Question

Solve the equation. Round to the nearest hundredth. Show work.


`4e^(5x) - e^(-x) = -3e^(2x)`

Answer 1

The value of x = -0.46

Explanation:

∵
`4e^(5x)-e^(-x)=-3e^(2x)` ÷
`e^(-x)`

∴
`(4e^(5x))/(e^(-x))-(e^(-x))/(e^(-x))=(-3e^(2x))/(e^(-x))`

* Subtract the power of the same bases

∴
`4e^(6x)-1=-3e^(3x)`

* Let
`e^(3x)=y`

∴
`e^(6x)=y^(2)`

∴ 4y² - 1 = -3y

∴ 4y² + 3y - 1 = 0 ⇒ factorize

∴ (4y - 1)(y + 1) = 0

∴ y + 1 = 0 ⇒ y = -1

∴ 4y - 1 = 0 ⇒ 4y = 1 ⇒ y = 1/4

∵
`y=e^(3x)`

∴ y = -1 refused (
`e^(ax)` never gives -ve value)

∴
`e^(3x)=1/4` ⇒ insert ln in both sides

∵
`lne^(ax)=axln(e)=ax` ⇒ ln(e) = 1

∴ 3xln(e) = ln(1/4)

∴ 3x = ln(1/4)

∴ x = [ln(1/4)] ÷ 3 = -0.46