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

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Professor PC Thomas Classes · Answer

Let $x-2>0$ i.e. $x>2$ Then, $(x-2)^{2}+x-2-2=0$\ $x^{2}-4 x+4+x-4=0$\ $x^{2}-3 x=0 herefore \quad x=0$\ (Not possible)\ Possible solution : $x=3$\ Now $x-2<0$ i.e. $x<2$\ Then $(x-2)^{2}-(x-2)-2=0$\ $x^{2}-5 x+4=0$\ $x^{2}-4 x-x+4=0$\ $\Rightarrow \quad(x-1)(x-4)=0$\ $x=1$ (Accepted) $x=4$\ (Not possible)\ Hence roots are 1 and 3 .\ $ herefore \quad$ Sum $=1+3=4$\