Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = x3 - 1
Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 29
Solve each equation in Exercises 15–34 by the square root property.
Verified step by step guidance1
Identify the equation given: \( (3x + 2)^2 = 9 \). This is a perfect setup for the square root property because the variable expression is squared.
Apply the square root property, which states that if \( A^2 = B \), then \( A = \pm \sqrt{B} \). Here, set \( 3x + 2 = \pm \sqrt{9} \).
Simplify the square root: \( \sqrt{9} = 3 \), so rewrite the equation as \( 3x + 2 = \pm 3 \). This gives two separate equations to solve: \( 3x + 2 = 3 \) and \( 3x + 2 = -3 \).
Solve each linear equation separately. For \( 3x + 2 = 3 \), subtract 2 from both sides to isolate the term with \( x \). For \( 3x + 2 = -3 \), do the same.
Finally, divide both sides of each equation by 3 to solve for \( x \). This will give you the two possible solutions for \( x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Square Root Property
The square root property states that if an equation is in the form (expression)^2 = k, then the solution can be found by taking the square root of both sides, resulting in expression = ±√k. This method is useful for solving quadratic equations that are already isolated as a perfect square.
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Imaginary Roots with the Square Root Property
Isolating the Squared Term
Before applying the square root property, it is essential to isolate the squared term on one side of the equation. This means manipulating the equation so that (3x + 2)^2 stands alone, allowing you to take the square root of both sides accurately.
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Solving Linear Equations After Taking Square Roots
After applying the square root property, you get two linear equations due to the ± sign. Solving these linear equations involves isolating the variable x by performing inverse operations such as addition, subtraction, multiplication, or division.
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Related Practice
Textbook Question
Simplify and write the result in standard form. √-49
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Textbook Question
Solve each radical equation in Exercises 11–30. Check all proposed solutions.
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Textbook Question
Solve each equation in Exercises 15–34 by the square root property. (4x - 1)2 = 16
Textbook Question
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 3x - 7 ≥ 13
Textbook Question
Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 20 - x/3 = x/2