JEE Main202331 Jan 2023Evening ShiftMathematicsQuadratic EquationActual
The equation e 4 x + 8 e 3 x + 13 e 2 x - 8 e x + 1 = 0 , x ∈ R has :
Options
- Afour solutions two of which are negative
- Btwo solutions and both are negative
- Cno solution
- Dtwo solutions and only one of them is negative
Correct answer
B. two solutions and both are negative
Step-by-step solution
Given, e 4 x + 8 e 3 x + 13 e 2 x - 8 e x + 1 = 0 ,   x ∈ R Now let e x = t we get, t 4 + 8 t 3 + 13 t 2 - 8 t + 1 = 0 Now divide complete equation by t 2 ⇒ t 2 + 1 t 2 + 8 t - 1 t + 13 = 0 ⇒ t - 1 t 2 + 8 t - 1 t + 15 = 0 Now let t - 1 t = z we get, ⇒ z 2 + 8 z + 15 = 0 ⇒ z = - 3 , - 5 So, t - 1 t = - 3 or t - 1 t = - 5 ⇒ t 2 + 3 t - 1 = 0 or t 2 + 5 t - 1 = 0 ⇒ t = - 3 ± 13 2 , - 5 ± 29 2 So, e x = - 3 + 13 2 , - 5 + 29 2 = α , β (rejecting negati