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

The given equation  ${|x-2|}^2+|x-2|-2=0$
There are two cases:
CaseI: If  $x\ge 2,then{(x-2)}^2+x-2-2=0$
 $$\begin{gathered} \Rightarrow x^2-3x=0 \\ \Rightarrow x(x-3)=0 \\ \Rightarrow x=0,3 \end{gathered}$$
 Here,0 is not possible.
 $\therefore x=3$
Case II if x<2, then
 $$\begin{gathered} {(x-2)}^2+x-2-2=0 \\ \Rightarrow x^2-5x+4=0 \\ \Rightarrow (x-1)(x-4)=0 \\ \Rightarrow x=1,4 \end{gathered}$$ 
Here, 4 is not possible.
 $\therefore x=1$ 
  The sum of roots=1+3=4
Aliter
Let  $|x-2|=y$
Then, we get  $y^2+y-2=0$
 $\Rightarrow (y-1)(y+2)=0\Rightarrow y=1,-2$
But-2 is not possible.
Hence,  $|x-2|=1\Rightarrow x=1,3$
  Sum of the roots=1+3=4