The sum of the roots of the equation x² - |2x - 3| = 0 is

x² - |2x - 3| = 0
|2x - 3| = 2x - 3, when 2x - 3 > 0 ⇒ x > 3/2
For the above case, the equation becomes
x² - x + 3/2 = 0
⇒ x = (1 ± √(1 - 4(3/2)))/2 = (1 ± √-5)/2
Since x is real there is no solution for x > 3/2.
Now, for x < 3/2, |2x - 3| = 3 - 2x
The equation becomes x² + 2x - 3 = 0
(x + 3)(x - 1) = 0
x = -3 or x = 1
Since x < 3/2, only x = -3 and x = 1 satisfy the condition.
The sum of the roots obtained is = -3 + 1 = -2