Answer
$(x-2)^2 + (y+2)^2 + (z-5)^2 = 8$
Work Step by Step
Midpoint: $(\frac{-4+0}{2},\frac{2+2}{2},\frac{3+7}{2}) = (-2,2,5)$
Distance between $P$ and $Q$: $\sqrt {(0+4)^2+(2-2)^2+(7-3)^2} = \sqrt {16 + 0 + 16} = \sqrt {32}$
Center = Midpoint
Radius = $\frac{(distance\ between\ P\ and\ Q)}{2}$
Equation of Sphere with Center $(a,b,c)$ and radius $r$
$(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2$
Plug in the center and radius:
$(x-2)^2 + (y+2)^2 + (z-5)^2 = (\frac{\sqrt {32}}{2})^2$
Simplify:
$(x-2)^2 + (y+2)^2 + (z-5)^2 = 8$
