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If $a=x+\sqrt{x^{2}+1}$, then what is $x$ equal to?
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The correct answer is:
$(1 / 2)\left(a-a^{-1}\right)$
$\begin{aligned} & a=x+\sqrt{x^{2}+1} \Rightarrow a-x=\sqrt{x^{2}+1} \\ & \Rightarrow x^{2}+1=(a-x)^{2} \Rightarrow x^{2}+1=a^{2}+x^{2}-2 a x \\ & \Rightarrow 2 a x=a^{2}-1 \Rightarrow 2 x=a-\frac{1}{a} \\ & \Rightarrow x=\frac{1}{2}\left(a-a^{-1}\right) \end{aligned}$
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