Discussion
If \(x + \frac{1}{x} = 2\), then the value of \(x^{4} + \frac{1}{x^{4}}\) is
Answer: D
The value is 2.
Square the given equation twice:
\[ \begin{aligned} x^2 + \frac{1}{x^2} &= 2^2 - 2 = 2 \\ x^4 + \frac{1}{x^4} &= 2^2 - 2 = 2 \end{aligned} \]
In fact \(x + \frac{1}{x} = 2\) has only one solution, \(x = 1\), and then every power gives \(1 + 1 = 2\). 34 is the answer to the common version where \(x + \frac{1}{x} = 2\sqrt{2}\) or \(x^2 + \frac{1}{x^2} = 6\), and 4 is \(\left(x + \frac{1}{x}\right)^2\), the result of squaring once without subtracting the 2.
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