So for my vector calculus homework I was given:
$$ (x^2+4x)+(y^2-6y)+(z^2+10z)=11 $$
and told to find the center and radius of the sphere this describes. Where do I even start? I know in general a sphere can be described as: $$ (x-h)^2+(y-k)^2+(z-l)^2=r^2 $$ but obviously this doesn't as easily map onto that, so how do I figure this out? Thanks!
Begin with $$(x^2+4x)+(y^2-6y)+(z^2+10z)=11$$
Complete the square within each bracket:
$$(x^2+4x+4)+(y^2-6y+9)+(z^2+10z+25)=11+4+9+25$$
Factor:
$$(x+2)^2+(y-3)^2+(z+5)^2=7^2$$
The center of this sphere is $(-2, 3, -5)$ and the radius is $7$.