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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.

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  • $\begingroup$ Hint: complete the square. For example, $x^2-4x = (x-2)^2 - 4$. $\endgroup$
    – peek-a-boo
    Aug 26, 2020 at 14:01
  • $\begingroup$ Complete the square for each of the variables. $\endgroup$
    – Andrew Chin
    Aug 26, 2020 at 14:01

2 Answers 2

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We need to complete the squares

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

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

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

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

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

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