Question:
If \[ a+\frac{1}{a}=-2 \] find:
\[ a^2+\frac{1}{a^2} \]
Solution:
Using identity:
\[ \left(a+\frac{1}{a}\right)^2 = a^2+\frac{1}{a^2}+2 \]
Substituting the given value:
\[ (-2)^2 = a^2+\frac{1}{a^2}+2 \]
\[ 4 = a^2+\frac{1}{a^2}+2 \]
\[ a^2+\frac{1}{a^2} = 4-2 \]
\[ =2 \]