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The sum of real roots of the equation |x...

The sum of real roots of the equation `|x|62+|x-2|-2=0i s ` a. 4 b. 1 c. 2 d. 2

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The correct Answer is:
4
Given, `|x-2|^(2)+|x-2|-2=0`
Case I When `c ge 2`
`implies(x-2)^(2)+(x-2)-2=0`
`impliesx^(2)+4-4x+x-2-2=0`
`impliesx^(2)-3x=0`
`impliesx(x-3)=0`
`impliesx=0,3" "[0 "is rejected"]`
`impliesx=3" "...(i)`
Case II When `x gt 2`
`implies{-(x-2))^(2)-(x-2)-3=0`
`(x-2)^(2)-x+2-2=0`
`impliesx^(2)+4-4x-x=0`
`impliesx^(2)--4x-(x-4)=0`
`impliesx(x-4)-1(x-4)=0`
`implies(x-1)(x-4)=0`
`impliesx=1,4" "["4 is rejected"]`
`impliesx=1" "...(ii)`
Hence, the sum of the roots is `3+1 =4.`
Alternate Solution
Given, `|x-2|^(2)+|x2|2=0`
`implies(|x-2|+2)(|x-2|-1)=0`
`therefore|x-2|=-2,1" "["neglacting"-2]`
`implies|x-2|=1impliesx=3,1`
`implies` Sum of the roots `=4`
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