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Question Number 22199 by chernoaguero@gmail.com last updated on 13/Oct/17

Commented by chernoaguero@gmail.com last updated on 13/Oct/17

$$\mathrm{Thank}\mathrm{you}\mathrm{guys}$$

Commented by Rasheed.Sindhi last updated on 13/Oct/17

$$\mathrm{x}=1$$

Commented by prakash jain last updated on 13/Oct/17

$$\frac{d}{dx}(x+2^x+6)=1+\left(\ln 2\right)2^x>0$$$$x+2^x+6\mathrm{is}\mathrm{increasing}\mathrm{function}$$$$\mathrm{so}\mathrm{only}1\mathrm{solution}.$$

Answered by Bruce Lee last updated on 13/Oct/17

$$\sqrt{\mathit{x}+2^{\mathit{x}}+6}=3$$$$2^{\mathit{x}}=3-\mathit{x}$$$$.2^{\mathit{x}}⩾1\iff 3-\mathit{x}⩾1\mid \to {\mathit{a}}^{\mathit{x}}⩾1\mathrm{\forall }\mathit{x}\in \mathit{R},\mathit{a}⩾1p$$$$or\mathit{x}⩽2$$$$+\mathit{c}\mathit{a}\mathit{s}\mathit{e}1.\mathit{x}=1$$$$\Rightarrow \sqrt{1+2^1+6}=3$$$$\Rightarrow \mathit{x}=1\mathit{i}\mathit{s}\mathit{a}\mathit{r}\mathit{o}\mathit{o}\mathit{t}$$$$+\mathit{c}\mathit{a}\mathit{s}\mathit{e}2.1<\mathit{x}⩽2\Rightarrow 2^{\mathit{x}}>2$$$$-1>-\mathit{x}⩾-2$$$$2>3-\mathit{x}⩾1$$$$2>2^{\mathit{x}}⩾1\mathit{n}\mathit{o}\mathit{t}\mathit{t}\mathit{r}\mathit{u}\mathit{e}$$$$+\mathit{c}\mathit{a}\mathit{s}\mathit{e}3.\mathit{x}<1\Rightarrow 2^{\mathit{x}}<2$$$$-\mathit{x}>-1$$$$3-\mathit{x}>2$$$$2^{\mathit{x}}>2\mathit{n}\mathit{o}\mathit{t}\mathit{t}\mathit{r}\mathit{u}\mathit{e}$$$$\mathit{S}\mathit{o}\mathit{x}=1\mathit{i}\mathit{s}\mathit{o}\mathit{n}\mathit{l}\mathit{y}\mathit{a}\mathit{r}\mathit{o}\mathit{o}\mathit{t}.$$

Commented by chernoaguero@gmail.com last updated on 13/Oct/17

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}$$

Answered by mrW1 last updated on 14/Oct/17

$$\mathrm{x}+2^{\mathrm{x}}+6=3^2=9$$$$2^{\mathrm{x}}=9-6-\mathrm{x}=3-\mathrm{x}$$$$2^{\mathrm{x}-3}=-\frac{\mathrm{x}-3}{8}$$$${\mathrm{e}}^{(\mathrm{x}-3)\ln 2}=-\frac{\mathrm{x}-3}{8}$$$$-(\mathrm{x}-3){\mathrm{e}}^{-(\mathrm{x}-3)\ln 2}=8$$$$-(\mathrm{x}-3)\ln {2\mathrm{e}}^{-(\mathrm{x}-3)\ln 2}=8\ln 2$$$$\Rightarrow -(\mathrm{x}-3)\ln 2=\mathrm{W}\left(8\ln 2\right)$$$$\Rightarrow \mathrm{x}=3-\frac{\mathrm{W}\left(8\ln 2\right)}{\ln 2}=3-2=1$$$$$$$$\mathrm{this}\mathrm{method}\mathrm{works}\mathrm{for}\mathrm{any}\mathrm{values}.$$$$\sqrt{\mathrm{x}+2^{\mathrm{x}}+\mathrm{a}}=\mathrm{b}$$$$\mathrm{x}+2^{\mathrm{x}}={\mathrm{b}}^2-\mathrm{a}=\mathrm{c}$$$$2^{\mathrm{x}}=-(\mathrm{x}-\mathrm{c})$$$$2^{\mathrm{x}-\mathrm{c}}=-\frac{\mathrm{x}-\mathrm{c}}{2^{\mathrm{c}}}$$$${\mathrm{e}}^{(\mathrm{x}-\mathrm{c})\ln 2}=-\frac{\mathrm{x}-\mathrm{c}}{2^{\mathrm{c}}}$$$$-(\mathrm{x}-\mathrm{c})\ln 2{\mathrm{e}}^{-(\mathrm{x}-\mathrm{c})\ln 2}=2^{\mathrm{c}}\ln 2$$$$-(\mathrm{x}-\mathrm{c})\ln 2=\mathrm{W}\left(2^{\mathrm{c}}\ln 2\right)$$$$\Rightarrow \mathrm{x}=\mathrm{c}-\frac{\mathrm{W}\left(2^{\mathrm{c}}\ln 2\right)}{\ln 2}$$

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