Question:
If \[ x+\frac{1}{x}=3 \] find:
\[ x^2+\frac{1}{x^2} \]
Solution:
Using identity:
\[ \left(x+\frac{1}{x}\right)^2 = x^2+\frac{1}{x^2}+2 \]
Substituting the given value:
\[ (3)^2 = x^2+\frac{1}{x^2}+2 \]
\[ 9 = x^2+\frac{1}{x^2}+2 \]
\[ x^2+\frac{1}{x^2} = 9-2 \]
\[ =7 \]