solve-for-x-x-1-x-1-ax-2-x-a-2-1-a-R-

Question Number 80144 by behi83417@gmail.com last updated on 31/Jan/20

solve for x :

\frac{\sqrt{x} + 1}{\sqrt{x + 1}} + {ax}^{2} = x (a^{2} + 1) [a \in R]

Commented by john santu last updated on 31/Jan/20

l e t \sqrt{x + 1} = \sec t

x + 1 = \sec^{2} t \Rightarrow x = \tan^{2} t

\frac{\tan t + 1}{\sec t} + a (\tan^{4} t) = (a^{2} + 1) \tan^{2} t

\cos t (\tan t + 1) + a \tan^{4} t = (a^{2} + 1) \tan^{2} t