Textbook Question
Solve and check each linear equation. 4x + 9 = 33
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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 1
Solve each equation in Exercises 1 - 14 by factoring.
Verified step by step guidance1
Start with the quadratic equation: \(x^2 - 3x - 10 = 0\).
Look for two numbers that multiply to the constant term \(-10\) and add up to the coefficient of the linear term \(-3\).
Express the quadratic as a product of two binomials using those two numbers: \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
Set each binomial equal to zero to find the solutions: \(x + a = 0\) and \(x + b = 0\).
Solve each equation for \(x\) to find the roots of the quadratic.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Quadratic Equations
Factoring quadratic equations involves rewriting the quadratic expression as a product of two binomials. This method is useful when the quadratic can be expressed as (x + a)(x + b) = 0, where a and b are numbers that multiply to the constant term and add to the coefficient of the linear term.
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Zero Product Property
The zero product property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for the variable, which is essential for finding the roots of the equation after factoring.
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Solving Quadratic Equations
Solving quadratic equations means finding the values of the variable that satisfy the equation. After factoring, these solutions are found by setting each factor equal to zero and solving the resulting linear equations, yielding the roots or solutions of the quadratic.
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Related Practice
Textbook Question
Graph each equation in Exercises 1–4. Let x= -3, -2. -1, 0, 1, 2 and 3. y = 2x-2
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Textbook Question
Plot the given point in a rectangular coordinate system. (1, 4)
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Textbook Question
In Exercises 1–8, add or subtract as indicated and write the result in standard form. (7 + 2i) + (1 - 4i)
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Textbook Question
Graph each equation in Exercises 1–4. Let x= -3, -2. -1, 0, 1, 2 and 3. y = x^2-3
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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. (1, 6]
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