Question

Explain what an extraneous solution is.

Short answer

An extraneous solution is a result obtained in the process of solving an equation that doesn't satisfy the original equation. An example is when solving the equation \(\sqrt{x} = x - 2\), squaring both sides leads to extraneous solutions.

Step-by-step solution

Define the Term

An extraneous solution is a solution obtained from the process of solving the equation that is not a true solution of the original equation. It usually results from the application of an operation that is not allowed or changes the solution set of the equation.

Provide an Example

For instance, consider the equation \(\sqrt{x} = x - 2\). Squaring both sides to vanish the square root leads to \(x = (x - 2)^2\), which simplifies to \(x = x^2 - 4x + 4\). Solving this quadratic equation would give solutions as x = 1 and x = 4. However, substituting x = 1 back into the original equation would result in a contradiction, which indicates that x = 1 is an extraneous solution.