Question

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

Answer 1

Answer:

The value of x = -0.46

Step-by-step explanation:

∵ $4e^{5x}-e^{-x}=-3e^{2x}$ ÷ $e^{-x}$

∴ $\frac{4e^{5x}}{e^{-x}}-\frac{e^{-x}}{e^{-x}}=\frac{-3e^{2x}}{e^{-x}}$

* 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