Answer :
Let's analyze the steps performed by the student to solve the quadratic equation [tex]\( x^2 - 2x - 8 = 0 \)[/tex].
### Step 1: Factor the polynomial
The student factored the polynomial [tex]\( x^2 - 2x - 8 \)[/tex] into [tex]\((x - 4)(x - 2)\)[/tex].
To check if this factoring is correct, we need to perform the multiplication:
[tex]\[ (x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8 \][/tex]
However, the original polynomial is [tex]\( x^2 - 2x - 8 \)[/tex], not [tex]\( x^2 - 6x + 8 \)[/tex]. Therefore, the correct factorization of the polynomial [tex]\( x^2 - 2x - 8 \)[/tex] should be checked again.
The correct form should factor into:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
To check, we multiply:
[tex]\[ (x - 4)(x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8 \][/tex]
This matches the original polynomial, so the correct factorization is:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
Thus, the student made an error in Step 1 by incorrectly factoring the polynomial into [tex]\((x - 4)(x - 2)\)[/tex] instead of the correct [tex]\((x - 4)(x + 2)\)[/tex].
### Step 2: Set each factor to zero
The student set each factor from Step 1 to zero:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x - 2 = 0 \][/tex]
Since Step 1 was incorrect, continuing from this step is based on a wrong factorization.
### Step 3: Solve for [tex]\( x \)[/tex]
The student solved the equations from Step 2:
[tex]\[ x - 4 = 0 \Rightarrow x = 4 \][/tex]
[tex]\[ x - 2 = 0 \Rightarrow x = 2 \][/tex]
Again, since Step 1 was wrong, no analysis needed at this step as it is based on incorrect factors.
### Conclusion
The student went wrong in Step 1 by improperly factoring the polynomial. The correct factorization should have been [tex]\((x - 4)(x + 2)\)[/tex].
Therefore, the correct answer is:
D. in Step 1
### Step 1: Factor the polynomial
The student factored the polynomial [tex]\( x^2 - 2x - 8 \)[/tex] into [tex]\((x - 4)(x - 2)\)[/tex].
To check if this factoring is correct, we need to perform the multiplication:
[tex]\[ (x - 4)(x - 2) = x^2 - 2x - 4x + 8 = x^2 - 6x + 8 \][/tex]
However, the original polynomial is [tex]\( x^2 - 2x - 8 \)[/tex], not [tex]\( x^2 - 6x + 8 \)[/tex]. Therefore, the correct factorization of the polynomial [tex]\( x^2 - 2x - 8 \)[/tex] should be checked again.
The correct form should factor into:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
To check, we multiply:
[tex]\[ (x - 4)(x + 2) = x^2 + 2x - 4x - 8 = x^2 - 2x - 8 \][/tex]
This matches the original polynomial, so the correct factorization is:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
Thus, the student made an error in Step 1 by incorrectly factoring the polynomial into [tex]\((x - 4)(x - 2)\)[/tex] instead of the correct [tex]\((x - 4)(x + 2)\)[/tex].
### Step 2: Set each factor to zero
The student set each factor from Step 1 to zero:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x - 2 = 0 \][/tex]
Since Step 1 was incorrect, continuing from this step is based on a wrong factorization.
### Step 3: Solve for [tex]\( x \)[/tex]
The student solved the equations from Step 2:
[tex]\[ x - 4 = 0 \Rightarrow x = 4 \][/tex]
[tex]\[ x - 2 = 0 \Rightarrow x = 2 \][/tex]
Again, since Step 1 was wrong, no analysis needed at this step as it is based on incorrect factors.
### Conclusion
The student went wrong in Step 1 by improperly factoring the polynomial. The correct factorization should have been [tex]\((x - 4)(x + 2)\)[/tex].
Therefore, the correct answer is:
D. in Step 1
