# How do you solve #e^x + e^(-x) = 3#?

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George C.
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Jan 1, 2016

#### Answer:

Express as a quadratic in

#x = ln((3+-sqrt(5))/2) =+-ln((3+sqrt(5))/2)#

#### Explanation:

Let

Then the equation becomes:

#t + 1/t = 3#

Multiplying both sides by

#t^2+1 = 3t#

Subtract

#t^2-3t+1 = 0#

Use the quadratic formula to find roots:

#t = (3+-sqrt(5))/2#

Note that due to the symmetry of the equation

Now

#e^x = (3+-sqrt(5))/2#

Taking natural logs of both sides we find:

#x = ln((3+-sqrt(5))/2) =+-ln((3+sqrt(5))/2)#

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