Solve the equation. Round to the nearest hundredth. Show work.[tex]2e^{8x} = 1 - e^{4x}[/tex]

Answer:The value of x = -0.17Step-by-step explanation:∵ [tex]2e^{8x}=1-e^{4x}[/tex]Let [tex]e^{4x}=y[/tex]∴ [tex]e^{8x}=y^{2}[/tex]∴ 2y² = 1 - y∴ 2y² + y - 1 =0 ⇒ factorize∴ (2y - 1)(y + 1) = 0∴ 2y - 1 = 0 ⇒ 2y = 1 ⇒ y = 1/2∴ y + 1 = 0 ⇒ y = -1∵ [tex]y=e^{4x}[/tex]  Note: [tex]e^{4x}=-1[/tex] ⇒ refused          ([tex]e^{ax}[/tex] never gives -ve values)∴ [tex]e^{4x}= 1/2[/tex] ⇒ insert ln in both sides∵ [tex]ln(e)^{ax}=axln(e)=ax[/tex] ⇒ ln(e) = 1∴ 4xln(e) = ln(1/2) ⇒ 4x = ln(1/2)∴ x = [ln(1/2)]/4 = -0.17