Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
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.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.