We can see from the fact that x^2 term is negative that the parabola goes to -infinity as |x| gets large (it forms an inverted-u shape).
We can also rearrange this standard form quadratic (y = ax^2 + bx + c) into vertex form (y = a(x - h)^2 + k):
= -2(x^2 -x/2 +3/2)
= -2(x - 1/4)^2 + 3/2-1/16
Since (h,k) is the vertex we just need to check whether k is positive or negative. It is positive, so the vertex lies above the x-axis. And since the parabola curves down, it will cross the x-axis at two points.