<s>[INST] Sum of real roots of the equation $x^2 + 5|x| + 6 = 0$? Sum of real roots of the equation $x^2 + 5|x| +6 =  0$
 [/INST] Hint
$|x|=x\forall x\ge0$
$|x|=-x\forall x<0$
Case 1 $x\ge 0$
$x^2+5x+6=(x+2)(x+3)=0\Rightarrow x=-2,-3$ but we already  assumed $x\ge 0$ so $(\Leftrightarrow)$
Case 2 $x<0$
then the equation becomes according to the definition of $|x|$
$x^2-5x+6=(x-2)(x-3)=0\Rightarrow x=2,3$ again $(\Leftrightarrow)$
$(\Leftrightarrow)$ is the sign  of contradiction
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