If x = \(\sqrt {\frac{\sqrt {10} + 1}{\sqrt {10} - 1}}\), then find the value of x2 - x - 1.
Answer & explanation
Correct answer: option 4
x = \(\sqrt {\frac{\sqrt {10} + 1}{\sqrt {10} - 1} × \frac{\sqrt {10} + 1}{\sqrt {10} + 1}}\) = \(\sqrt {\frac{(\sqrt {10} + 1)^2}{10 - 1}}\)
= \(\frac{(\sqrt {10} + 1)}{3}\)
Therefore, x2 - x - 1
= (\(\frac{(\sqrt {10} + 1)}{3}\))2 - (\(\frac{(\sqrt {10} + 1)}{3}\)) - 1
= \(\frac{11 + 2\sqrt {10}}{9}\) - \(\frac{(\sqrt {10} + 1)}{3}\) - 1 = \(\frac{11 + 2\sqrt {10} - 3\sqrt {10} - 3 - 9}{9}\)
= \(\frac{-1 - \sqrt {10}}{9}\) = -\(\frac{(\sqrt {10} + 1)}{9}\)