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In this video,I'll work through some problems.So first let's say we were asked to find equations for the sphere centered at negative 1,0,2,and radius square root 7.So remember that in general,the equation for a sphere centered at X0,Y0,Z0,and with radius r is x minus X0 squared plus my Y minus Y0 squared plus Z minus Z0 squared equals r squared.So in our case,that becomes X plus 1 squared plus Y squared or plus Z minus 2 squared equals 7.If you like you could expand this out as X squared plus 2x plus 1 plus Y squared plus Z squared minus 4Z plus 4 equals 7.And then rewrite this as X squared plus 2X plus Y squared plus Z squared minus 4Z is equal to 2.But for most purposes,it'll be a lot easier to just leave it in this form.So the only reason I did that expansion is to prepare for the next problem,which is--so now let's say we were given the equation for a sphere in that expanded out form.Now we want to find the center and radius of the sphere,and then it asks us to also find the two points on the sphere that have the highest and the lowest z-coordinate.So for part a,we'll complete the square.So x plus 2 squared is x squared plus 4x plus 4y,y minus 1 squared is y squared minus 2y plus 1,and z minus 3 squared is z squared minus 6z plus 9.So to complete the square,we need to add 14 to both sides of the equation.So then we get x plus 2 squared plus y minus 1 squared plus z minus 3 squared equals 25,which of course is 5 squared.So the center is C equals negative 2,1,3,and the radius is a equals to 5 Then for part b,so if we have a sphere and let's say this is the center point,has center at x0,y0,z0,and we can see that the points with the highest and lowest set coordinates are going to be the points directly above and directly below the center.Since this has radius r,these are going to have coordinates x0,y0,z0 plus the radius and x0,y0 z0 minus the radius.So,now the point our answer is points with the highest and respectively lowest z-coordinates are negative 2,1,8 or let's say respectively negative 2,1,negative 2.

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