Let's walk through Caroline's work step by step to identify where the first error occurred:
1. Original equation (standard form):
[tex]\[
f(x) = -2x^2 + 12x - 15
\][/tex]
2. Factor out [tex]\(-2\)[/tex] from the quadratic and linear terms:
[tex]\[
f(x) = -2(x^2 - 6x) - 15
\][/tex]
So far, this is correct. Factoring out [tex]\(-2\)[/tex] is appropriate here.
3. Complete the square inside the parentheses:
To do this, take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-6\)[/tex], halve it to get [tex]\(-3\)[/tex], and then square it to get [tex]\(9\)[/tex].
Caroline added and subtracted this squared number inside the parentheses:
[tex]\[
f(x) = -2(x^2 - 6x + 9 - 9) - 15
\][/tex]
This step should be correct if balanced properly in the next steps.
4. Separate the perfect square trinomial:
[tex]\[
f(x) = -2((x - 3)^2 - 9) - 15
\][/tex]
Now, distribute the [tex]\(-2\)[/tex] correctly:
Here is where the mistake starts to appear. The inside the parentheses should be separated as follows:
[tex]\(-2(x - 3)^2\)[/tex] is correct, but the [tex]\( -9 \)[/tex] part is still inside multiplied by [tex]\(-2\)[/tex]:
[tex]\[
f(x) = -2(x - 3)^2 + 2 \cdot 9 - 15
= -2(x - 3)^2 + 18 - 15
= -2(x-3)^2 + 3
\][/tex]
Caroline's incorrect calculations show:
[tex]\[
f(x) = -2(x-3)^2 - 9 - 15
= -2(x-3)^2 - 24
\][/tex]
Identifying the first error:
Given that Caroline wrote:
[tex]\[
f(x) = -2(x^2 - 6x + 9) - 9 - 15
\][/tex]
She subtracted 9 instead of adding [tex]\( -2 \times 9 = 18 \)[/tex]. This disrupts the balance after completing the square.
Therefore, the correct choice is:
C. She subtracted the wrong value to maintain balance after completing the square.
