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Question 24

The sum of all real values of $$x$$ satisfying $$\left(x+\frac{1}{x}-17\right)^2=x+\frac{1}{x}+17$$ is


Correct Answer: 35

Solution

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Put $$t=x+\frac{1}{x}-17$$. Then $$t^2=t+34$$, so $$t=\frac{1\pm\sqrt{137}}{2}$$. Therefore $$x+\frac{1}{x}=\frac{35\pm\sqrt{137}}{2}$$, producing two reciprocal pairs of roots. The sum of the roots from the two quadratic equations is $$\frac{35+\sqrt{137}}{2}+\frac{35-\sqrt{137}}{2}=35$$.

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