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Question Number 144329 by gsk2684 last updated on 24/Jun/21

$$\mathrm{find}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{solutions}$$$$\mathrm{of}\sqrt{6-\cos \mathrm{x}+{7\sin }^2\mathrm{x}}+\cos \mathrm{x}=0$$

Commented by MJS_new last updated on 25/Jun/21

$$\mathrm{let}c=\cos x\Rightarrow -1⩽c⩽1$$$$\sqrt{13-c-7c^2}+c=0\Rightarrow c<0$$$$\sqrt{13-c-7c^2}=-c$$$$13-c-7c^2=c^2$$$$8c^2+c-13=0\wedge c<0\Rightarrow c=-\frac{1+\sqrt{417}}{16}<-1$$$$\Rightarrow \mathrm{no}\mathrm{solution}$$$$$$$$[-1+\sqrt{7}⩽\sqrt{6-\cos x+{7\sin }^{2}x}+\cos x⩽\frac{-1+2\sqrt{730}}{14}]$$

Commented by gsk2684 last updated on 25/Jun/21

$$\mathrm{thank}\mathrm{you}\mathrm{mr}\mathrm{MJS}$$

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