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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 109
Chapter 2, Problem 109

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 4x - 5

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To find the x-intercepts of the graph of the equation y = x^2 - 4x - 5, set y = 0 because the x-intercepts occur where the graph crosses the x-axis (i.e., where y = 0). This gives the equation 0 = x^2 - 4x - 5.
Rewrite the equation as x^2 - 4x - 5 = 0. This is a quadratic equation, and we will solve it using factoring, the quadratic formula, or completing the square. In this case, factoring is a good approach.
Factor the quadratic equation x^2 - 4x - 5. Look for two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of x). These numbers are -5 and 1, so the equation factors as (x - 5)(x + 1) = 0.
Apply the Zero Product Property, which states that if a product of two factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero: x - 5 = 0 and x + 1 = 0.
Solve each equation for x. For x - 5 = 0, add 5 to both sides to get x = 5. For x + 1 = 0, subtract 1 from both sides to get x = -1. Therefore, the x-intercepts are x = 5 and x = -1.

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Key Concepts

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

X-Intercept

The x-intercept of a graph is the point where the graph intersects the x-axis. This occurs when the value of y is zero. To find the x-intercept(s) of an equation, you set y equal to zero and solve for x. In the context of the given equation, this means solving the quadratic equation x^2 - 4x - 5 = 0.
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Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not zero. The solutions to a quadratic equation can be found using factoring, completing the square, or the quadratic formula. Understanding how to manipulate and solve these equations is essential for finding x-intercepts.
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Introduction to Quadratic Equations

Graphing Quadratics

Graphing a quadratic function involves plotting a parabolic curve that opens upwards or downwards, depending on the sign of the leading coefficient (a). The vertex of the parabola represents the maximum or minimum point, while the x-intercepts indicate where the graph crosses the x-axis. Recognizing the shape and key features of quadratic graphs is crucial for matching equations to their corresponding graphs.
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Related Practice
Textbook Question

If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

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Textbook Question

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = x2 - 2x + 2

Textbook Question

Solve and graph the solution set on a number line: (2x−3)/4 ≥ 3x/4 + 1/2

Textbook Question

When 4 times a number is subtracted from 5, the absolute value of the difference is at most 13. Use interval notation to express the set of all numbers that satisfy this condition.

Textbook Question

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). y = - (x + 1)2 + 4

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Textbook Question

In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)