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