Using Discriminants to Find the Number of Roots of a Quadratic Curve

In general, we could apply the formula to work out the solutions of a quadratic function ax²+bx+c=0. 

The b²-4ac part is called the discriminant and the value of a discriminant could allow us to know the number of real roots that a quadratic function has. In other words, how many times does a quadratic curve cross the horizontal x axis in a graph?<o:p></o:p>

If b²-4ac=0, then a quadratic function has one real root and the graph of the function would be a curve just touch but not cross the x axis. In other words, the x axis is a tangent at the touching point and the touching point is also the minimum or maximum point of the function.<o:p></o:p>

If b²-4ac>0, then there are two real roots for the quadratic function and the corresponding graph would be a quadratic curve crosses over x axis twice.<o:p></o:p>

If b²-4ac<0, then there is no real roots for the quadratic function and a quadratic curve does not intersect or touch the horizontal axis at all in the graph. We could say that all points lying on this particular curve are either below the x axis or above the x axis.<o:p></o:p>