Question 21

If $$x^2 + (2 + \sqrt{3})x - 1 = 0$$ and $$x^2 + \frac{1}{x^2} = a + b\sqrt{c}$$, then $$(a + b + c)$$ is

Correct Answer: 16

Solution

Dividing the equation by $$x$$ gives $$x - \frac{1}{x} = -(2 + \sqrt{3})$$. Squaring both sides, $$x^2 + \frac{1}{x^2} - 2 = (2 + \sqrt{3})^2 = 7 + 4\sqrt{3}$$, so $$x^2 + \frac{1}{x^2} = 9 + 4\sqrt{3}$$. Comparing, $$a = 9$$, $$b = 4$$ and $$c = 3$$, so $$a + b + c = 16$$.

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