The number of real roots of the equation e ^ 4x - e ^ 3x - 4 e ^ 2x...

We have, ${e}^{4x} - {e}^{3x} - 4{e}^{2x} - {e}^{x} + 1 = 0$

Let ${e}^{x} = t$, we get

${t}^{4} - {t}^{3} - 4{t}^{2} - t + 1 = 0$

Divide with ${t}^{2}$ on both sides

$⇒ {t}^{2} - t - 4 - \dfrac{1}{t} + \dfrac{1}{{t}^{2}} = 0$

$⇒ {\left(t + \dfrac{1}{t}\right)}^{2} - \left(t + \dfrac{1}{t}\right) - 6 = 0$

Let $\alpha = t + \dfrac{1}{t} \ge 2$, we get

${\alpha}^{2} - \alpha - 6 = 0$

$⇒ \left(\alpha - 3\right)\left(\alpha + 2\right) = 0$

$⇒ \alpha = 3, - 2$ (reject)

$⇒ t + \dfrac{1}{t} = 3 ⇒ {t}^{2} - 3t + 1 = 0$

$⇒ {e}^{x} = \dfrac{3 \pm \sqrt{5}}{2}$

$⇒ {e}^{x} = \dfrac{3 \pm 2 . 23}{2}\left(∵ \sqrt{5} \approx 2 . 23\right)$

$⇒ {e}^{x} = \dfrac{5 . 23}{2}$ or ${e}^{x} = \dfrac{0 . 77}{2}$

$⇒ {e}^{x} = 2 . 615$ or ${e}^{x} = 0 . 385$

$⇒$ The number of real roots $= 2 .$