The sum of all the real roots of the equation x - 2 2 + x - 2 - 2 = 0 is

Given:

${\left|x-2\right|}^2+\left|x-2\right|-2=0 ....\left(1\right)$

Putting $y=\left|x-2\right|$, then we can rewrite $\left(1\right)$ as

$y^2+y-2=0$

$\Rightarrow y^2+2y-y-2=0$

$\Rightarrow y\left(y+2\right)-\left(y+2\right)=0$

$\Rightarrow \left(y-1\right)\left(y+2\right)=0$

$\Rightarrow y=1, -2$

Since, $y\ge 0$, so

$y=\left|x-2\right|=1$

$\Rightarrow x-2=\pm \mathrm{ }1$

$\Rightarrow x=3\mathrm{ }\mathrm{o}\mathrm{r}\mathrm{ }1$

Required sum is $=3+1=4$