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

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

WBJEEWBJEE 2009Complex Numbers and Quadratic Equations

Solution:

Given, $\left|x-2\right|^{2}+\left|x-2\right|-2=0$
$\therefore y^{2}+y-2=0$, where $y=\left|x-2\right|$
$\Rightarrow \left(y+2\right)\left(y-1\right)=0$
$y=1 [\because y=\left|x-2\right|=-2$ not possible]
$\Rightarrow \left|x-2\right|=1$
$\Rightarrow x-2=\pm1$
$\Rightarrow x=3, 1$
$\therefore $ Sum of roots is $4$