Textbook Question
Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. -2(x + 3) = -6(x + 7)
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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.6.41
Solve each quadratic equation using the zero-factor property. See Example 5. x² - 5x + 6 = 0
Verified step by step guidance1
Start with the quadratic equation given: \(x^{2} - 5x + 6 = 0\).
Factor the quadratic expression on the left side. Look for two numbers that multiply to \(6\) (the constant term) and add up to \(-5\) (the coefficient of \(x\)).
Write the factored form as \((x - a)(x - b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
Apply the zero-factor property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero: \(x - a = 0\) and \(x - b = 0\).
Solve each equation for \(x\) to find the solutions to the 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 polynomial equation of degree two, generally written as ax² + bx + c = 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
Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. For example, x² - 5x + 6 factors into (x - 2)(x - 3). This step is essential to apply the zero-factor property to find the roots.
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6:08Factoring
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 principle allows solving equations like (x - 2)(x - 3) = 0 by setting each factor equal to zero.
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6:08Factoring
Related Practice
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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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
9x² - 12x + 4 = 0
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Example 10. 10 ≤ 2x + 4 ≤ 16
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