The number of real solutions of the equation e 4 x + 4 e 3 x - 58 e 2 x + 4 e x + 1 = 0 is _____.

Step-by-step solution

Given equation e 4 x + 4 e 3 x - 58 e 2 x + 4 e x + 1 = 0 Let f x = e 2 x e 2 x + 1 e 2 x + 4 e x + 1 e x - 58 Now let e x + 1 e x = t i.e. t 2 + 4 t - 58 = 0 t = - 4 ± 16 + 4 × 58 2 = - 4 ± 2 62 2 t = - 2 + 2 62 is possible and t = - 2 - 2 62 is not possible as t ≥ 2 i.e. e x + 1 e x = - 2 + 2 62 e 2 x - - 2 + 2 62 e x + 1 = 0 For real solution, discriminant 248 - 8 62 > 0 Also - b 2 a = - 2 + 2 62 2 = 62 - 1 > 0 Hence, both roots of the equation are positive real roots.