Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 27

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 guidance

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.

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.

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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