Question

Use the equation e^x+e^-x/e^x-e^-x=t tosolve for x

Answer 1

Explanation:

I assume that the equation is

`e^x+(e^(-x))/(e^x) -e^(-x)=t \\`

`e^x+(1)/(e^(2x))-(1)/(e^x) = t`

`e^(3x)+1-e^x=e^(2x)t`

`e^(3x)-te^(2x)-e^x=-1`

Let u = e^x

`u^3-tu^2-u=-1`

`u(u^2-tu-1) = -1`

if u=1, then
`u^2-tu-1 =-1 \longleftrightarrow u^2-tu=0`

then u = 0 (reject) or u = t

So far, we have u = 1 and u = t

if u =1, e^x =1 , then x = 0

if u = t, then e^x = t or x = lnt

Answer 2

Solve the rational equation by combining expressions and isolating the variable

x = ln ( t − √ t ^2 + 4 )/2

x = ln ( t +√ t ^2 + 4 )/2