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:

Analyze the student's work

Using the Quadratic Functions and x-intercepts knowledge points

$$
LATEXBLOCK0
$$

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).

Answer:

• (A) He incorrectly solved the equation \(x+3 = x+1\).
• (B) He factored incorrectly.
• (C) He set the factored expressions equal to each other. (Correct answer)