Problem
ALG-B1-M01-P010 The Substitution t=x+1/x
#10
★★☆☆☆ Level 2 of 5
Let \(x\ne0\) and \(x+\frac1x=3\). Find \(x^2+\frac1{x^2}\).
Square the given equality.
Squaring, \[ \left(x+\frac1x\right)^2=x^2+2+\frac1{x^2}=9. \] Hence \(x^2+\frac1{x^2}=7\).
This substitution often unlocks longer expressions.