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Question Number 151164 by puissant last updated on 18/Aug/21

Commented by puissant last updated on 18/Aug/21

solve in R

$${solve}\:{in}\:\mathbb{R} \\ $$

Answered by MJS_new last updated on 18/Aug/21

let t=e^(πx)  ⇔ x=((ln t)/π)  t+((√2)/t)−1−(√2)=0  t^2 −(1+(√2))t+(√2)=0  (t−1)(t−(√2))=0  t=1∨t=(√2)  ⇔  x=0∨x=((ln 2)/(2π))

$$\mathrm{let}\:{t}=\mathrm{e}^{\pi{x}} \:\Leftrightarrow\:{x}=\frac{\mathrm{ln}\:{t}}{\pi} \\ $$$${t}+\frac{\sqrt{\mathrm{2}}}{{t}}−\mathrm{1}−\sqrt{\mathrm{2}}=\mathrm{0} \\ $$$${t}^{\mathrm{2}} −\left(\mathrm{1}+\sqrt{\mathrm{2}}\right){t}+\sqrt{\mathrm{2}}=\mathrm{0} \\ $$$$\left({t}−\mathrm{1}\right)\left({t}−\sqrt{\mathrm{2}}\right)=\mathrm{0} \\ $$$${t}=\mathrm{1}\vee{t}=\sqrt{\mathrm{2}} \\ $$$$\Leftrightarrow \\ $$$${x}=\mathrm{0}\vee{x}=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{2}\pi} \\ $$

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