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Question Number 193175 by Humble last updated on 06/Jun/23

$$pleasesolveforxif$$$$2x^2=2^x$$

Answered by Frix last updated on 06/Jun/23

$$\mathrm{Obviously}x=1$$

Answered by Frix last updated on 06/Jun/23

$$2x^2=2^x$$$$\mathrm{Let}x=\pm \frac{1}{\sqrt{2}{\mathrm{e}}^t}\iff t=-(\ln (\pm x)+\frac{\ln 2}{2})$$$${\mathrm{e}}^{-2t}=2^{\pm \frac{1}{\sqrt{2}{\mathrm{e}}^t}}$$$$-2t=\pm \frac{\ln 2}{\sqrt{2}{\mathrm{e}}^t}$$$$t{\mathrm{e}}^t=\mp \frac{\ln 2}{2\sqrt{2}}$$$$t=W(\mp \frac{\ln 2}{2\sqrt{2}})$$$$t\approx \{\begin{array}{l}.200534631257\\ -2.19024886644\vee -.34657359028\end{array}$$$$\Rightarrow$$$$x=\{\begin{array}{l}\approx -.578620636081\\ \approx 6.31972235584\vee 1\end{array}$$

Answered by aba last updated on 06/Jun/23

$$\mathrm{x}=1\mathrm{is}\mathrm{solution}$$$${2\mathrm{x}}^2-2^{\mathrm{x}}=0\Rightarrow {\left(\sqrt{2}\mathrm{x}\right)}^2-{\left({\left(\sqrt{2}\right)}^{\mathrm{x}}\right)}^2=0\Rightarrow (\sqrt{2}\mathrm{x}-{\left(\sqrt{2}\right)}^{\mathrm{x}})(\sqrt{2}\mathrm{x}+{\left(\sqrt{2}\right)}^{\mathrm{x}})=0$$$$\Rightarrow \sqrt{2}\mathrm{x}-{\left(\sqrt{2}\right)}^{\mathrm{x}}=0\vee \sqrt{2}\mathrm{x}+{\left(\sqrt{2}\right)}^{\mathrm{x}}=0$$$$\Rightarrow \mathrm{x}.{\left(\sqrt{2}\right)}^{-\mathrm{x}}=\frac{1}{\sqrt{2}}\vee \mathrm{x}.{\left(\sqrt{2}\right)}^{-\mathrm{x}}=-\frac{1}{\sqrt{2}}$$$$\Rightarrow -\mathrm{xln}\left(\sqrt{2}\right).{\mathrm{e}}^{-\mathrm{xln}\left(\sqrt{2}\right)}=-\frac{\sqrt{2}\ln \left(\sqrt{2}\right)}{2}\vee -\mathrm{xln}\left(\sqrt{2}\right).{\mathrm{e}}^{-\mathrm{xln}\left(\sqrt{2}\right)}=\frac{\sqrt{2}\ln \left(\sqrt{2}\right)}{2}$$$$\Rightarrow -\mathrm{xln}\left(\sqrt{2}\right)=\mathrm{W}(-\frac{\sqrt{2}\ln \left(\sqrt{2}\right)}{2})\vee -\mathrm{xln}\left(\sqrt{2}\right)=\mathrm{W}\left(\frac{\sqrt{2}\ln \left(\sqrt{2}\right)}{2}\right)$$$$\Rightarrow \mathrm{x}=-\frac{2}{\ln \left(2\right)}\mathrm{W}(-\frac{\sqrt{2}\ln \left(2\right)}{4})\vee \mathrm{x}=-\frac{2}{\ln \left(2\right)}\mathrm{W}\left(\frac{\sqrt{2}\ln \left(2\right)}{4}\right)$$$$\Rightarrow \mathrm{x}=-\frac{2}{\ln \left(2\right)}\mathrm{W}(-\frac{\ln \left(2\right)}{2\sqrt{2}})\approx 6.31972\vee \mathrm{x}=-\frac{2}{\ln \left(2\right)}\mathrm{W}\left(\frac{\ln \left(2\right)}{2\sqrt{2}}\right)\approx -0.578621$$

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