Problem 2 - Algebra 1: Concepts and Skills

Question

Explain what an extraneous solution is.

Step-by-Step Solution

Answer

Extraneous solutions are results that emerge during the process of solving an equation but do not satisfy the original equation. They're often found in algebraic or mathematical equations when methods such as squaring both sides or dealing with rational equations are used. Always check solutions in the original equation to ensure they're not extraneous.

Step 1: Definition of Extraneous Solutions

Extraneous solutions are solutions to an equation that emerge from the process of solving but do not satisfy the original problem. These are often the result of mathematical operations involving squaring both sides of an equation, or when solving rational equations.

Step 2: Give an example of Extraneous Solutions

For example, consider the equation \( \sqrt{x-3} + 1 = x \). After squaring both sides to eliminate the square root, we get \(x-3+2\sqrt{x-3}+1=x^2\). Simplifying and bringing terms to one side, we get \(x^2-2x+2=0\). Solving for \(x\) gives us \(x=1±i\). However, if we substitute these solutions back into the original problem, we find that they do not hold true. Therefore, these are extraneous solutions.

Step 3: Importance of Checking for Extraneous Solutions

While solving mathematical or algebraic equations, it is very important to always check potential solutions against the original problem. This 'checking back' step is fundamental to ensure the solutions are not extraneous.