Textbook Question
Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 4x2 - 23x - 6 | = 0
Ch. 1 - Equations and Inequalities
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 2, Problem 95
Answer each question. Find the values of a, b, and c for which the quadratic equation. has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)
Verified step by step guidance1
Recall that if a quadratic equation \(ax^2 + bx + c = 0\) has solutions (roots) \(r_1\) and \(r_2\), then it can be factored as \(a(x - r_1)(x - r_2) = 0\).
Given the solutions \(4\) and \(5\), write the factored form of the quadratic as \(a(x - 4)(x - 5) = 0\).
Expand the factored form by first multiplying the binomials: \((x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20\).
Multiply the expanded expression by \(a\) to get \(a x^2 - 9a x + 20a = 0\), which matches the general form \(ax^2 + bx + c = 0\).
Identify the coefficients: \(a\) is the leading coefficient, \(b = -9a\), and \(c = 20a\). You can choose any nonzero value for \(a\) to get specific values for \(b\) and \(c\).
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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 of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions (roots) of the equation are the values of x that satisfy it, often found using factoring, completing the square, or the quadratic formula.
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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 and setting each equal to zero to find the roots.
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07:30Introduction to Factoring Polynomials
Forming Quadratic Equations from Roots
Given the roots of a quadratic equation, you can construct the equation by reversing the factoring process. If the roots are r₁ and r₂, the quadratic can be written as a(x - r₁)(x - r₂) = 0, which expands to ax² - a(r₁ + r₂)x + a(r₁r₂) = 0, allowing you to identify a, b, and c.
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Related Practice
Textbook Question
Simplify each power of i. i-13
Textbook Question
Solve each inequality. Give the solution set using interval notation. 3x+6 / x-5 > 0
Textbook Question
Answer each question. Find the values of a, b, and c for which the quadratic equation. has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.)
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Textbook Question
Solve each inequality. Give the solution set using interval notation. (x+7) / (2x+1) ≤1
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Textbook Question
Solve each equation. 4(x+1)4-13(x+1)2=-9