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

Answer

4

Explanation

The given equation is `|x-2|^(2)+|x-2|-2=0`
There are two cases:
Case If `xge2` then `(x-2)^(2)+x-2-2=0`
`impliesx^(2)-3x=0`
`impliesx(x-3)=0`
`impliesx=0,3`
Here 0 is not possible
`:.x=3`
Case II If `xlt2` then
`(x-2)^(2)-x+2-2=0`
`impliesx^(2)-5x+4=0`
`implies(x-1)(x-4)=0`
`impliesx=-1,4`
Here 4 is not possible.
`:.x=1`
`:. ` The sum of roots `=1+3=4`
Aliter
Let `|x-2|=y`
Then we get `y^(2)+y-2=0`
`implies(y-1)(y+2)=0impliesy=1,-2`
But `-2` is not possible.
Hence `|x-2|=1impliesx=1,3`
`:.` Sum of the roots `=1+3=4`