Question

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

• A. 2
• B. 3
• C. 4
• D. None of these

Answer 1

The correct option is C 4

The given equation is $|x-2{|}^2+|x-2|-2=0$
Let us assume $|x-2|=m$
$\begin{array}{cc}then& m^2+m-2=0\\ & (m-1)(m+2)=0\end{array}$
Only admissible value is
$\begin{array}{ccc}& m=1& (\because m\ne -2asm\ge 0)\\ \therefore & |x-2|=1& \\ \Rightarrow & x-2=1\Rightarrow x=3& \\ or& -(x-2)=1\Rightarrow x=1& \\ Hence& x=1,3& \end{array}$
$\therefore$ Sum of the roots of equation = 1+ 3 = 4