Question

Find the sum of the roots of the equation $|x^2-x-6|=x+2$.

Answer 1

Given equation $|x^2-x-6|=x+2$
$\Rightarrow x^2-x-6=\pm (x+2)$ (by definition of absolute value function)

Case 1

For $x^2-x-6>0$ that is for $x\ge 3$ or $x\le -2$

$x^2-x-6=x+2$
$\Rightarrow (x-4)(x+2)=0$
$\Rightarrow x=4,-2$
$\therefore$ Roots are $4,-2$ for $x^2-x-6\ge 0$
Case 2

For $x^2-x-6<0$ that is for $x\ge -2$ or $x\le 3$

$x^2-x-6=-x-2$

$x^2=4$
$\Rightarrow x=-2,2$
$\therefore$ Roots are $-2,2$ for $x^2-x-6<0$
So, sum of roots $=-2+2+4=4$ for the given equation.