How do I convert this equation to the standard form of a circle?

Question

I'm looking for the radius of the sphere of this: $4x^2 + 4y^2 +4z^2 -16x - 24y + 8z= 44$.

I have to get it into standard form in order to find the radius. So I factored out a 4 and simplified it to:

$$x(x-4) + y(y-6) +z(z+2) =11$$

I am not sure what else I can do from here. Any ideas? Apparently the answer is 5. Which means my 11 has to get to 25 somehow. Though I don't know what I can do. Thank you in advance.

Answer 1

We need to complete the squares

$$4x^2 + 4y^2 +4z^2 -16x - 24y + 8z= 44$$

$$x^2 + y^2 +z^2 -4x - 6y + 2z= 11$$

$$(x-2)^2-4 + (y-3)^2-9 +(z+1)^2-1 = 11$$

$$(x-2)^2 + (y-3)^2 +(z+1)^2 = 25$$

Answer 2

For each of the variables $x, y, z$, you want to get something of the form $(x - x_0)^2$, by adding some constant if necessary. Of course, you also should add the same constant to the right-hand side.

So we want $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 11 + C = R^2$, for some number $C$.