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Ch. 2 - Functions and Graphs
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 2 - Functions and GraphsProblema 55
Capitolo 3, Problema 55

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 – 10x – 6y – 30 = 0

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Start with the given equation: \(x^{2} + y^{2} - 10x - 6y - 30 = 0\).
Group the \(x\) terms and \(y\) terms together and move the constant to the other side: \(\left(x^{2} - 10x\right) + \left(y^{2} - 6y\right) = 30\).
Complete the square for the \(x\) terms: take half of \(-10\), which is \(-5\), then square it to get \(25\). Add \(25\) inside the \(x\) group.
Complete the square for the \(y\) terms: take half of \(-6\), which is \(-3\), then square it to get \(9\). Add \(9\) inside the \(y\) group.
Since you added \(25\) and \(9\) to the left side, add the same amounts to the right side to keep the equation balanced: \(30 + 25 + 9\).

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Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)² or (y - k)² by adding and subtracting appropriate constants. This technique helps transform the general form of a circle's equation into its standard form, making it easier to identify key features like the center and radius.
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Solving Quadratic Equations by Completing the Square

Standard Form of a Circle's Equation

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Writing the equation in this form allows for straightforward identification of the circle's geometric properties and simplifies graphing.
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Circles in Standard Form

Graphing Circles

Graphing a circle involves plotting its center (h, k) on the coordinate plane and using the radius r to mark points at a distance r in all directions. Understanding how to interpret the standard form equation aids in accurately sketching the circle and visualizing its position and size.
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Circles in Standard Form
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