Which error did Mathieu make?
A. He factored incorrectly.
B. He did not use the constant as the $x$-intercept.
C. He set the factored expressions equal to each other.
D. He incorrectly solved the equation $x+3=x+1$.
Answer (1)
The problem asks to identify Mathieu's error from a list of options.
Analyze each option to determine the type of error it represents.
Option 3, 'He set the factored expressions equal to each other,' is a common mistake when solving equations.
Therefore, the most probable error Mathieu made is setting the factored expressions equal to each other.
Explanation
Understanding the Problem We are asked to identify the error Mathieu made from a list of options. The options are:
He factored incorrectly.
He did not use the constant as the x -intercept.
He set the factored expressions equal to each other.
He incorrectly solved the equation x + 3 = x + 1 .
Analyzing the Options Let's analyze each option to determine the type of error it represents.
He factored incorrectly: This suggests an error in algebraic manipulation, specifically in factoring a polynomial or expression.
He did not use the constant as the x -intercept: This implies a misunderstanding of how constants relate to x -intercepts in a function or equation. The constant term of a polynomial is related to the y-intercept, not the x-intercept. The x-intercept is found by setting y = 0 and solving for x .
He set the factored expressions equal to each other: This could occur when solving an equation where terms are moved to one side and then factored. Setting the factors equal to zero is the correct procedure, but setting the expressions equal to each other is generally incorrect unless there's a specific reason to do so (e.g., comparing two functions).
He incorrectly solved the equation x + 3 = x + 1 : This indicates an error in basic algebra. Let's solve it: Subtracting x from both sides gives 3 = 1 , which is a contradiction. This means there's no solution to the equation, but the error is in the interpretation, not necessarily in the solving process itself.
Identifying the Most Likely Error Without more context about Mathieu's work, it's difficult to pinpoint the exact error. However, based on the options, the most likely error is related to a misunderstanding of fundamental algebraic principles.
Option 3, "He set the factored expressions equal to each other," is a common mistake when solving equations. For example, if Mathieu had an equation that simplified to ( x + 3 ) ( x + 1 ) = 0 , the correct next step is to set each factor equal to zero: x + 3 = 0 or x + 1 = 0 . Setting the factors equal to each other, x + 3 = x + 1 , is incorrect in this context.
Conclusion Therefore, the most probable error Mathieu made is setting the factored expressions equal to each other.
Examples
When solving quadratic equations, students often make mistakes in factoring or in applying the zero-product property. For instance, consider the equation x 2 + 5 x + 6 = 0 . The correct approach is to factor it as ( x + 2 ) ( x + 3 ) = 0 , and then set each factor to zero, yielding x = − 2 or x = − 3 . A common error is to set the factors equal to each other, i.e., x + 2 = x + 3 , which leads to an incorrect conclusion. Understanding these algebraic manipulations is crucial in various fields, such as physics, engineering, and economics, where solving equations is a fundamental skill.
