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Extraneous Solutions

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Extraneous Solutions

Study Guide

Key Definition

An extraneous solution is a root of a transformed equation that is not a root of the original equation because it was excluded from the domain of the original equation.

Important Notes

Check if the solution is valid in the original equation's domain
Extraneous solutions arise from operations like squaring or multiplying both sides by expressions that could be zero
Always verify potential solutions by substituting back into the original equation
Division by zero is undefined, leading to exclusions from the domain
Solving rational equations often requires checking for extraneous solutions

Mathematical Notation

$\\frac{a}{b}$ represents a fraction

$x^2$ represents x squared

$\\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

Extraneous solutions occur due to transformations that alter the equation's domain, such as squaring both sides or multiplying by variables.

Remember

Always check solutions by substituting back into the original equation to ensure they are not extraneous.

Quick Reference

Extraneous Solution:A solution that does not satisfy the original equation.

Understanding Extraneous Solutions

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Video explanation of this concept

Beginner

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Beginner Explanation

Extraneous solutions are results that do not satisfy the original equation, often due to domain restrictions.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the extraneous solution of $\\frac{1}{x - 2} + \\frac{1}{x + 2} = \\frac{4}{(x - 2)(x + 2)}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are solving an equation to find out how much time you need to complete a project. Solve the equation $\\frac{1}{t-3} + \\frac{1}{t+3} = \\frac{2}{t^2-9}$. Find all real solutions and identify any extraneous solutions.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find all real solutions of the equation $\\frac{x+4}{x-2} = \\frac{x-2}{x+4}$ and identify any extraneous solutions.

Challenge Quiz

Single Choice Quiz

Advanced

For the equation $\\sqrt{x+1} = x - 1$, identify the extraneous solution.

Recap

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