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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 incorrectly combined the constant terms.
B. She incorrectly factored out the value of a.
C. She subtracted the wrong value to maintain balance after completing the square.
D. She squared the wrong value when completing the square.

Sagot :

GinnyAnswer
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.
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