JEE Main 2021 (Online) 31st August Morning ShiftQuestion 1
The number of real roots of the equation $e^{4x}+2e^{3x}-e^{x}-6=0$ is :
Quadratic Equation · Equations Reducible to Quadratic
Solution
Concept. With $t=e^x>0$ the quartic factorises into two quadratics. Formulas used. (1) $t^4+2t^3-t-6=(t^2+t-3)(t^2+t+2)$Step 1 - Factorise. By (1). The factor $t^2+t+2$ has $D<0$, so no real root. Step 2 - The other factor. $t^2+t-3=0$, $t=\frac{-1\pm\sqrt{13}}{2}$. Only $\frac{-1+\sqrt{13}}{2}>0$. Step 3 - Count. One positive $t$, so one real $x$. Answer: (B)