To write the equation [tex]\(x^2 + 6x + y^2 - 8y = 19\)[/tex] in standard form, which is typically the equation of a circle [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we need to complete the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms.
1. Rewrite the equation grouping [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
[tex]\[
x^2 + 6x + y^2 - 8y = 19
\][/tex]
2. Complete the square for the [tex]\(x\)[/tex]-terms:
[tex]\[
x^2 + 6x
\][/tex]
To complete the square:
- Take half of the coefficient of [tex]\(x\)[/tex], which is [tex]\(6\)[/tex], giving [tex]\(3\)[/tex].
- Square this value, [tex]\(3^2 = 9\)[/tex].
- Rewrite the [tex]\(x\)[/tex]-terms as:
[tex]\[
x^2 + 6x = (x + 3)^2 - 9
\][/tex]
3. Complete the square for the [tex]\(y\)[/tex]-terms:
[tex]\[
y^2 - 8y
\][/tex]
To complete the square:
- Take half of the coefficient of [tex]\(y\)[/tex], which is [tex]\(-8\)[/tex], giving [tex]\(-4\)[/tex].
- Square this value, [tex]\((-4)^2 = 16\)[/tex].
- Rewrite the [tex]\(y\)[/tex]-terms as:
[tex]\[
y^2 - 8y = (y - 4)^2 - 16
\][/tex]
4. Substitute these completed squares back into the original equation:
[tex]\[
(x + 3)^2 - 9 + (y - 4)^2 - 16 = 19
\][/tex]
5. Combine the constants on the right side:
[tex]\[
(x + 3)^2 + (y - 4)^2 - 25 = 19
\][/tex]
6. Add 25 to both sides of the equation to isolate the completed squares:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 44
\][/tex]
Thus, the equation [tex]\(x^2 + 6x + y^2 - 8y = 19\)[/tex] in standard form is:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 44
\][/tex]
The correct answer is choice 1:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 44
\][/tex]
