Question

The sum of the roots of the equation $\left|x^2-x-6\right|=x+2$ is

• A. $0$
• B. $-2$
• C. $2$
• D. $4$

Answer 1

Answer: (D)

$\left|x^2-x-6\right|=\left|\left(x+2\right)\left(x-3\right)\right|$

Case
(i) $x<-2$ or $x\ge 3$ $x^2-x-6=x+2\Rightarrow x^2-2x-8=0$
$(x-4)(x+2)=0\Rightarrow x=-2,4$
$\Rightarrow x=4$ in this case
Case (ii) $-2\le x<3$ $-x^2+x+6=x+2\Rightarrow x^2=4\Rightarrow x=\pm 2$
$\Rightarrow x=-2,2$ in this case
$\therefore$ The roots are -2,2,4 $\Rightarrow$ sum of roots $=4$