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]
