SOLUTION: solve for x if (2e^x-e^(-x))/5=t

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Question 1080329: solve for x if (2e^x-e^(-x))/5=t
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let u=e%5Ex,
2u-1%2Fu=5t
2u%5E2-1=5ut
2u%5E2-5ut-1=0
Using the quadratic equation,
u=%28-%28-5t%29%2B-sqrt%28%28-5t%29%5E2-4%282%29%28-1%29%29%29%2F%282%282%29%29
u=%285t%2B-sqrt%2825t%5E2%2B8%29%29%2F4
e%5Ex=%285t+%2B-+sqrt%2825t%5E2%2B8%29%29%2F4
The solution with the negative sign would give a negative right hand side which would have no solution so we remove that portion,
e%5Ex=%285t+%2B+sqrt%2825t%5E2%2B8%29%29%2F4
x=ln%28%285t+%2B+sqrt%2825t%5E2%2B8%29%29%2F4%29