Question

The value(s) of $x,$ which satisfy the equation $|x-2{|}^2+|x-2|-2=0$ is/are

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (B), (D)

The correct options are
B $1$
D $3$

$|x-2{|}^2+|x-2|-2=0$

Case I $:x\ge 2$
$\Rightarrow (x-2{)}^2+(x-2)-2=0$
$\Rightarrow x^2-3x=0\Rightarrow x(x-3)=0$
$\Rightarrow x=0,3(0$ is rejected because $x\ge 2)$
$\therefore x=3$

Case II: $x<2$
$\{-(x-2){\}}^2-(x-2)-2=0$
$\Rightarrow x^2-5x+4=0$
$\Rightarrow (x-1)(x-4)=0$
$\Rightarrow x=1,4(4$ is rejected because $x<2$ )
$\therefore x=1$