Question

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

• A. $2$
• B. $3$
• C. $4$
• D. $1$

Answer 1

Answer: (C)

$4$

Given equation is $|x-2{|}^2+|x-2|-2=0$

Let $|x-2|=t$

Therefore, the given equation becomes $t^2+t-2=0$

$⟹t^2+2t-t-2=0$

$⟹t(t+2)-1(t+2)=0$

$⟹(t-1)(t+2)=0$

$⟹t-1=0$ or $t+2=0$

$⟹t=1$ or $t=-2$

$⟹|x-2|=1$

$⟹\pm (x-2)=1$

$⟹(x-2)=1$ or $(x-2)=-1$

$⟹x=3$ or $x=1$

Therefore, the sum of the real roots of the given equation is $3+1=4$