Question

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

Answer 1

Let $f\left(x\right)=|x-2|$
For $x>2,f\left(x\right)=x-2$
for $x<2,f\left(x\right)=2-x$
Thus, for $x>2$ equation becomes $(x-2{)}^2+(x-2)-2=0$
$x^2-3x=0$
Thus, the root of equation $x>2$ is $3$
For $x<2$ the equation becomes $(x-2{)}^2+(2-x)-2=0$
$x^2-5x+4=0$
$(x-4)(x-1)=0$
Root which is less than $2$ is $1$
Thus, roots of given equations are $3$, $1$
Sum will be $3+1=4$