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e^x + 3e^-x = 4 Find the exact solutions to the equations

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Konrad509idn
[tex]e^x+3e^{-x}=4|\cdot e^x\\ (e^x)^2+3=4e^x\\ (e^x)^2-4e^x+3=0\\ (e^x)^2-e^x-3e^x+3=0\\ e^x(e^x-1)-3(e^x-1)=0\\ (e^x-3)(e^x-1)=0\\ e^x-3=0 \vee e^x-1=0\\ e^x=3 \vee e^x=1\\ x=\ln 3 \vee x=0 [/tex]
e^x+3e^-x=4

e^x+3/(e^x)=4

p=e^x

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Continued...

p+3/p=4

p^2+3=4p

p^2-4p+3=0

(p-1)(p-3)=0

Therefore p=1 and p=3.

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When p=1,

e^x=1

Therefore, x=ln1=0

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When p=3,

e^x=3

Therefore x=ln3
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