Answer
The surface is a sphere with a radius $r=\frac{3 \sqrt{6}}{4}$ and centered at $C(1 / 2,-3 / 4,-5 / 4)$
Work Step by Step
We need to describe the surface whose equation is
\[
0=-2+2 z^{2}+2 y^{2}+2 x^{2}-2 x-3 y+5 z
\]
If we multiply this equation by $\left(\frac{1}{2}\right)$, then the result will be:
\[
1=-x-3 / 2 y+5 / 2 z+z^{2}+x^{2}+y^{2}
\]
This is the equation of a sphere. We complete
squares to find its center and radius.
\[
\begin{array}{l}
1=\left(-3 / 2 y+y^{2}\right)+\left(\frac{5}{2} z+z^{2}\right)+\left(-x+x^{2}\right) \\
\Rightarrow \left(-\frac{3}{2} y+y^{2}+\frac{9}{16}\right)+\left(\frac{5}{2} z+z^{2}+\frac{25}{10}\right)+\left(1 / 4-x+x^{2}\right)=\frac{9}{16}+\frac{25}{16}+1+\frac{1}{4}+\frac{9}{16} \\
\Rightarrow (5 / 4+z)^{2}+(3 / 4+y)^{2}+(-1 / 2+x)^{2}=\frac{54}{16}=\frac{27}{8} \\
\Rightarrow \quad (5 / 4+z)^{2}+(3 / 4+y)^{2}+(-1 / 2+x)^{2}=\left(\frac{3 \sqrt{3}}{2 \sqrt{2}}\right)^{2}=\left(\frac{3 \sqrt{6}}{9}\right)^{2}
\end{array}
\]
And therefore, the surface is a sphere of radius $r=\frac{3 \sqrt{6}}{4}$ and is concentrated at $C(1 / 2,-3 / 4,-5 / 4)$
