Solve the equation. Round to the nearest hundredth. Show work.[tex]4e^{5x} - e^{-x} = -3e^{2x}[/tex]

Answer:The value of x = -0.46Step-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