Let's go through Caroline's steps to rewrite the quadratic equation in vertex form by completing the square:
1. Starting with the quadratic function:
[tex]\[
f(x) = -2x^2 + 12x - 15
\][/tex]
2. Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[
f(x) = -2(x^2 - 6x) - 15
\][/tex]
3. Complete the square for the expression inside the parentheses. To complete the square for [tex]\(x^2 - 6x\)[/tex], take half of the coefficient of [tex]\(x\)[/tex], square it, and add and subtract this value inside the parentheses:
[tex]\[
x^2 - 6x \quad \Rightarrow \quad \left( x^2 - 6x + 9 \right) - 9
\][/tex]
where 9 is [tex]\(\left(\frac{-6}{2}\right)^2\)[/tex].
4. Substitute this back into the equation:
[tex]\[
f(x) = -2(x^2 - 6x + 9 - 9) - 15
\][/tex]
5. Separate the completed square and simplify:
[tex]\[
f(x) = -2\left((x - 3)^2 - 9\right) - 15
\][/tex]
6. Distribute the [tex]\(-2\)[/tex] inside the parentheses:
[tex]\[
f(x) = -2(x - 3)^2 + 18 - 15
\][/tex]
7. Combine the constant terms:
[tex]\[
f(x) = -2(x - 3)^2 + 3
\][/tex]
As we can see, Caroline's mistake occurred in step 4. After correctly completing the square inside the parentheses, she incorrectly added [tex]\(-9\)[/tex] after factoring [tex]\(-2\)[/tex] out of the expression [tex]\( (x^2 - 6x + 9) \)[/tex].
Therefore, the first error in Caroline's work is:
B. She subtracted the wrong value to maintain balance after completing the square.
By maintaining the balance correctly, the correct rewritten function should be:
[tex]\[
f(x) = -2(x - 3)^2 + 3
\][/tex]
