[tex]\displaystyle \\
\texttt{Se da: }x+ \frac{1}{x} = 3 \\ \\
\texttt{Se cere: } x^2+ \frac{1}{x^2} = ? \\ \\
\texttt{Rezolvare: } \\ \\
x+ \frac{1}{x} = 3 ~~~~~| \text{Ridicam la puterea a 2-a.} \\ \\
\left(x+ \frac{1}{x} \right)^2= 3^2 \\ \\
\left(x+ \frac{1}{x} \right)^2= 9 \\ \\
x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 9 \\ \\
x^2 + \frac{2x}{x} + \left(\frac{1}{x}\right)^2 = 9 \\ \\
x^2 + 2+ \frac{1}{x^2} = 9 \\ \\
x^2 + \frac{1}{x^2} = 9-2 [/tex]
[tex]\displaystyle \\
\boxed{x^2 + \frac{1}{x^2} = 7 } [/tex]