[tex]\displaystyle \\
\texttt{Se da: } a^2+\frac{1}{a^2}=7\\\\
\texttt{Se cere: }a+\frac{1}{a} =x~~\texttt{(Am notat valoarea cautata cu } x ) \\\\
\texttt{Rezolvare: } \\ \\
a+ \frac{1}{a} = x ~~~~~| \text{Ridicam la puterea a 2-a.} \\ \\
\left(a+ \frac{1}{a} \right)^2= x^2 \\\\
a^2 + 2 \cdot a \cdot \frac{1}{a} +\left(\frac{1}{a}\right)^2=x^2\\\\
a^2 +\frac{2a}{a}+\left(\frac{1}{a}\right)^2=x^2\\\\
a^2+2+\frac{1}{a^2}=x^2\\\\
a^2+\frac{1}{a^2}=x^2-2
[/tex]
[tex]\displaystyle \\
\texttt{Dar } a^2+\frac{1}{a^2}=7 ~~~\text{(din enunt)} \\ \\
7=x^2 - 2 \\
x^2 = 7+2 \\
x^2 = 9 \\ \\
x_{12} = \pm \sqrt{9} \\ \\
x_1 = +\sqrt{9} = 3 \\ \\
x_2 = -\sqrt{9} = -3 \\ \\
\texttt{Dar } x \texttt{ este: } a+\frac{1}{a} \\ \\
\texttt{Rezulta solutiile: } \\ \\
S_1:~~~~~ \boxed{a+\frac{1}{a} = 3} \\ \\
S_2:~~~~~ \boxed{a+\frac{1}{a} = -3} [/tex]
