Sum of real roots of the equation $x^2 + 5|x| + 6 = 0$?

Question

Sum of real roots of the equation $x^2 + 5|x| +6 = 0$

Answer 1

Hint: It $r$ is a solution then so is $-r$

Note: After seeing Ryan's answer, I realized that the solution set is empty. Thus, Ryan's answer is the correct answer.

Now if we are looking for solutions inside $\mathbb{C}$, my hint can be used to deduce that either the solution set is infinite (hence the sum is undefined) or the sum is zero

Answer 2

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

Answer 3

Another hint: all the terms are non-negative, and the constant term is actually positive.