<s>[INST] Are there nice ways to solve $(2+x)^{0.25}-(2-x)^{0.25}=1$ Are there nice and elegant ways to solve this equation?
$(2+x)^{0.25}-(2-x)^{0.25}=1$
Thanks.
 [/INST] Let $t = 2 + x$, and $u = 2 - x$, then $u + t = 4$, and $t^{1/4} - u^{1/4} = 1$. Again let $p = t^{1/4}$, and $q = u^{1/4}$, then $p - q = 1$, and $p^4 + q^4 = 4$. So $(q+1)^4 + q^4 = 4$ and this gives: $2q^4 + 4q^3 + 6q^2 + 4q - 3 = 0$. Solving this we have:
$q = \dfrac{-1 + \sqrt{2\sqrt{10} - 3}}{2}$, and $u = q^4 = \dfrac{\left(-1 + \sqrt{2\sqrt{10} - 3}\right)^4}{16}$. So $x = 2 - u = 2 - \dfrac{\left(-1 + \sqrt{2\sqrt{10} - 3}\right)^4}{16}$
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