Let's carefully work through completing the square to identify the first error in Caroline's work.
Given the function:
[tex]\[ f(x) = -2x^2 + 12x - 15 \][/tex]
Step 1: Factor out the coefficient of [tex]\(x^2\)[/tex] from the quadratic and linear terms.
[tex]\[ f(x) = -2(x^2 - 6x) - 15 \][/tex]
Step 2: Complete the square inside the parentheses. To complete the square, add and subtract the square of half the linear coefficient inside the parentheses.
[tex]\[ x^2 - 6x \][/tex]
Half of the linear coefficient [tex]\( -6 \)[/tex] is [tex]\( -3 \)[/tex]. Squaring this gives [tex]\( 9 \)[/tex].
So add and subtract [tex]\( 9 \)[/tex] inside the parentheses:
[tex]\[ -2(x^2 - 6x + 9 - 9) - 15 \][/tex]
Separate the terms:
[tex]\[ -2[(x - 3)^2 - 9] - 15 \][/tex]
Step 3: Distribute the [tex]\(-2\)[/tex] across the terms inside the parentheses:
[tex]\[ -2(x - 3)^2 + 18 - 15 \][/tex]
Step 4: Combine the constant terms:
[tex]\[ -2(x - 3)^2 + 3 \][/tex]
So, the correct form of the function after completing the square is:
[tex]\[ f(x) = -2(x - 3)^2 + 3 \][/tex]
Now, let's identify the first error in Caroline's work. In Caroline's work:
[tex]\[ f(x) = -2(x^2 - 6x + 9) - 9 - 15 \][/tex]
This was incorrect. Caroline added and subtracted [tex]\(9\)[/tex] correctly within the parentheses, but after multiplying by [tex]\(-2\)[/tex], she:
[tex]\[ -2(x - 3)^2 - 9 - 15 \][/tex]
She incorrectly subtracted [tex]\(9\)[/tex] directly, forgetting to account for the [tex]\(-2\)[/tex] factor that modified it. This resulted in:
[tex]\[ -2(x - 3)^2 - 24 \][/tex]
The issue first occurred when Caroline subtracted the value after completing the square. She should have added [tex]\(18\)[/tex] (since [tex]\(-2 \times -9 = 18\)[/tex]) and then subtracted [tex]\(15\)[/tex]. Therefore, the first error was:
A. She subtracted the wrong value to maintain balance after completing the square.
