Let S denote the set of all values of the parameter a for which $x+\sqrt{x^2+a}=a$           

has no solution, then S equals

If a=0 (1) holds for all $x\le 0$. Write (1) as 

$\begin{array}{l}{x}^2+a=a^2-2ax+x^2\\ \Rightarrow 2ax=a(a-1)\Rightarrow x=(a-1)/2\end{array}$

Substituting this (1) we get $a-1+|a+1|=2a$

This holds if $a\ge -1$.

$\therefore$ The equation has no solution if $a<-1$