Solve the following equation: 2 ^ 2 x+3 +1=0 - Vidyadip

Question

Solve the following equation: $2\sin^{2}\text{x}+\sqrt{3}\cos\text{x}+1=0$

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Answer

We have, $2\sin^{2}\text{x}+\sqrt{3}\cos\text{x}+1=0$ $\Rightarrow2(1-\cos^{2}\text{x})+\sqrt{3}\cos\text{x}+1=0$ $\Rightarrow2\cos^{2}\text{x}-\sqrt{3}\cos\text{x}-3=0$ factorise it,we get, $\Rightarrow2\cos^{2}\text{x}-2\sqrt{3}\cos\text{x}+\sqrt{3}\cos\text{x}-3=0$ $\Rightarrow2\cos\text{x}(\cos\text{x}-\sqrt{3})+\sqrt{3}(\cos\text{x}-\sqrt{3})=0$ $\Rightarrow(2\cos\text{x}+\sqrt{3})(\cos\text{x}-\sqrt{3})=0$ $\Rightarrow\text{Either}$ $\cos\text{x}=-\frac{\sqrt{3}}{2}$ or $\cos\text{x}=\sqrt{3}$ [This is not possible as$-1<\cos\text{x}<1$] $\Rightarrow\cos\text{x}=\cos\Big(\pi-\frac{\pi}{6}\Big)$ $\Rightarrow\cos\text{x}=\cos\frac{5\pi}{6}$ $\Rightarrow\text{x}=2\text{n}\pi\pm\frac{5\pi}{6},\text{n}\in\text{z}$

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