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Ch. 2 - Graphs and Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 2 - Graphs and FunctionsProblem 38
Chapter 3, Problem 38

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

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Start with the given equation: \(x^2 + y^2 + 4x + 4y + 8 = 0\).
Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 + 4y) = -8\).
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 \((\frac{4}{2})^2 = 4\).
Rewrite the equation including the completed squares: \((x^2 + 4x + 4) + (y^2 + 4y + 4) = -8 + 4 + 4\).
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.

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Key Concepts

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

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.
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Circles in Standard Form

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.
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Solving Quadratic Equations by Completing the Square

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.
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The Natural Log
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