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

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.

• Hint: complete the square. For example, $x^2-4x = (x-2)^2 - 4$. Aug 26, 2020 at 14:01
• Complete the square for each of the variables. Aug 26, 2020 at 14:01

We need to complete the squares

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

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

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

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

• thanks homie i forgot about ompleting the square. i keep having to relearn it over and over lol Aug 26, 2020 at 14:08
• @DoctorReality That's fine! Now you have revised the method. You are welcome! Bye
– user
Aug 26, 2020 at 14:09

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 $$C$$.