How can i solve this equation?

Every time i try to solve this equation i get the wrong solution set:

#(e^y-e^-y) /2=1#

1 Answer
Daniel L.
Oct 30, 2017

This equation has one solution: #y=ln(1+sqrt(2))#. See explanation.

Explanation:

First we can substitute the expression #e^y# with a new variable:

#t=e^y#, so that we get:

#(t-1/t)/2=1#

If we multiply this by #2# we get:

#t-1/t=2#

If we multiply this by #t# we het a quadratic equation:

#t^2-1=2t#

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

If we use the quadratic formula we get:

#Delta=(-2)^2-4*1*(-1)#

#Delta= 4+4=8#

#t_1=(2-2sqrt(2))/2=1-sqrt(2)#

#t_1# is a negative number, so it is not in the domain of the expression.
(#e^x# is always positive).

#t_2=(2+2sqrt(2))/2=1+sqrt(2)#

This expression lies in the domain, so we can calculate the corresponding value of #y#

#e^y=1+sqrt(2) => y=ln(1+sqrt(2))#

Answer: This equation has one solution:

# y=ln(1+sqrt(2))#

Related questions
Impact of this question
315 views around the world
You can reuse this answer
Creative Commons License