If $2x + \dfrac{1}{x} = 3$, $x^2 + \dfrac{1}{4x^2} = ?$

Squaring $2x+\frac1x=3$ gives $4x^2+\frac1{x^2}+4=9$, so $4x^2+\frac1{x^2}=5$. Dividing by $4$ yields $x^2+\frac1{4x^2}=\frac54$.