Example 4 Solve the equation
$$
x^{2}-x-1000 \sqrt{1+8000 x}=1000 .
$$
Solution: Clearly, $x \neq 0$. The original equation can be transformed into
$$
\begin{array}{l}
x^{2}-x=1000(1+\sqrt{1+8000 x}) \\
=\frac{8 \times 10^{6} x}{\sqrt{1+8000 x}-1} .
\end{array}
$$

Then $x-1=\frac{8 \times 10^{6}}{\sqrt{1+8000 x}-1}$.
Let $a=10^{3}, t=\sqrt{1+8 a x}-1$.
Thus, $x=\frac{t^{2}+2 t}{8 a}$, and we have $\frac{t^{2}+2 t}{8 a}-1=\frac{8 a^{2}}{t}$.
Rearranging gives $t^{3}+2 t^{2}-8 a t-64 a^{3}=0$.
Factoring yields
$$
(t-4 a)\left[t^{2}+(4 a+2) t+16 a^{2}\right]=0 \text {. }
$$

Clearly, $t^{2}+(4 a+2) t+16 a^{2}>0$.
Therefore, $t=4 a, x=\frac{t^{2}+2 t}{8 a}=2 a+1=2001$.