Complete the square to find the center and the radius of the circle:

3x^2+3y^2-12x+24y+15=0

3x^2+3y^2-12x+24y+15=0

3(x^2 - 4x + ...) + 3(y^2 + 8y + ...) = -15
3(x^2 - 4x + 4) + 3(y^2 + 8y + 16) = -15 + 12 + 48
3(x-2)^2 + 3(y+4)^2 = 45
(x-2)^2 + 3(y+4)^2 = 9

take it from there.

(I suppose I could have divided each term by 3 at the start)

Thank you so much, i'm so stressed if its not too much trouble check out some of my other questions;)

3x^2 + 3y^2 - 12x + 24y + 15 = 0.

Divide both sides of Eq by 3 to reduce
the coefficients of x^2 and y^2 to one:

x^2 + y^2 - 4x + 8y + 5 = 0
Complete the squares:
x^2 -4x + (-4/2)^2 + y^2 + 8y +(8/2)^2,
The terms that were added to complete
the square should be added to rt side
also:

x^2 -4x + 4 + y^2 +8y +16 = 4 +16 - 5,
Write the perfect squares as binomials
squared:
(x - 2)^2 + (x + 4)^2 = 15,
(x - h)^2 + (y - k)^2 = r^2 = STD Form.
C(h , k) = C(2 , -4), r^2 = 15, r =
sqrt(15) = 3.89.

Of course 45÷3 = 15 and not 9 like I had.

Silly me!

To complete the square and find the center and radius of the circle, we need to rewrite the equation of the circle in the standard form, which is:

(x - h)^2 + (y - k)^2 = r^2

Where (h, k) represents the center of the circle, and r represents the radius.

Let's start by rearranging the equation:

3x^2 + 3y^2 - 12x + 24y + 15 = 0

First, let's group the terms with x and y together:

(3x^2 - 12x) + (3y^2 + 24y) + 15 = 0

Now, let's complete the square for the x-terms and the y-terms separately.

For the x-terms:
1. Divide the coefficient of x by 2 and square it: (12/2)^2 = 36.
2. Add 36 inside the parenthesis: (3x^2 - 12x + 36) + (3y^2 + 24y) + 15 = 36.

For the y-terms:
1. Divide the coefficient of y by 2 and square it: (24/2)^2 = 144.
2. Add 144 inside the parenthesis: (3x^2 - 12x + 36) + (3y^2 + 24y + 144) + 15 = 36 + 144.

Now, let's simplify the equation:

(3x^2 - 12x + 36) + (3y^2 + 24y + 144) + 15 = 36 + 144

(3x - 6)^2 + (3y + 12)^2 + 15 = 180

Next, let's isolate the terms involving x and y:

(3x - 6)^2 + (3y + 12)^2 = 180 - 15

(3x - 6)^2 + (3y + 12)^2 = 165

Finally, we have the equation in standard form. By comparing it to the standard form equation, we can determine the center and radius of the circle.

Center: The center of the circle is obtained by taking the opposite signs of the values inside the parentheses. In this case, the center is (6, -12).

Radius: The radius is determined by taking the square root of the value on the right side of the equation (165 in this case). The radius of the circle is √165.

Therefore, the center of the circle is (6, -12), and the radius is √165.