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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$.
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?
Flashcards
Solving an Equation
Solving an Equation
The process of isolating a variable by performing the same operations on both sides of the equation to maintain equality.
Squaring Both Sides
Squaring Both Sides
To get rid of the square root on one side of the equation, we square both sides. This is because squaring the square root of a number cancels it out.
Solution to an Equation
Solution to an Equation
The solution to an equation is the value that makes the equation true. If the value doesn't satisfy the original equation, it's not the solution.
Why the method is wrong?
Why the method is wrong?
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Correcting the Method
Correcting the Method
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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.
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