique is essential for rewriting the given equation of the circle in standard form, allowing us to identify the center and radius more easily.

Quadratic Terms

Quadratic terms are expressions that include variables raised to the second power, such as x² and y². In the context of a circle's equation, these terms represent the geometric properties of the circle, and understanding their role is crucial for manipulating the equation to find the center and radius.