The sum of all the real roots of the equation $|\mathrm{x}-2{|}^2+|\mathrm{x}-2|-2=0$ is

The given equation is $|\mathrm{x}-2{|}^2+|\mathrm{x}-2|-2=0$

There are two cases: 

Case I: If $\mathrm{x}\ge 2$, then $(\mathrm{x}-2{)}^2+\mathrm{x}-2-2=0$

             $\begin{array}{rl}\Rightarrow {\mathrm{x}}^2-3\mathrm{x}& =0\\ \Rightarrow \mathrm{x}(\mathrm{x}-3)& =0\\ \Rightarrow \mathrm{x}& =0,3\end{array}$

Here, 0 is not possible. 

x=3

Case II : If x <2, then

             $\begin{array}{rl}\Rightarrow (\mathrm{x}-2{)}^2-\mathrm{x}+2-2& =0\\ \Rightarrow {\mathrm{x}}^2-5\mathrm{x}+4& =0\\ \Rightarrow (\mathrm{x}-1)(\mathrm{x}-4)& =0\\ \mathrm{x}& =1,4\end{array}$

Here, 4 is not possible. 

x=1

$\therefore$The sum of roots =1+3 = 4