Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.6.42
Solve each quadratic equation using the zero-factor property. See Example 5.
x² + 2x - 8 = 0
Verified step by step guidance1
Start with the quadratic equation given: \(x^{2} + 2x - 8 = 0\).
Factor the quadratic expression on the left side. Look for two numbers that multiply to \(-8\) and add to \(2\).
Rewrite the quadratic as a product of two binomials: \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
Apply the zero-factor property, which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\). Set each binomial equal to zero: \(x + a = 0\) and \(x + b = 0\).
Solve each linear equation for \(x\) to find the solutions to the original quadratic equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the discriminant.
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Introduction to Quadratic Equations
Zero-Factor Property
The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by factoring them into binomials set equal to zero.
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6:08Factoring
Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. For example, x² + 2x - 8 factors into (x + 4)(x - 2). This step is essential before applying the zero-factor property to find the solutions.
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6:08Factoring
Related Practice
Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
9x² - 12x + 4 = 0
Textbook Question
Solve each quadratic equation using the square root property. See Example 6. x² - 27 = 0
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. 2x² - x - 28 = 0
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. -5 < 5 + 2x < 11
Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 2(x - 8) = 3x - 16