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

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. – Per Manne Oct 19 '12 at 15:27
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.) – Thomas Andrews Oct 19 '12 at 15:45
Not sure that you can do this analytically, seeing the results: tinyurl.com/95l2p8f – Eric Angle Oct 19 '12 at 15:47
@EricAngle, given that solution, you might be able to prove analytically there is only one real solution. Given that the real solution is $x=1$ – Thomas Andrews Oct 19 '12 at 15:50
Alpha 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. – Ross Millikan Oct 19 '12 at 15:51