Question

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places:

$\frac{e^x-e^{-x}}{2}=2$.

Short answer

The values of x which satisfies the given equation is$x=-0.236andx=1.444$

Step-by-step solution

Given information

We are given an equation$\frac{e^x-e^{-x}}{2}=2$

multiply both sides by ex

we get,

$$\begin{gathered} e^{2x}-1=4e^x \\ {e}^{2x}-4e^x-1=0 \end{gathered}$$

Substitute ex=t and solve for t

We get,

$$\begin{gathered} t^2-4t-1=0 \\ t=\frac{4\pm \sqrt{16+4}}{2} \\ t=\frac{4\pm \sqrt{20}}{2} \\ t=\frac{4\pm 2\sqrt{5}}{2} \\ t=2\pm \sqrt{5} \end{gathered}$$

Now again substitute 

we get

$$\begin{gathered} e^x=t \\ {e}^x=2-\sqrt{5} \\ x=-0.236 \\ And \\ {e}^x=2+\sqrt{5} \\ x=1.444 \end{gathered}$$