6. Solve the equation: $5 x^{2}+x-x \sqrt{5 x^{2}-1}-2=0$.
$$
\begin{array}{l}
5 x^{2}-1-x \sqrt{5 x^{2}-1}=-x+1 \Rightarrow\left(\sqrt{5 x^{2}-1}-\frac{1}{2} x\right)^{2}=\left(\frac{1}{2} x-1\right)^{2}, \\
\text { If } \sqrt{5 x^{2}-1}-\frac{1}{2} x=\frac{1}{2} x-1 \Rightarrow 5 x^{2}-1=x^{2}-2 x+1 \Rightarrow x_{1}=\frac{1}{2}, x_{2}=-1 \text {, }
\end{array}
$$

Upon inspection, both are extraneous roots, so the equation has no solution at this point;
If $\sqrt{5 x^{2}-1}-\frac{1}{2} x=1-\frac{1}{2} x \Rightarrow 5 x^{2}-1=1 \Rightarrow x= \pm \frac{\sqrt{10}}{5}$, which satisfies the equation.
Therefore, the solution set of the original equation is $\left\{\frac{\sqrt{10}}{5},-\frac{\sqrt{10}}{5}\right\}$.