[tex]\it a+\dfrac{1}{a} =\dfrac{5}{2} \Rightarrow \left(a+\dfrac{1}{a}\right)^2 = \left(\dfrac{5}{2}\right)^2\Rightarrow a^2+2\cdot a\cdot\dfrac{1}{a} +\dfrac{1}{a^2} = \dfrac{25}{4} \Rightarrow
\\\;\\ \\\;\\
a^2+\dfrac{1}{a^2} = \dfrac{25}{4} -2 = \dfrac{25-8}{4} = \dfrac{17}{4}
[/tex]
