Question 195270
Without drawing the graph of the equation, determine how many points the parabola has in common with the x-axis and whether its vertex lies above, on, or below the x-axis. 


y = 2x^2 + x + 3


Here, the discriminant will tell us how many real or imaginary roots we have.


b^2 – 4ac > 0 gives two real solutions which are different (crosses the x-axis in two places)


b^2 – 4ac = 0 gives two real solutions which are equal (crosses the x-axis in one place at the vertex)


b^2 – 4ac < 0 gives two complex solutions which are different (doesn’t cross the x-axis).


For our equation, a = 2, b = 1, c = 3


b^2 – 4ac = 1 – 24 = – 23


Thus we have two complex roots which are different and therefore the parabola doesn’t cross the x-axis so there are no points in common with the x-axis, and the vertex lies above the x-axis.