Ch. 2 - Functions and Graphs

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 2 - Functions and Graphs

Problem 105

Chapter 3, Problem 105

Exercises 103–105 will help you prepare for the material covered in the next section. Solve by completing the square: y² – 6y — 4 = 0.

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Start with the given quadratic equation: \(y^2 - 6y - 4 = 0\).

Move the constant term to the right side to isolate the terms involving \(y\): \(y^2 - 6y = 4\).

To complete the square, take half of the coefficient of \(y\) (which is \(-6\)), divide by 2 to get \(-3\), then square it to get \((-3)^2 = 9\).

Add 9 to both sides of the equation to maintain equality: \(y^2 - 6y + 9 = 4 + 9\).

Rewrite the left side as a perfect square trinomial: \((y - 3)^2 = 13\).

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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 solve quadratic equations by transforming the equation into a perfect square trinomial. This involves adding and subtracting a specific value to create a binomial squared, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.

Isolating the Variable

Isolating the variable means rearranging the equation so that the variable term stands alone on one side. This step is crucial before completing the square, as it simplifies the process and helps in accurately solving the equation.

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Let (x1, y₁) = (7, 2) and (x2, y2) = (1, −1). Find √[(x2 − x1)² + (y2 − y₁)²]. Express the - answer in simplified radical form.

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Textbook Question

In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)

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Textbook Question

Exercises 103–105 will help you prepare for the material covered in the next section. Use a rectangular coordinate system to graph the circle with center (1, -1) and radius 1.

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