Podcast
Questions and Answers
What is the first step in solving the equation $ ext{√}5 + x + 10 = 6$?
What is the first step in solving the equation $ ext{√}5 + x + 10 = 6$?
The first step is to isolate the variable $x$ by moving the constant term to the other side of the equation.
Explain why the statement $ ext{√}5 + x = -4$ leads to an incorrect solution.
Explain why the statement $ ext{√}5 + x = -4$ leads to an incorrect solution.
The expression $ ext{√}5$ is positive, and adding $x$ cannot equal a negative number like -4, which means there are no real solutions.
What error is made when transforming the equation to $5 + x = 16$?
What error is made when transforming the equation to $5 + x = 16$?
The error arises from incorrectly manipulating the equation; it misapplies properties of equality after reaching an impossible condition.
Calculate the correct value of $x$ from the original equation $ ext{√}5 + x + 10 = 6$.
Calculate the correct value of $x$ from the original equation $ ext{√}5 + x + 10 = 6$.
Signup and view all the answers
What does the final value of $x = 11$ suggest about the validity of the proposed method?
What does the final value of $x = 11$ suggest about the validity of the proposed method?
Signup and view all the answers
Study Notes
Analysis of the Solution Method
- The equation presented is √(5+x) + 10 = 6.
- The solution method attempts to isolate the square root term, then square both sides to eliminate the radical.
Step-by-Step Evaluation
-
Step 1: √(5+x) + 10 = 6. The first step attempts to isolate the square root term.
- Subtracting 10 from both sides gives √(5+x) = -4.
-
Step 2: √(5+x) = -4
- Squaring both sides to eliminate the radical results in (5+x) = 16.
-
Step 3: 5+x = 16
- Solving for x, we have x = 11.
Validity of the Solution
-
Crucial Error: The method is flawed because a square root cannot yield a negative result.
- The critical step where √(5+x) = -4 is invalid. The square root of any real number must be a non-negative value. A square root can never equal a negative number.
- Incorrect Conclusion: Consequently, x=11 does not satisfy the original equation, despite seemingly following algebraic manipulations.
Conclusion
- The solution method provides an invalid intermediate step and will lead to an incorrect solution.
- The presented solution is incorrect. The given solution method is demonstrably flawed and does not produce the correct solution to the original equation.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
This quiz examines the solution method of a specific algebraic equation involving square roots. It focuses on isolating terms, squaring both sides, and identifying flaws in the logical process that lead to incorrect conclusions about the solution. Test your understanding of algebraic principles and error analysis.
