Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = 9 - x2
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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 27
Solve each radical equation in Exercises 11–30. Check all proposed solutions. √(2x + 3) + √(x - 2) = 2
Verified step by step guidance1
Identify the equation: \(\sqrt{2x + 3} + \sqrt{x - 2} = 2\). Our goal is to solve for \(x\).
Isolate one of the square root terms. For example, subtract \(\sqrt{x - 2}\) from both sides to get: \(\sqrt{2x + 3} = 2 - \sqrt{x - 2}\).
Square both sides of the equation to eliminate the square root on the left. This gives: \(\left(\sqrt{2x + 3}\right)^2 = \left(2 - \sqrt{x - 2}\right)^2\).
Simplify both sides: the left side becomes \(2x + 3\), and the right side expands using the formula \((a - b)^2 = a^2 - 2ab + b^2\) to \(4 - 4\sqrt{x - 2} + (x - 2)\).
Rearrange the equation to isolate the remaining square root term, then square both sides again to eliminate it. After that, solve the resulting polynomial equation for \(x\). Finally, check all proposed solutions in the original equation to discard any extraneous roots.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Radical Equations
Radical equations involve variables within a root, typically a square root. Solving them requires isolating the radical expression and then eliminating the root by raising both sides of the equation to the appropriate power, often squaring. This process can introduce extraneous solutions, so checking all proposed solutions is essential.
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Expanding Radicals
Domain Restrictions
The domain of a radical equation is limited by the requirement that the expression inside the square root must be non-negative. For example, in √(2x + 3), the expression 2x + 3 must be ≥ 0. Identifying these restrictions helps determine valid solutions and avoid invalid or extraneous answers.
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3:51Domain Restrictions of Composed Functions
Checking Solutions
After solving a radical equation, substituting the solutions back into the original equation is crucial to verify their validity. Squaring both sides can introduce extraneous solutions that do not satisfy the original equation. Checking ensures only true solutions are accepted.
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Related Practice
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Solve each equation in Exercises 15–34 by the square root property.
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