Ch. 1 - Equations and Inequalities

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

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 19

Chapter 2, Problem 19

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $$\sqrt{2x + 19}$ - 8 = x$

Verified step by step guidance

Start by isolating the square root term on one side of the equation. Add 8 to both sides to get: $\sqrt{2x + 19} = x + 8$.

Next, square both sides of the equation to eliminate the square root. This gives: $(\sqrt{2x + 19})^2 = (x + 8)^2$, which simplifies to $2x + 19 = (x + 8)^2$.

Expand the right side of the equation: $(x + 8)^2 = x^2 + 16x + 64$. So the equation becomes $2x + 19 = x^2 + 16x + 64$.

Rearrange the equation to set it equal to zero by subtracting $2x + 19$ from both sides: $0 = x^2 + 16x + 64 - 2x - 19$, which simplifies to $0 = x^2 + 14x + 45$.

Solve the quadratic equation $x^2 + 14x + 45 = 0$ using factoring, completing the square, or the quadratic formula. After finding the solutions, check each one by substituting back into the original equation to verify they do not produce extraneous results.

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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 are equations in which the variable is inside a root, such as a square root. Solving these often involves isolating the radical expression and then eliminating the root by raising both sides to the appropriate power.

Checking for Extraneous Solutions

When solving radical equations, squaring both sides can introduce extraneous solutions that do not satisfy the original equation. It is essential to substitute all solutions back into the original equation to verify their validity.

Isolating the Radical Expression

Before eliminating the radical, the radical term must be isolated on one side of the equation. This step simplifies the process of removing the root and helps avoid errors during the solution process.

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