Problem 5 - Intermediate Algebra

Question

Fill in the blanks. Proposed solutions of a radical equation that don’t satisfy it are called _____ solutions.

Step-by-Step Solution

Answer

These solutions are called 'extraneous solutions.'

Step 1: Understanding the Question

The exercise asks you to identify what solutions that don't satisfy a radical equation are called. In mathematical literature, there is a specific term for such solutions.

Step 2: Definition and Conceptual Understanding

When solving radical equations, it is possible to find solutions that, when substituted back into the original equation, do not hold true. These are solutions that appear valid based on algebraic manipulations but fail the initial condition of the equation.

Step 3: Technical Term Identification

The specific technical term used to describe solutions that don't satisfy the original equation when re-substituted back is 'extraneous solutions.' This occurs often with radical and rational equations during the solving process.

Key Concepts

Extraneous Solutions

When solving equations, you might encounter solutions that seem correct after performing algebraic manipulations but don't satisfy the original equation when checked. These are known as extraneous solutions. Such solutions are common when dealing with radical equations, where operations like squaring both sides can introduce values not initially valid for the original equation.
It's important to remember:

• Always substitute solutions back into the original equation to verify their validity.
• Extraneous solutions often arise from the method used to isolate variables.

Detecting these solutions ensures that only genuine and correct solutions are accepted.

Rational Equations

Rational equations involve fractions that can pose tricky situations similar to radical equations. In these equations, the variable appears in the denominator, thereby introducing the concept of restrictions where certain values make the equation undefined.

• Recognize the restrictions before solving: Identify values that make any denominators zero, as these are inadmissible solutions.
• Perform algebraic manipulations: Clear the fractions by finding a common denominator.

Be cautious when solving rational equations as these steps can sometimes lead to extraneous solutions, making validation crucial.

Algebraic Manipulations

Algebraic manipulations are techniques used to simplify, solve or rearrange equations. In radical and rational equations, these manipulations often involve:

• Isolating radicals by raising both sides of the equation to get rid of the square root.
• Clearing fractions by multiplying through by a common denominator in rational equations.

While these steps help to simplify equations, they can also introduce extraneous solutions. Thus, verification through substitution is a key step that ensures the solution satisfies the original equation, confirming it's not extraneous.