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+8x-2y-8=0

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Start with the given equation: \(x^2 + y^2 + 8x - 2y - 8 = 0\).

Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(\left(x^2 + 8x\right) + \left(y^2 - 2y\right) = 8\).

Complete the square for the \(x\) terms: take half of 8, which is 4, then square it to get 16. Add 16 inside the parentheses for \(x\), and also add 16 to the right side to keep the equation balanced.

Complete the square for the \(y\) terms: take half of -2, which is -1, then square it to get 1. Add 1 inside the parentheses for \(y\), and also add 1 to the right side to keep the equation balanced.

Rewrite the equation as perfect square trinomials: \(\left(x + 4\right)^2 + \left(y - 1\right)^2 = \) (sum of constants on the right side). From this, identify the center as \((-4, 1)\) and the radius as the square root of the right side.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 an equation to this form helps identify the circle's key features and makes graphing straightforward.

Identifying the Center and Radius from the Equation

Once the equation is in standard form, the center is given by the coordinates (h, k), and the radius is the square root of the constant on the right side. Understanding this allows you to graph the circle accurately and interpret its geometric properties.

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