$x^4-4x=1\Leftrightarrow x^4+2x^2+1=2x^2+4x+2$$\Leftrightarrow (x^2+1)^2=2(x+1)^2$
$\Leftrightarrow (x^2+1)^2-2(x+1)^2=0\Leftrightarrow (x^2-\sqrt{2}x-\sqrt 2+1)(x^2+\sqrt 2x+\sqrt 2+1)=0$
$\Leftrightarrow x=\frac{1}{\sqrt 2}\pm \frac{\sqrt{4\sqrt2-2}}{2}$