Answer:
The value of x = -0.46
Step-by-step explanation:
∵ [tex]4e^{5x}-e^{-x}=-3e^{2x}[/tex] ÷ [tex]e^{-x}[/tex]
∴ [tex]\frac{4e^{5x}}{e^{-x}}-\frac{e^{-x}}{e^{-x}}=\frac{-3e^{2x}}{e^{-x}}[/tex]
* Subtract the power of the same bases
∴ [tex]4e^{6x}-1=-3e^{3x}[/tex]
* Let [tex]e^{3x}=y[/tex]
∴ [tex]e^{6x}=y^{2}[/tex]
∴ 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
∵ [tex]y=e^{3x}[/tex]
∴ y = -1 refused ([tex]e^{ax}[/tex] never gives -ve value)
∴ [tex]e^{3x}=1/4[/tex] ⇒ insert ln in both sides
∵ [tex]lne^{ax}=axln(e)=ax[/tex] ⇒ ln(e) = 1
∴ 3xln(e) = ln(1/4)
∴ 3x = ln(1/4)
∴ x = [ln(1/4)] ÷ 3 = -0.46
