$2x+ \frac{2}{x} =3 \Rightarrow x+ \frac{1}{x} = \frac{3}{2} \Rightarrow (x+ \frac{1}{x})^2 = (\frac{3}{2})^2 \Rightarrow x^2+ 2x\frac{1}{x}+ \frac{1}{x^2} = \frac{9}{4} $
$\Rightarrow x^2+\frac{1}{x^2} = \frac{9}{4}-2= \frac{1}{4} \Rightarrow 5x^2+\frac{5}{x^2} = \frac{5}{4} $
$ \Box $
