Is it possible to obtain the solution of
$$e^{\sqrt{x^{2} - x - 1}} = |x|$$
in closed form?
I know that $x$ must be somewhere between $\displaystyle\frac{\sqrt{5} + 1}{2}$ and $2$ after trying some substitutions. WolframAlpha gave me $x \approx 1.75036$. But apart from that I really have no idea how to start solving this.