To find the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 + 4x + 3 \)[/tex], we follow these steps:
1. Set the function equal to zero:
[tex]\[
0 = x^2 + 4x + 3
\][/tex]
2. Factor the quadratic equation:
[tex]\[
0 = (x + 3)(x + 1)
\][/tex]
3. Set each factor equal to zero:
[tex]\[
x + 3 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
4. Solve each equation for [tex]\( x \)[/tex]:
[tex]\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\][/tex]
[tex]\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts are [tex]\( x = -3 \)[/tex] and [tex]\( x = -1 \)[/tex].
Now, let's analyze Mathieu's steps:
1. [tex]\( 0 = x^2 + 4x + 3 \)[/tex]
2. [tex]\( 0 = (x + 3)(x + 1) \)[/tex]
So far, Mathieu's work is correct.
3. [tex]\( x + 3 = x + 1 \)[/tex]
This is where Mathieu made an error. Instead of setting each factor equal to zero, he set the factored expressions equal to each other.
4. [tex]\( x = x - 2 \)[/tex]
5. [tex]\( 0 = -2 \)[/tex]
6. Concluded that there are no [tex]\( x \)[/tex]-intercepts.
Because of his error in step 3, his subsequent steps and conclusion are incorrect.
Error Identified:
Mathieu made an error in step 3. He should have set each factor equal to zero rather than setting the factored expressions equal to each other.
The correct identification of the error:
He set the factored expressions equal to each other.
