Solve the following
e^-x (square root((3e^2x) - 5/4)) = e^x
e^-x √(3e^(2x) - 5/4) = e^x
multiply both sides by e^x
√(3e^(2x) - 5/4) = e^2x
square both sides
3e^(2x) - 5/4 = e^4x
times 4
12 e^2x - 5 = 4e^4x
4e^4x - 12 e^2x + 5 = 0
(2e^2x - 1)(2e^2x - 5) = 0
e^2x = 1/2 OR e^2x = 5/2
if e^2x = 1/2
ln both sides
2x lne = ln1 - ln2
2x = 0-ln2
x = -(1/2)ln 2 = appr -.3466
if e^2x = 5/2
2x = ln5 - ln2
x = (ln5 - ln2)/2 = appr .458
I checked the last answer in the original equation, it worked