<s>[INST] How to find this kind of relationship: $x-\frac{1}{x}=A$ and $x+\frac{1}{x}=\sqrt{A^2+4}$? Could someone explain how to get from: $x-\frac{1}{x}=A$ to $x+\frac{1}{x}=\sqrt{A^2+4}$ ? It is one of the Algebra II tricks.
Thanks.
 [/INST] Start by squaring both sides:
$$\begin{align}x-\frac{1}{x}&=A\\\left(x-\frac{1}{x}\right)^2&=A^2\\x^2-2+\frac{1}{x^2}&=A^2.\end{align}$$
Then try adding $4$ to both sides and "reversing" the processes above.
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