For IIT aspirants #4

Simplifying the above equation we get $x^2=( 2^{x-1} - 2^{|x-3|+2})/(2^{x+1} - 2^{|x-3|+4})$ . Now on differentiating the term $( 2^{x-1} - 2^{|x-3|+2})/(2^{x+1} - 2^{|x-3|+4})$ We find its derivative is always $0$ hence its a constant function now put any value of x say 1 to find its value as $1/4$ . Then solving $x^2=1/4$ we get the roots as $\boxed{0.5}$ and $\boxed{-0.5}$ . Hence it has no negative integeral solutions.

First take all the terms to the other side. As x is negative, simplify the modulus with the given condition. Then take all the terms to the left hand side. Now multiply by 2^(x+1) on both sides. Now, if we factor terms out, we get the answer as x=-0.5 or x=3. Neither of which are negative integers