Have any idea of a way to solve this equation? I don't have any software to solve this.

Question

So here is my problem. I have no idea about how to solve this equation and i am not even working on it. It seems gigantic. I've been Googling, but can't narrow it down. I have tried to use auxiliary but that doesn't help. Thank in advance. $${\left( {x + \frac{1}{{{x^3}}}} \right)^4} + {\left( {\frac{{{{\left( {2x + \sqrt {{x^2} - 1} } \right)}^9} + {{\left( {2x - \sqrt {{x^2} - 1} } \right)}^9}}}{{512}}} \right)^4} = 32$$

Answer 1

Over the field of complex numbers (assuming square roots are understood in terms of principal value) the equation is equivalent to: $$ 268435456 + 1073741824 x^4 + 1610612736 x^8 - 7516192768 x^{12} + 268442017 x^{16} - 1084752 x^{18} + 76230072 x^{20} - 2987714160 x^{22} + 72439538652 x^{24} - 1148594915664 x^{26} + 12375815932296 x^{28} - 93118758691824 x^{30} + 498320849428038 x^{32} - 1917207853906032 x^{34} + 5323216549904136 x^{36} - 10626091566758352 x^{38} + 15044570043647580 x^{40} - 14690301220036848 x^{42} + 9380736135506232 x^{44} - 3517842960288720 x^{46} + 586426328170081 x^{48} = 0. $$

The only roots over the field of real numbers are $x=\pm1$. Over the field of complex numbers additional roots are: $$ x = \pm 0.873154,\quad \pm 0.0567931 \pm 0.60749 i,\quad \pm 0.170289 \pm 0.591186 i,\quad \pm 0.277682 \pm 0.555457 i,\quad \pm 0.36861 \pm 0.511883 i,\quad \pm 0.463319 \pm 0.469405 i,\quad \pm 0.549576 \pm 0.400337 i,\quad \pm 0.612468 \pm 0.362964 i,\quad \pm 0.71957 \pm 0.310851 i,\quad \pm 0.815616 \pm 0.228228 i,\quad \pm 0.902606 \pm 0.132738 i,\quad \pm 0.97717 \pm 0.0600534 i $$