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
Answer by J2R2R(94)   (Show Source):
You can put this solution on YOUR website! 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.
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