TITLE: Exploring the quadratic equation $x^2 + \lvert x\rvert - 6 = 0$
QUESTION [7 upvotes]: This question and the described solution are copied from a test-paper :
For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.

there is only one root
sum of the roots is + 1
sum of the roots is zero
the product of the roots is +4

Answer : (3)
Answer Explanation : 
If x > 0 |x| = x.
Given equation will be $x^2 + x - 6 = 0$⇒ x = 2,- 3 ⇒ x = 2
If x < 0 |x| = - x.
Given equation will b e  $x^2$ - x - 6 = 0 ⇒ x = -2, 3 ⇒x = - 2
Sum of roots is 2 - 2 = 0
Now I have a doubt on the statment "If x < 0 |x| = - x." I think modulus means that |x| is always positive ?! Also I can see that (2) seems to be the correct option isn't ?!
Please post your views.

REPLY [6 votes]: $f(x)=x^2+|x|-6$ is an even function—that is, $f(x)=f(-x)$ for all $x$, or the graph of $y=f(x)$ is symmetric over the y-axis—so if $f(c)=0$ then $f(-c)=0$, so the sum of the zeros of $f$ must be 0.