# equation of a sphere

• Oct 1st 2010, 05:19 AM
greencheeseca
equation of a sphere
i have to show this equation represents a sphere..

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

but i'm stuck on the z variable as it won't come down to an even square. it comes down to a (x+2)(x-2).
• Oct 1st 2010, 05:20 AM
mr fantastic
Quote:

Originally Posted by greencheeseca
i have to show this equation represents a sphere..

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

but i'm stuck on the z variable as it won't come down to an even square. it comes down to a (x+2)(x-2).

Note that x^2 - 8x = (x - 4)^2 - 16 and y^2 + 4y = (y + 2)^2 - 4.
• Oct 1st 2010, 05:31 AM
greencheeseca
Quote:

Originally Posted by mr fantastic
Note that x^2 - 8x = (x - 4)^2 - 16 and y^2 + 4y = (y + 2)^2 - 4.

ah, sorry i should have write (z+2)(z-2).. the other variables work out as even squares..
• Oct 1st 2010, 08:18 AM
greencheeseca
so, what can i do about the z variable? i can't get it to work out as an even square..
• Oct 1st 2010, 09:50 AM
earboth
Quote:

Originally Posted by greencheeseca
i have to show this equation represents a sphere..

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

but i'm stuck on the z variable as it won't come down to an even square. it comes down to a (x+2)(x-2).

1. The general equation of a sphere with the center \$\displaystyle C(x_C, y_C, z_C)\$ and the radius r is

\$\displaystyle (x-x_C)^2+(y-y_C)^2+(z-z_C)^2=r^2\$

2. Transform your equation until it has the form of the general equation:

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

Completing the square(s):

\$\displaystyle x^2-8x + 16+y^2+4y+4 +z^2 = -4+16+4\$

3. I'll leave the rest for you.
• Oct 1st 2010, 01:34 PM
mr fantastic
Quote:

Originally Posted by greencheeseca
i have to show this equation represents a sphere..

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

but i'm stuck on the z variable as it won't come down to an even square. it comes down to a (x+2)(x-2).

Quote:

Originally Posted by greencheeseca
so, what can i do about the z variable? i can't get it to work out as an even square..

The only term with z in it is z^2 and I don't see what your problem could be with it. I have already shown you how to deal with the x-terms and y-terms.
• Oct 2nd 2010, 02:44 AM
HallsofIvy
Quote:

Originally Posted by greencheeseca
i have to show this equation represents a sphere..

\$\displaystyle x^2 + y^2 + z^2 = 8x - 4y - 4\$

but i'm stuck on the z variable as it won't come down to an even square. it comes down to a (x+2)(x-2).

You apparently are trying to do this as
\$\displaystyle (x^2- 8x)+ (y^2+ 4y)+ (z^2- 4)= 0\$
which is wrong for two reasons. First, moving that "- 4" to the other side of the equation makes it "\$\displaystyle z^2+ 4\$" which cannot be factored, not "\$\displaystyle z^2- 4\$".

But you don't want to do that to begin with!

Instead leave that constant, 4, on the right and do it as \$\displaystyle (x^2- 8x)+ (y^2- 4y)+ (z^2)= -4\$

Complete the square in x and y. The "z" term is already a perfect square! The numbers you need to add on the left to make the perfect squares, and, of course, add on the right, will give you a positive number on the right.