At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To convert the quadratic equation [tex]\( y = -x^2 - 8x - 29 \)[/tex] to its vertex form by completing the square, follow these steps:
1. Factor out the coefficient of [tex]\(x^2\)[/tex] term:
Since the coefficient of [tex]\(x^2\)[/tex] is [tex]\(-1\)[/tex], factor that out from the [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] terms:
[tex]\[ y = -\left(x^2 + 8x\right) - 29 \][/tex]
2. Complete the square:
To complete the square, we need to add and subtract a specific value inside the parentheses. The value to add and subtract is calculated by taking half of the coefficient of [tex]\(x\)[/tex], squaring it:
[tex]\[ \left(\frac{8}{2}\right)^2 = 16 \][/tex]
Add and subtract 16 inside the parentheses:
[tex]\[ y = -\left(x^2 + 8x + 16 - 16\right) - 29 \][/tex]
3. Rewrite the completed square:
The expression [tex]\(x^2 + 8x + 16\)[/tex] can be written as a perfect square:
[tex]\[ y = -\left((x + 4)^2 - 16\right) - 29 \][/tex]
4. Distribute the negative sign:
Distribute the negative sign through the completed square:
[tex]\[ y = - (x + 4)^2 + 16 - 29 \][/tex]
5. Simplify the constants:
Combine the constants:
[tex]\[ y = - (x + 4)^2 - 13 \][/tex]
Therefore, the vertex form of the given quadratic equation is:
[tex]\[ \boxed{y = -(x+4)^2 - 13} \][/tex]
So, the correct answer is:
A. [tex]\( y = -(x+4)^2 - 13 \)[/tex]
1. Factor out the coefficient of [tex]\(x^2\)[/tex] term:
Since the coefficient of [tex]\(x^2\)[/tex] is [tex]\(-1\)[/tex], factor that out from the [tex]\(x^2\)[/tex] and [tex]\(x\)[/tex] terms:
[tex]\[ y = -\left(x^2 + 8x\right) - 29 \][/tex]
2. Complete the square:
To complete the square, we need to add and subtract a specific value inside the parentheses. The value to add and subtract is calculated by taking half of the coefficient of [tex]\(x\)[/tex], squaring it:
[tex]\[ \left(\frac{8}{2}\right)^2 = 16 \][/tex]
Add and subtract 16 inside the parentheses:
[tex]\[ y = -\left(x^2 + 8x + 16 - 16\right) - 29 \][/tex]
3. Rewrite the completed square:
The expression [tex]\(x^2 + 8x + 16\)[/tex] can be written as a perfect square:
[tex]\[ y = -\left((x + 4)^2 - 16\right) - 29 \][/tex]
4. Distribute the negative sign:
Distribute the negative sign through the completed square:
[tex]\[ y = - (x + 4)^2 + 16 - 29 \][/tex]
5. Simplify the constants:
Combine the constants:
[tex]\[ y = - (x + 4)^2 - 13 \][/tex]
Therefore, the vertex form of the given quadratic equation is:
[tex]\[ \boxed{y = -(x+4)^2 - 13} \][/tex]
So, the correct answer is:
A. [tex]\( y = -(x+4)^2 - 13 \)[/tex]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.