Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x²+y²+4x+4y+8=0

Verified step by step guidance

1. Start with the given equation: \(x^2 + y^2 + 4x + 4y + 8 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 4y) = -8\).
3. Complete the square for both \(x\) and \(y\) terms. For \(x^2 + 4x\), add and subtract \((\frac{4}{2})^2 = 4\). For \(y^2 + 4y\), add and subtract \((\frac{4}{2})^2 = 4\).
4. Rewrite the equation including the completed squares: \((x^2 + 4x + 4) + (y^2 + 4y + 4) = -8 + 4 + 4\).
5. Express the perfect squares as binomials: \((x + 2)^2 + (y + 2)^2 = 0\). From here, analyze the radius and center to determine if the graph is a circle, a point, or nonexistent.

Key Concepts

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties clearly.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a perfect square form. This technique is essential to transform the given equation into the standard circle form by grouping x and y terms and adding constants appropriately.

Determining the Nature of the Graph

After rewriting the equation, the value of r^2 determines the graph's nature: if r^2 > 0, it's a circle; if r^2 = 0, it's a single point; if r^2 < 0, the graph does not exist in the real plane. This helps classify the equation's graph accurately.