HMMT November 2011 E5

Find all real values of $x$ for which

$\frac{1}{\sqrt{x}+\sqrt{x-2}}+\frac{1}{\sqrt{x+2}+\sqrt{x}}=\frac{1}{4}$

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{257}{16}

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