Question

analyzing student work

mathieu is finding the \\(x\\)-intercepts of the function \\(f(x) = x^2 + 4x + 3\\).

mathieus work

1. \\(0 = x^2 + 4x + 3\\)
2. \\(0 = (x + 3)(x + 1)\\)
3. \\(x + 3 = x + 1\\)
4. \\(x = x - 2\\)
5. \\(0 = -2\\)
6. there are no \\(x\\)-intercepts.

options:

• he incorrectly solved the equation \\(x + 3 = x + 1\\).
• he factored incorrectly.
• he set the factored expressions equal to each other.

Explanation:

<pre_analysis>
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"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Functions",
"x-intercepts"
],
"new_concepts": [
"Zero Product Property",
"Analyzing Student Errors"
],
"current_concepts": [
"Quadratic Functions",
"x-intercepts",
"Zero Product Property",
"Analyzing Student Errors"
]
}
</pre_analysis>

<reasoning>

Analyze the student's work

Using the Quadratic Functions and x-intercepts knowledge points
\[

$$\begin{aligned} &f(x) = x^2 + 4x + 3 \\ &\text{Step 1: } 0 = x^2 + 4x + 3 \quad \text{(Correct setup for finding x-intercepts)} \\ &\text{Step 2: } 0 = (x+3)(x+1) \quad \text{(Correct factoring since } 3 \times 1 = 3 \text{ and } 3 + 1 = 4\text{)} \\ &\text{Step 3: } x+3 = x+1 \quad \text{(Incorrect step)} \end{aligned}$$

\]

Identify the mathematical error

To solve the factored equation \(0 = (x+3)(x+1)\), we must apply the Zero Product Property, which states that if \(ab = 0\), then \(a = 0\) or \(b = 0\).
Therefore, the correct step is:
\[
x+3 = 0 \quad \text{or} \quad x+1 = 0
\]
Instead of setting each factor equal to zero, Mathieu set the two factored expressions equal to each other:
\[
x+3 = x+1
\]

Match with the given options

Let's evaluate the visible options on the right side of the image:

• Option 1: "He incorrectly solved the equation \(x+3 = x+1\)." (While he did solve this incorrectly, the primary conceptual error occurred in setting them equal to each other in Step 3).
• Option 2: "He factored incorrectly." (Incorrect, the factoring in Step 2 is correct).
• Option 3: "He set the factored expressions equal to each other." (Correct, this describes the exact conceptual error made in Step 3).

</reasoning>

<answer>
<mcq-option>(A) He incorrectly solved the equation \(x+3 = x+1\).</mcq-option>
<mcq-option>(B) He factored incorrectly.</mcq-option>
<mcq-correct>(C) He set the factored expressions equal to each other.</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Functions"
]
}
</post_analysis>

Answer:

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"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Quadratic Functions",
"x-intercepts"
],
"new_concepts": [
"Zero Product Property",
"Analyzing Student Errors"
],
"current_concepts": [
"Quadratic Functions",
"x-intercepts",
"Zero Product Property",
"Analyzing Student Errors"
]
}
</pre_analysis>

<reasoning>

Analyze the student's work

Using the Quadratic Functions and x-intercepts knowledge points
\[

$$\begin{aligned} &f(x) = x^2 + 4x + 3 \\ &\text{Step 1: } 0 = x^2 + 4x + 3 \quad \text{(Correct setup for finding x-intercepts)} \\ &\text{Step 2: } 0 = (x+3)(x+1) \quad \text{(Correct factoring since } 3 \times 1 = 3 \text{ and } 3 + 1 = 4\text{)} \\ &\text{Step 3: } x+3 = x+1 \quad \text{(Incorrect step)} \end{aligned}$$

\]

Identify the mathematical error

To solve the factored equation \(0 = (x+3)(x+1)\), we must apply the Zero Product Property, which states that if \(ab = 0\), then \(a = 0\) or \(b = 0\).
Therefore, the correct step is:
\[
x+3 = 0 \quad \text{or} \quad x+1 = 0
\]
Instead of setting each factor equal to zero, Mathieu set the two factored expressions equal to each other:
\[
x+3 = x+1
\]

Match with the given options

Let's evaluate the visible options on the right side of the image:

• Option 1: "He incorrectly solved the equation \(x+3 = x+1\)." (While he did solve this incorrectly, the primary conceptual error occurred in setting them equal to each other in Step 3).
• Option 2: "He factored incorrectly." (Incorrect, the factoring in Step 2 is correct).
• Option 3: "He set the factored expressions equal to each other." (Correct, this describes the exact conceptual error made in Step 3).

</reasoning>

<answer>
<mcq-option>(A) He incorrectly solved the equation \(x+3 = x+1\).</mcq-option>
<mcq-option>(B) He factored incorrectly.</mcq-option>
<mcq-correct>(C) He set the factored expressions equal to each other.</mcq-correct>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Quadratic Functions"
]
}
</post_analysis>