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Question 2

If $$\frac{x^2 - x + 1}{x^2 + x + 1} = \frac{11}{13}$$, then the value of $$x + \frac{1}{x}$$ is

Solution

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Dividing the numerator and the denominator by $$x$$ turns the equation into $$\frac{x + \frac{1}{x} - 1}{x + \frac{1}{x} + 1} = \frac{11}{13}$$. Writing $$t = x + \frac{1}{x}$$ gives $$13(t - 1) = 11(t + 1)$$, so $$2t = 24$$ and $$t = 12$$.

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