Solve: 2 ^2 x + 3 x = 0 - Vidyadip

Question

Solve: $2 \cos^2 x + 3 \sin x = 0$

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Answer

$2 \cos^2 x + 3 \sin x = 0$
$\Rightarrow 2 (1 - \sin^2 x) + 3 \sin x = 0$
$\Rightarrow 2 \sin^2 x - 3 \sin x - 2 = 0$
$\Rightarrow 2 \sin^2\ x - 4 \sin x + \sin x - 2 = 0$
$\Rightarrow 2 \sin x (\sin x - 2) +1 (\sin x - 2 ) = 0$
$\Rightarrow (\sin x - 2) (2 \sin x +1) = 0$
$\Rightarrow 2 \sin x + 1 = 0 [\because \sin x \neq 2 \quad \therefore \sin x-2 \neq 0]$
$\Rightarrow \quad \sin x=-\frac{1}{2}$
$\Rightarrow \quad \sin x=\sin \left(-\frac{\pi}{6}\right) \Rightarrow x=n \pi+(-1)^{n}\left(-\frac{\pi}{6}\right), n \in Z \Rightarrow x=n \pi+(-1)^{n+1} \frac{\pi}{6}, \quad n \in Z.$

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