(i) $\left(x^{2}-3 x-\sqrt{x}+4\right)$ contains a term with $\sqrt{x}$, i.e., $x^{\frac{1}{2}}$, where $\frac{1}{2}$ is not a integer.
Therefore, it is not a quadratic polynomial.
$\therefore x^{2}-3 x-\sqrt{x}+4=0$ is not a quadratic equation.
(ii)
$\begin{array}{l}
x-\frac{6}{x}=3 \\
\Rightarrow x^{2}-6=3 x \\
\Rightarrow x^{2}-3 x-6=0
\end{array}$
$\left(x^{3}-3 x-6\right)$ is not quadratic polynomial; therefore, the given equation is quadratic.