Problem 117 - College Algebra Essentials

Question

What is an extraneous solution to a radical equation?

Step-by-Step Solution

Answer

An extraneous solution is a solution that comes about during the problem-solving process but doesn't satisfy the original problem. They often emerge when solving radical equations due to squaring or cubing both sides of an equation, which can introduce solutions that don't actually satisfy the original equation.

Step 1: Define Extraneous Solution

An extraneous solution is a solution that emerges from the process of solving the problem but does not satisfy the original problem.

Step 2: Describe Radical Equation

A radical equation is an equation in which the variable appears under a radical, such as a square root, cube root, etc.

Step 3: Show an Example

Let's illustrate this on an example. Suppose you solve the equation \(\sqrt{x} = x - 2\). You could square both sides to remove the square root. Then, you get \(x = (x - 2)^2\), which simplifies to \(x = x^2 - 4x + 4\), and finally to \(x^2 - 5x + 4 = 0\). The solutions to this equation are \(x = 1\) and \(x = 4\). But if you substitute these solutions into the original equation, \(\sqrt{x} = x - 2\), you can see that \(x = 4\) works, but \(x = 1\) doesn't because \(\sqrt{1} \neq 1 - 2\). Thus, \(x = 1\) is an extraneous solution.