Textbook Question
Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 7
Solve each equation in Exercises 1 - 14 by factoring.
Verified step by step guidance1
First, rewrite the equation so that all terms are on one side, setting the equation equal to zero: \(3x^2 - 2x - 8 = 0\).
Next, look for two numbers that multiply to the product of the coefficient of \(x^2\) term and the constant term (i.e., \(3 \times (-8) = -24\)) and add up to the coefficient of the \(x\) term (which is \(-2\)).
Use these two numbers to split the middle term \(-2x\) into two terms, then group the terms in pairs to factor by grouping.
Factor out the greatest common factor (GCF) from each group, which should give you a common binomial factor.
Set each factor equal to zero and solve for \(x\) to find the solutions of the equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rearranging Equations to Standard Form
To solve quadratic equations by factoring, first rewrite the equation so that one side equals zero. This involves moving all terms to one side, resulting in a standard form ax² + bx + c = 0, which is essential for applying factoring techniques.
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Standard Form of Line Equations
Factoring Quadratic Expressions
Factoring involves expressing a quadratic polynomial as a product of two binomials. This step simplifies solving the equation by setting each factor equal to zero, allowing us to find the roots of the quadratic.
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Zero Product Property
The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is used after factoring to set each factor equal to zero and solve for the variable.
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Related Practice
Textbook Question
A new car worth \$36,000 is depreciating in value by \$4000 per year. a. Write a formula that models the car's value, y, in dollars, after x years. b. Use the formula from part (a) to determine after how many years the car's value will be \$12,000. c. Graph the formula from part (a) in the first quadrant of a rectangular coordinate system. Then show your solution to part (b) on the graph.
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Textbook Question
In Exercises 1–14, express each interval in set-builder notation and graph the interval on a number line. (2, ∞)
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Textbook Question
Solve and check each linear equation. 11x - (6x - 5) = 40
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Textbook Question
In Exercises 6–8, use the graph and determine the x-intercepts if any, and the y-intercepts if any. For each graph, tick marks along the axes represent one unit each.
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Textbook Question
In Exercises 1–8, add or subtract as indicated and write the result in standard form. 8i - (14 - 9i)
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