6.004. $x^{4}-\frac{50}{2 x^{4}-7}=14$.
Solution.

Domain of definition: $x \neq \pm 4 \sqrt{\frac{7}{2}}$.

Let $2 x^{4}-7=z$, then $\frac{z+2}{2}-\frac{50}{z}=14 \Rightarrow$

$\Rightarrow z^{2}-21 z-100=0 \Leftrightarrow z_{1}=-4, z_{2}=25$.

To find $x$, solve the two equations $2 x^{4}-7=-4$ or $2 x^{4}-7=25$, solving which, we get $x_{1}=-\sqrt[4]{\frac{3}{2}}, x_{2}=\sqrt[4]{\frac{3}{2}}, x_{3}=-2, x_{4}=2$.

Answer: $x_{1,2}= \pm \sqrt[4]{\frac{3}{2}}, x_{3,4}= \pm 2$.