Question:
If \[ x^2+\frac{1}{x^2}=102, \] then \[ x-\frac{1}{x}= \]
(a) 8
(b) 10
(c) 12
(d) 13
Solution:
Using identity:
\[ \left(x-\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}-2 \]
Substituting the given value:
\[ \left(x-\frac{1}{x}\right)^2 = 102-2 \]
\[ \left(x-\frac{1}{x}\right)^2 = 100 \]
\[ x-\frac{1}{x} = \sqrt{100} \]
\[ =10 \]
Hence, the correct answer is:
\[ \boxed{10} \]