Textbook Question
Solve each equation. 5x+4= 3x-4
Ch. 1 - Equations and Inequalities
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 2, Problem 11
Use Choices A–D to answer each question. A. 3x2 - 17x - 6 = 0 B. (2x + 5)2 = 7 C. x2 + x = 12 D. (3x - 1)(x - 7) = 0 Only one of the equations does not require Step 1 of the method for completing the square described in this section. Which one is it? Solve it.
Verified step by step guidance1
Step 1: Understand the method of completing the square. The first step usually involves making the coefficient of \(x^2\) equal to 1 by dividing the entire equation by that coefficient if it is not already 1.
Step 2: Examine each equation to see if the coefficient of \(x^2\) is 1. For equation A: \(3x^2 - 17x - 6 = 0\), the coefficient of \(x^2\) is 3, so Step 1 is required.
Step 3: For equation B: \((2x + 5)^2 = 7\), the equation is already a perfect square on the left side, so the coefficient of \(x^2\) is effectively 1 after expansion, and Step 1 is not needed.
Step 4: For equation C: \(x^2 + x = 12\), the coefficient of \(x^2\) is 1, so Step 1 is not needed here either.
Step 5: For equation D: \((3x - 1)(x - 7) = 0\), this is a factored form, not a quadratic in standard form, so completing the square is not the first method to use; instead, solve by setting each factor equal to zero.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square Method
Completing the square is a technique used to solve quadratic equations by transforming them into a perfect square trinomial. This involves isolating the constant term, dividing the coefficient of x by 2, squaring it, and adding it to both sides. It simplifies solving quadratics that are not easily factorable.
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Solving Quadratic Equations by Completing the Square
Identifying When Completing the Square is Needed
Not all quadratic equations require completing the square. If the equation is already factored or can be solved by simple algebraic manipulation, the initial step of isolating the quadratic and linear terms (Step 1) may be unnecessary. Recognizing these cases saves time and effort.
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Solving Quadratic Equations by Factoring
Factoring involves expressing a quadratic equation as a product of binomials set to zero. If the quadratic is already factored, like (3x - 1)(x - 7) = 0, you can directly find the roots by setting each factor equal to zero, bypassing the need for completing the square.
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Related Practice
Textbook Question
Solve each problem. See Example 1. Michael must build a rectangular storage shed. He wants the length to be 6 ft greater than the width, and the perimeter will be 44 ft. Find the length and the width of the shed.
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Textbook Question
Solve each equation. | 5 - 3x | = 3
Textbook Question
Solve each problem. Alcohol Mixture Barak wishes to strengthen a mixture that is 10% alcohol to one that is 30% alcohol. How much pure alcohol should he add to 12 L of the 10% mixture?
Textbook Question
Solve each equation. |7 - 3x| = 3
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Textbook Question
Use the following facts. If x represents an integer, then x+1 represents the next consecutive integer. If x represents an even integer, then x+2 represents the next consecutive even integer. If x represents an odd integer, then x+2 represents the next consecutive odd integer. Find two consecutive even integers whose product is 224.