Question:
If \[ x+\frac{1}{x}=5, \] then \[ x^2+\frac{1}{x^2}= \]
(a) 25
(b) 10
(c) 23
(d) 27
Solution:
Using identity:
\[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \]
Substituting the given value:
\[ (5)^2 = x^2+\frac{1}{x^2}+2 \]
\[ 25 = x^2+\frac{1}{x^2}+2 \]
\[ x^2+\frac{1}{x^2} = 25-2 \]
\[ =23 \]
Hence, the correct answer is:
\[ \boxed{23} \]