Question

Solve the equation. (Round your answer to four decimal places.) e2x − ex − 12 = 0 x =

Answer 1

x = 1.3863

Step-by-step explanation:

We can rewrite our equation as

`(e^x)^2-e^x-12=0`

which can be factored as

`(e^x+3)(e^x-4)=0`

Therefore, the two equations we must solve are

`\begin{gathered} e^x+3=0 \\ e^x-4=0 \end{gathered}`

The first equation gives

`\begin{gathered} e^x+3=0 \\ \Rightarrow e^x=-3 \end{gathered}`

Since the natural logarithm can never give a negative number, the above equation is undefined.

Now the second equation gives

`e^x-4=0`
`e^x=4`

taking the natural log of both sides gives

`\ln[e^x]=\ln[4]`
`\boxed{x=\ln[4].}`

which we evaluate to get (rounded to the nearest four decimal places)

`\boxed{x=1.3862.}`

which is our answer!