If x + $\frac{1}{x}$ = 8, find x$^2$ + $\frac{1}{x^2}$

Analyze the question: We're given x + 1/x = 8 and asked for x² + 1/x². This is a classic algebra identity question.

Concept: Squaring a sum gives

(x + 1/x)² = x² + 2(x)(1/x) + 1/x² = x² + 2 + 1/x²

The middle term simplifies because x · (1/x) = 1, so it becomes 2.

Solution:

1. Square both sides of x + 1/x = 8: (x + 1/x)² = 8² x² + 2 + 1/x² = 64

2. Subtract 2 from both sides: x² + 1/x² = 64 − 2 = 62

Answer: x² + 1/x² = 62, which is option B.

Tip: Whenever you see x + 1/x = k, the answer for x² + 1/x² is always k² − 2. The common mistake is choosing 64 (option A) by forgetting to subtract the 2. Likewise, if you see x − 1/x = k, then x² + 1/x² = k² + 2.