00:01
We want to find the center and radius of the sphere that has the equation 3x squared plus 3y squared plus 3z squared equals 6z plus 1.
00:08
So if we were to show the standard form of the equation of a sphere, the easiest way to find the center and radius is if it's in this form, x minus h squared plus y minus k squared plus z minus p squared equals the radius squared.
00:46
The center, if it's in that form, would be the three dimensional point h, k, p and the radius would be r.
01:03
Okay, so we need to manipulate our equation we have up there to get it into this form.
01:09
So we have 3x squared plus 3y squared plus 3z squared.
01:15
I'm going to subtract the 6z from both sides and then we would have minus 6z and that would be equal to 1.
01:24
So we can see that as far as these terms go, the 3x squared and 3y squared, those are the only x and y variables respectively that we have.
01:39
So we shouldn't have to mess with anything over there, but the problem is in the z's we have z squared and a z.
01:46
So we need to manipulate that a bit and we can do that by completing the square.
01:50
So we'll bring down our 3x squared plus 3y squared plus, then we're going to factor out a 3 out of those next two terms.
01:57
Factor out a 3, that leaves us with z squared minus 2z.
02:02
Then i'm going to leave a space so we can complete the square.
02:05
So we want to complete the square with this binomial here.
02:10
We need to turn that into a trinomial, that would be a perfect square trinomial.
02:14
In order to do that we take half of that coefficient with z, we take half of that, so we divide negative 2 by 2 and then we square the result to figure out what we need to add.
02:28
That would give us negative 1 squared which is 1...