Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Intro to Quadratic Equations

College Algebra

1. Equations & Inequalities

Intro to Quadratic Equations

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1:50 minutes

Problem 109

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^2 - 4x - 5

Verified step by step guidance

To find the x-intercepts of the equation \( y = x^2 - 4x - 5 \), set \( y = 0 \).

This gives the equation \( x^2 - 4x - 5 = 0 \).

Factor the quadratic equation. Look for two numbers that multiply to -5 and add to -4.

The equation can be factored as \((x - 5)(x + 1) = 0\).

Set each factor equal to zero: \(x - 5 = 0\) and \(x + 1 = 0\), then solve for \(x\).

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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.

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.

Graphing Quadratics

Graphing a quadratic function involves plotting a parabola, which can open upwards or downwards depending on the sign of the leading coefficient (a). The vertex, axis of symmetry, and x-intercepts are key features of the graph. Knowing how to identify these elements helps in matching the equation to its corresponding graph, as well as understanding the overall shape and behavior of the function.