Question

Number of real roots of the equation e^4x + e^3x - 7e^2x - 13e^x = 6 is

Answer 1

Let e^x = t

Then equation converts into

t⁴ + t³ -7t² - 13t = 6

⇒ (t + 1) (t³ - 7t - 6) = 0

⇒ (t + 1)² (t² - t - 6) = 0

⇒ (t + 1)² (t + 2) (t - 3) = 0

⇒ t = -1, -2, 3

\(\because\) e^x > 0

\(\therefore\) t \(\neq\) -1, -2 (\(\because\) t = e^x)

\(\therefore\) t = 3

⇒ e^x = 3

⇒ x = ln 3

Number of real roots is 1.