To convert the given equation of a circle from its general form to the standard form, we will complete the square for both [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Here is the step-by-step process:
1. Start with the given equation:
[tex]\[
3x^2 + 3y^2 + 30x - 24y - 12 = 0
\][/tex]
2. Divide the entire equation by 3 to simplify it:
[tex]\[
x^2 + y^2 + 10x - 8y - 4 = 0
\][/tex]
3. Group the [tex]\(x\)[/tex]-terms and [tex]\(y\)[/tex]-terms:
[tex]\[
(x^2 + 10x) + (y^2 - 8y) = 4
\][/tex]
4. Complete the square for the [tex]\(x\)[/tex]-terms:
[tex]\[
x^2 + 10x \rightarrow (x + 5)^2 - 25
\][/tex]
5. Complete the square for the [tex]\(y\)[/tex]-terms:
[tex]\[
y^2 - 8y \rightarrow (y - 4)^2 - 16
\][/tex]
6. Substitute the completed squares back into the equation:
[tex]\[
(x + 5)^2 - 25 + (y - 4)^2 - 16 = 4
\][/tex]
7. Rearrange the equation to form the standard equation of a circle:
[tex]\[
(x + 5)^2 + (y - 4)^2 = 4 + 25 + 16
\][/tex]
[tex]\[
(x + 5)^2 + (y - 4)^2 = 45
\][/tex]
So, the standard form of the equation is:
[tex]\[
(x + 5)^2 + (y - 4)^2 = 45
\][/tex]
Thus, the standard form of the equation is
[tex]\(
(x + \boxed{5})^2 + (y - \boxed{4})^2 = \boxed{45}
\)[/tex]
