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Q.
The sum of the roots of the equation $\left|x^{2} - x - 6\right|=x+2$ is
NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations
Solution:
$\left|x^{2} - x - 6\right|=\left|\left(x + 2\right) \left(x - 3\right)\right|$
Case
(i) $x<-2$ or $x \geq 3$ $x^{2}-x-6=x+2 \Rightarrow x^{2}-2 x-8=0$
$(x-4)(x+2)=0 \Rightarrow x=-2,4$
$\Rightarrow x=4$ in this case
Case (ii) $-2 \leq x<3$ $-x^{2}+x+6=x+2 \Rightarrow x^{2}=4 \Rightarrow x=\pm 2$
$\Rightarrow x=-2,2$ in this case
$\therefore $ The roots are -2,2,4 $\Rightarrow $ sum of roots $=4$
