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Caroline rewrote a quadratic equation in vertex form by completing the square, but her work has errors.

[tex]\[
\begin{aligned}
f(x) & = -2x^2 + 12x - 15 \\
& = -2\left(x^2 - 6x\right) - 15 \\
& = -2\left(x^2 - 6x + 9\right) - 9 - 15 \\
& = -2(x - 3)^2 - 9 - 15 \\
& = -2(x - 3)^2 - 24
\end{aligned}
\][/tex]

Identify the first error in her work.

A. She subtracted the wrong value to maintain balance after completing the square.
B. She incorrectly combined the constant terms.
C. She incorrectly factored out the value of [tex]a[/tex].

Sagot :

GinnyAnswer
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.
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