Discussion
If \(x = 2 + \sqrt{3}\), find the value of \(x^{2} + 1/x^{2}\)
Answer: C
The value is 14.
Since \((2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1\), the reciprocal is \(1/x = 2 - \sqrt{3}\). So \(x + 1/x = 4\):
\[ \begin{aligned} x^{2} + \tfrac{1}{x^{2}} &= \left(x + \tfrac{1}{x}\right)^{2} - 2 \\ &= 16 - 2 = 14 \end{aligned} \]
16 would be the result of forgetting to subtract 2. 6, 15 and 5 do not follow from the identity.
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