+1678 Define \[f(x) = \frac{1}{x + \frac{1}{x}}\] and \[g(x) = \frac{1}{x - \frac{1}{x}}.\] Find the square of the largest real solution to the equation \[ (g(x))^2 - (f(x))^2 = 1.\]
+14968 Find the square of the largest real solution to the equation \((g(x))^2 - (f(x))^2 = 1.\)
Hello wiseowl!
\((g(x))^2 - (f(x))^2 = 1\\ \dfrac{1}{(x-\dfrac{1}{x})^2}-\dfrac{1}{(x+\dfrac{1}{x})^2}=1\)
\(\dfrac{1}{(\dfrac{x^2-1}{x})^2}-\dfrac{1}{(\dfrac{x^2+1}{x})^2}=1\)
\(\dfrac{x^2}{(x^2-1)^2}-\dfrac{x^2}{(x^2+1)^2}=1\\ \)
\(\dfrac{4x^4}{(x-1)^2(x+1)^2(x^2+1)^2}=1\)
Calculated by WolframAlpha.
\(x=-\sqrt{\sqrt{2}-1}\\ x=\sqrt{\sqrt{2}-1}\\ x=-\sqrt{1+\sqrt{2}}\\ x=\sqrt{1+\sqrt{2}}\\\)
\(\color{blue}x\in \{-1.55377,-0.64359,0.64359,1.55377\}\)
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