Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The numerical coefficient in the term -7yz² is ___________.
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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.6.51
Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0
Verified step by step guidance1
Start with the quadratic equation: \(25x^{2} + 30x + 9 = 0\).
Check if the quadratic can be factored into the form \((ax + b)(cx + d) = 0\) by looking for two numbers that multiply to \(25 \times 9 = 225\) and add to \(30\).
Rewrite the middle term \$30x$ as a sum of two terms using the numbers found in the previous step, then group the terms to factor by grouping.
Factor out the greatest common factor (GCF) from each group to express the quadratic as a product of two binomials.
Apply the zero-factor property: set each binomial equal to zero, \(ax + b = 0\) and \(cx + d = 0\), then solve each equation for \(x\) to find the solutions.
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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, where a, b, and c are constants and a ≠ 0. Solving these equations involves finding 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 two 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
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x - 2 ≤ 1 + x
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 2x - 2 = 0
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Textbook Question
Solve each quadratic equation using the square root property. See Example 6. x² = 16
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7.
-3x² + 6x + 5 = 0
Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. 4x + 7 ———— ≤ 2x + 5 -3
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