The Discriminant of a Quadratic Equation
The discriminant of an equation is the expression found underneath the square root sign in the quadratic equation (b2-4ac). If this discriminant is greater than zero, there will be two solutions. The parabola will cross the x axis two times. If the discriminant equals zero, there will be only one solution. The parabola just touches the x axis one time. If the discriminant is less than zero, you will have two complex solutions, which will be conjugates of each other. The parabola never touches the x axis.
If the discriminant, b squared, minus 4ac equals 0, then there’s only one real solution. That means that your parabola just touches the x-axis one time, your vertex is on the x-axis. If b squared, minus 4ac is less than 0—now remember that this is what’s under a square root sign, so if you take the square root of a negative number (less than 0 would be negative)—then you’re going to get an imaginary number, so you would have two complex solutions, and they would be conjugates of each other.
One solution would be a plus bi, while the other solution would be a minus bi. The real part would be the same, and the imaginary part those would be opposites of each other, so what that means for your parabola is that it never touches the x-axis. There aren’t any real solutions, that means that it never actually crosses the x-axis. All of these parabolas are drawn concave up, but of course the same would be true for a concave down parabola.