Question

Solve the following equation:

\[2 \cos^2 x - 5 \cos x + 2 = 0\]

Answer 1

\[2 \cos^2 x - 5 \cos x + 2 = 0\]

\[ \Rightarrow 2 \cos^2 x - 4 \cos x - \cos x + 2 = 0\]

\[ \Rightarrow 2 \cos x ( \cos x - 2) - 1 ( \cos x - 2) = 0\]

\[ \Rightarrow (\cos x - 2) ( 2 \cos\theta - 1) = 0\]

\[\Rightarrow ( \cos x - 2 ) = 0\] or

\[( 2 \cos x - 1) = 0\]

\[\cos x = 2\]  is not possible.

\[\therefore 2 \cos x - 1 = 0 \]

\[ \Rightarrow \cos x = \frac{1}{2} \]

\[ \Rightarrow \cos x = \cos \frac{\pi}{3} \]

\[ \Rightarrow x = 2n\pi \pm \frac{\pi}{3}, n \in Z\]