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

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

Identify the first error in her work.

A. She incorrectly combined the constant terms.

B. She incorrectly factored out the value of [tex]\( a \)[/tex].

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
To identify the first error in Caroline's work, we need to carefully go through the steps she took in completing the square.

Let's rewrite and analyze the quadratic function step-by-step:

1. Original function:
[tex]\[ f(x) = -2x^2 + 12x - 15 \][/tex]

2. Factoring out the leading coefficient (not the error yet):
[tex]\[ f(x) = -2(x^2 - 6x) - 15 \][/tex]

3. Completing the square:
[tex]\[ -2(x^2 - 6x + 9) - 9 - 15 \][/tex]
Here, Caroline added and subtracted 9 inside the parenthesis to complete the square. The completed square is [tex]\((x - 3)^2\)[/tex], but we must also account for the impact of the factor [tex]\(-2\)[/tex] when maintaining balance:

Correct step should involve [tex]\(-2 \cdot 9\)[/tex]:
[tex]\[ -2(x^2 - 6x + 9) - 18 - 15 \][/tex]

4. Simplifying, we correctly get:
[tex]\[ -2(x - 3)^2 - 33 \][/tex]

Returning to Caroline's work, she incorrectly wrote:
[tex]\[ -2(x^2 - 6x + 9) - 9 - 15 \][/tex]
Instead of:
[tex]\[ -2(x^2 - 6x + 9) - 18 - 15 \][/tex]

So the first error in her work occurs in the third step. She subtracted the wrong value when maintaining balance after completing the square. Therefore, the correct answer is:

C. She subtracted the wrong value to maintain balance after completing the square.
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