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Question

The root of the equation $${\left| x \right|^2} + \left| x \right| - 6 = 0$$ are -


A
only one real number
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B
real and sum =1or1
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C
real and sum =0
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D
real and product =0
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Solution

The correct option is C real and sum =$$0$$
For $$x>0$$
$${\left|x\right|}^{2}+\left|x\right|-6=0$$ becomes 
$${x}^{2}+x-6=0$$
$${x}^{2}+3x-2x-6=0$$
$$x\left(x+3\right)-2\left(x+3\right)=0$$
$$\left(x+3\right)\left(x-2\right)=0$$
$$\Rightarrow\,x=-3,2$$
Roots are real and sum$$=-3+2=-1$$
For $$x<0$$
$${\left|x\right|}^{2}+\left|x\right|-6=0$$ becomes 
$${x}^{2}-x-6=0$$
$${x}^{2}-3x+2x-6=0$$
$$x\left(x-3\right)+2\left(x-3\right)=0$$
$$\left(x-3\right)\left(x+2\right)=0$$
$$\Rightarrow\,x=3,-2$$
Roots are real and sum$$=3-2=1$$
Hence real and sum $$=-1\,or \,1$$

Mathematics

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