[tex]\texttt{Formule folosite: } \\
cos\,2x = cos^2x - sin^2x\\
sin^2x+cos^2x = 1~~\Longrightarrow~~ sin^2 = 1-cos^2 \\ \\
\texttt{Rezolvare:} \\
3cos\,2x + 2cos\,x = 5 \\
3(cos^2x - sin^2x)+ 2cos\,x = 5 \\
3(cos^2x - (1-cos^2x))+ 2cos\,x = 5 \\
3(cos^2x - 1+cos^2x)+ 2cos\,x = 5 \\
3(2cos^2x - 1)+ 2cos\,x - 5 =0\\
6cos^2x - 3+ 2cos\,x - 5 =0 \\
6cos^2x + 2cos\,x - 8 =0 ~~~~~|:2 \\
3cos^2x + cos\,x - 4 =0 [/tex]
[tex]\displaystyle \\
\texttt{Rezolvam ecuatia de gradul 2} \\
Notam: ~~\boxed{t = cos\;x} \\
3t^2 + t - 4 =0 \\ \\
t_{12} = \frac{-1 \pm \sqrt{1 - 4\times 3 \times(-4)} }{2\times 3}= \\ \\
= \frac{-1 \pm \sqrt{1+48}}{6}= \frac{-1 \pm \sqrt{49}}{6} = \frac{-1\pm7}{6}\\ \\
t_1 = \frac{-1 + 7}{6} = \frac{6}{6} = \boxed{1} \\ \\
t_2 = \frac{-1 - 7}{6} = \frac{-8}{6} = \boxed{\frac{-4}{3}} ~~\texttt{Solutie eliminata deoarece }t \ \textless \ -1 [/tex]
[tex]\\ \\
\text{t2 se elimina deoarece cosinusul nu poate fi mai mic decat minus unu } \\ \\
t1 = 1\\\Longrightarrow ~~cos\;x=1 \\
\Longrightarrow ~~ \boxed{x = 0 + 2k\pi, ~~k \in N } [/tex]
