discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots which are complex conjugates of each other. In these expressions is the imaginary unit. Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. Geometric interpretation The function is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of , , and . As shown in Figure 1, if , the parabola has a minimum point and opens upward. If , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The -coordinate of the vertex will be located at , and the -coordinate of the vertex may be found by substituting this -value into the function. The -intercept is located at the point . The solutions of the quadratic equation correspond to the roots of the function , since they are the values of for which . As shown in Figure 2, if , , and are real numbers and the domain of is the set of real numbers, then the roots of are exactly the -coordinates of the points where the graph touches the -axis. As shown in Figure 3, if the discriminant is positive, the graph touches the -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the -axis. Quadratic factorization The term is a factor of the polynomial if and only if is a root of the quadratic equation It follows from the quadratic formula that In the special case where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as Graphical solution The solutions of the quadratic equation may be deduced from the graph of the quadratic function which is a parabola. If the parabola intersects the -axis in two points, there are two real roots, which are the -coordinates of these two points (also called -intercept). If the parabola is tangent to the -axis, there is a double root, which is the -coordinate of the contact point between the graph and parabola. If