Problem 117 - Algebra and Trigonometry

Question

What is an extraneous solution to a radical equation?

Step-by-Step Solution

Answer

Extraneous solutions are solutions that emerge from the solving process but are not valid when plugged back into the original equation. It often occurs when radical equations are being solved and can be identified by checking the solution in the original equation.

Step 1: Definition

An extraneous solution is a solution that results from the solving process but does not satisfy the original problem. It often occurs in algebra when squaring both sides of an equation during the solving process.

Step 2: Why extraneous solution occurs

Extraneous solutions often occur when solving equations involving radicals or fractions. When square roots are involved in an equation, squaring both sides eliminate the square root, leading to a simpler equation. However, this process can introduce solutions that weren't there in the original equation. These are called extraneous solutions.

Step 3: Checking for extraneous solutions

To check if the solution is extraneous, plug the calculated value back to the original equation. If it makes the equation true it is a valid solution, but if it does not, it is an extraneous solution. It is important to always check for extraneous solutions when solving radical equations.

Step 4: Example of an extraneous solution

Consider the equation \(\sqrt{x + 2} = x - 2\). After squaring both sides, we get \(x + 2 = x^2 - 4x + 4\) or \(x^2 - 5x + 2 = 0\). Solving this we get \(x = 1, 2\). After checking back, \(x = 1\) is an extraneous solution because \(1 + 2\) doesn't equal to \(1 - 2\).