Extraneous Solutions - Varsity Tutors

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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 xRemember 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

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BeginnerIntermediateAdvanced

Beginner Explanation

Extraneous solutions are results that do not satisfy the original equation, often due to domain restrictions.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$x = 2$B$x = -2$C$x = 0$D$x = 4$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

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.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$x = 0$B$x = 1$C$x = 2$D$x = 3$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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