Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5.
x² + 2x - 8 = 0
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.6.50
Solve each quadratic equation using the zero-factor property. See Example 5.
9x² - 12x + 4 = 0
Verified step by step guidance1
Recognize that the given equation is a quadratic equation in the form \(ax^{2} + bx + c = 0\), where \(a = 9\), \(b = -12\), and \(c = 4\).
Check if the quadratic can be factored easily. Notice that \(9x^{2} - 12x + 4\) is a perfect square trinomial because \(9x^{2} = (3x)^{2}\) and \(4 = 2^{2}\), and the middle term \(-12x\) equals \(2 \times 3x \times (-2)\).
Rewrite the quadratic as a square of a binomial: \((3x - 2)^{2} = 0\).
Apply the zero-factor property, which states that if a product of factors equals zero, then at least one of the factors must be zero. Here, set the factor equal to zero: \(3x - 2 = 0\).
Solve the linear equation \(3x - 2 = 0\) by isolating \(x\): add 2 to both sides and then divide both sides by 3 to find the solution for \(x\).
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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 its solutions are the values of x that satisfy the equation.
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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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Factoring Quadratic Expressions
Factoring involves rewriting a quadratic expression as a product of two binomials. This step is essential before applying the zero-factor property, as it breaks down the quadratic into simpler expressions that can be individually set to zero.
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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 square root property. See Example 6. x² - 27 = 0
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 4x² - 4x + 1 = 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