Key idea. Square the given sum before trying to find x itself.

1. Squaring the given equality gives

   \[\left(x+\frac1x\right)^2=9.\]

2. The middle term is \(2x\cdot(1/x)=2\), so

   \[x^2+2+\frac1{x^2}=9.\]

3. Subtract 2. Thus \(x^2+1/x^2=7\). We did not need the individual value of x.

Remember. A symmetric expression can often be found directly from another symmetric expression.