To convert the given general form of the circle equation into the standard form, we'll go through the process step-by-step.
Given equation:
[tex]\[ 3x^2 + 3y^2 + 30x - 24y - 12 = 0 \][/tex]
1. Simplify the equation: Divide the entire equation by 3 to make it easier to work with.
[tex]\[ x^2 + y^2 + 10x - 8y - 4 = 0 \][/tex]
2. Rearrange the terms: Group the [tex]\(x\)[/tex] terms together and the [tex]\(y\)[/tex] terms together.
[tex]\[ (x^2 + 10x) + (y^2 - 8y) = 4 \][/tex]
3. Complete the square for [tex]\(x\)[/tex]:
- Take the coefficient of [tex]\(x\)[/tex], which is 10, divide it by 2 to get 5, and then square it to get 25.
- Add and subtract 25 within the [tex]\(x\)[/tex] terms.
[tex]\[ (x^2 + 10x + 25 - 25) + (y^2 - 8y) = 4 \][/tex]
[tex]\[ (x + 5)^2 - 25 + (y^2 - 8y) = 4 \][/tex]
4. Complete the square for [tex]\(y\)[/tex]:
- Take the coefficient of [tex]\(y\)[/tex], which is -8, divide it by 2 to get -4, and then square it to get 16.
- Add and subtract 16 within the [tex]\(y\)[/tex] terms.
[tex]\[ (x + 5)^2 - 25 + (y^2 - 8y + 16 - 16) = 4 \][/tex]
[tex]\[ (x + 5)^2 - 25 + (y - 4)^2 - 16 = 4 \][/tex]
5. Combine constants on the right side:
[tex]\[ (x + 5)^2 + (y - 4)^2 - 41 = 4 \][/tex]
[tex]\[ (x + 5)^2 + (y - 4)^2 = 45 \][/tex]
Thus, the standard form of the equation of the circle is:
[tex]\[ (x + 5)^2 + (y - 4)^2 = 45 \][/tex]
So the filled standard form equation is:
[tex]\[ (x + \boxed{5})^2 + (y + \boxed{-4})^2 = \boxed{45} \][/tex]
