\[ e^x + 3e^{-x} = 4 \]

Find the exact solutions to the equation.

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

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.

When p=1,
e^x=1
Therefore, x=ln1=0

When p=3,
e^x=3
Therefore x=ln3

Answered by Anonymous | 2024-06-10

[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
]

Answered by konrad509 | 2024-06-24

The exact solutions to the equation e x + 3 e − x = 4 are x = ln ( 3 ) and x = 0 . We solved this by rewriting the equation in a quadratic form and applying the quadratic formula. Finally, we found the solutions by substituting back to e x and taking logarithms.
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Answered by konrad509 | 2024-09-27