To rewrite the given equation [tex]\(x^2 + y^2 + 16x - 8y + 76 = 0\)[/tex] in standard form, we will use the method of completing the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
1. Group the [tex]\(x\)[/tex] terms and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 + 16x + y^2 - 8y + 76 = 0
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex] terms:
- Take the coefficient of [tex]\(x\)[/tex], which is 16, divide by 2, and square it:
[tex]\[
\left(\frac{16}{2}\right)^2 = 8^2 = 64
\][/tex]
- Add and subtract 64 inside the equation:
[tex]\[
x^2 + 16x + 64 - 64
\][/tex]
- Rewrite as a perfect square:
[tex]\[
(x + 8)^2 - 64
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex] terms:
- Take the coefficient of [tex]\(y\)[/tex], which is -8, divide by 2, and square it:
[tex]\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\][/tex]
- Add and subtract 16 inside the equation:
[tex]\[
y^2 - 8y + 16 - 16
\][/tex]
- Rewrite as a perfect square:
[tex]\[
(y - 4)^2 - 16
\][/tex]
4. Rewrite the entire equation with the completed squares:
[tex]\[
(x + 8)^2 - 64 + (y - 4)^2 - 16 + 76 = 0
\][/tex]
5. Simplify the constants:
[tex]\[
-64 - 16 + 76 = -4
\][/tex]
So the equation becomes:
[tex]\[
(x + 8)^2 + (y - 4)^2 - 4 = 0
\][/tex]
6. Move the constant term to the right side to get the standard form:
[tex]\[
(x + 8)^2 + (y - 4)^2 = 4
\][/tex]
Therefore, the equation in standard form is:
[tex]\[
(x - (-8))^2 + (y - 4)^2 = 4
\][/tex]
In the format:
[tex]\[
(x - [?])^2 + (y - [\quad])^2 = [\quad]
\][/tex]
The values are:
[tex]\[
(x - (-8))^2 + (y - 4)^2 = 4
\][/tex]
So, the completed answer is:
[tex]\[
(x - (-8))^2 + (y - 4)^2 = 4
\][/tex]
