To convert the general form of the equation [tex]\( 3x^2 + 3y^2 + 30x - 24y - 12 = 0 \)[/tex] into the standard form of a circle's equation, follow these steps:
1. Simplify the equation by dividing everything by 3:
[tex]\[
x^2 + y^2 + 10x - 8y - 4 = 0
\][/tex]
2. Move the constant term to the right side of the equation:
[tex]\[
x^2 + y^2 + 10x - 8y = 4
\][/tex]
3. Complete the square for the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms:
- For the [tex]\(x\)[/tex] terms: [tex]\( x^2 + 10x \)[/tex]
- Take half of the coefficient of [tex]\(x\)[/tex], which is 10, to get 5.
- Square 5 to get 25.
- Rewrite the [tex]\(x\)[/tex] terms as: [tex]\( (x + 5)^2 - 25 \)[/tex]
- For the [tex]\(y\)[/tex] terms: [tex]\( y^2 - 8y \)[/tex]
- Take half of the coefficient of [tex]\(y\)[/tex], which is -8, to get -4.
- Square -4 to get 16.
- Rewrite the [tex]\(y\)[/tex] terms as: [tex]\( (y - 4)^2 - 16 \)[/tex]
4. Rewrite the equation using the completed squares:
[tex]\[
(x + 5)^2 - 25 + (y - 4)^2 - 16 = 4
\][/tex]
5. Combine the constants on the right side:
[tex]\[
(x + 5)^2 + (y - 4)^2 - 41 = 4
\][/tex]
6. Add 41 to both sides to isolate the completed squares:
[tex]\[
(x + 5)^2 + (y - 4)^2 = 45
\][/tex]
Thus, the standard form of the circle's equation is:
[tex]\[
(x + \boxed{5})^2 + (y - \boxed{4})^2 = \boxed{45}
\][/tex]
