[tex]\it a+\dfrac{1}{a} = 4 \Rightarrow \left(a+\dfrac{1}{a}\right)^2 = 4^2 \Rightarrow a^2+ 2\cdot a \cdot \dfrac{1}{a} +\left(\dfrac{1}{a}\right)^2 = 16\Rightarrow
\\\;\\ \\\;\\
\Rightarrow a^2+2+ \dfrac{1}{a^2} =16 |_{-2} \Rightarrow a^2 + \dfrac{1}{a^2} = 14 [/tex]
