171.
Which of the following is the value of (x+ 1/x)²?
Answer: C
Apply \( (p+q)^{2} = p^{2} + 2pq + q^{2} \) with \( p = x \) and \( q = \tfrac{1}{x} \).
\[ \left(x + \frac{1}{x}\right)^{2} = x^{2} + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^{2}} \]
\[ = x^{2} + \frac{1}{x^{2}} + 2 \]
The middle term simplifies to the constant 2 because \( x \) cancels with \( \tfrac{1}{x} \) - which is why this identity is so useful for problems linking \( x + \tfrac{1}{x} \) to \( x^{2} + \tfrac{1}{x^{2}} \).
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