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Solve for $x$ $$\big(x^3+\frac{1}{x^3}+1\big)^4=3\big(x^4+\frac{1}{x^4}+1\big)^3$$

let $x+\frac{1}{x}=t$ the equation equivalent to $(t^3-3t+1)^4=3(t^4-4t^2+3)^3$ but it's very complicated. Thanks.

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    By inspection, $x=1$ is a solution.2012-10-19
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    That should be $3(t^4-4t^2+3)^3$ on the right. (Not that I've checked the math that got $t^4-4t^2+3$, but clearly the exponent is wrong.)2012-10-19
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    Not sure that you can do this analytically, seeing the results: http://tinyurl.com/95l2p8f2012-10-19
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    @EricAngle, given that solution, you might be able to prove analytically there is only one real solution. Given that the real solution is $x=1$2012-10-19
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    [Alpha](http://www.wolframalpha.com/input/?i=solve+%28t^3-3t%2B1%29^4%3D3%28t^4-4t^2%2B3%29^3) finds 12 roots for the equation in $t$, with only $t=2$ corresponding to a real solution for $x$. The other three real $t$ solutions are less than $2$ in absolute value.2012-10-19

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