Answer
$(x-1)^2+(y-2)^2+(z-3)^2=14$
Work Step by Step
Since the sphere is centered at $(1,2,3)$ and passes through $(0,0,0)$, we know the radius is the distance between these two points.
Using the distance formula, we get
$$r=\sqrt{(1-0)^2+(2-0)^2+(3-0)^2}
\\=\sqrt{(1)^2+(2)^2+(3)^2}
\\=\sqrt{1+4+9}\\=\sqrt{14}.$$
Hence we have a sphere centered at $(1,2,3)$ and with radius $r=\sqrt{14}$.
Thus the equation of the sphere is
$$(x-1)^2+(y-2)^2+(z-3)^2=14.$$
