Relationship Between Quadratic Equations and Quadratic Functions

1. Graph of a Quadratic Function and the Solutions of a Quadratic Equation
   The -coordinates of the intersection points between the graph of the quadratic function and the -axis are the real roots of the quadratic equation .
2. Relationship Between the Graph of a Quadratic Function and the -axis
   The number of intersection points between the graph of the quadratic function and the -axis is equal to the number of real solutions of the quadratic equation . Therefore, for the quadratic function , the relationship between the graph of the quadratic function and the solutions of the quadratic equation can be determined based on the sign of the discriminant , as follows:

Solutions of	Two distinct roots	One repeated root (double root)	Two distinct complex roots
Graph of and -axis	Intersects at two distinct points	Touches at one point (tangent)	Does not intersect
Graph of
Graph of

Relationship Between the Graph of a Quadratic Function and a Line

The number of intersection points between the graph of the quadratic function and the line is equal to the number of real solutions of the quadratic equation obtained by solving the system , that is:

The relationship between the graph of the quadratic function and the line can be determined based on the sign of the discriminant of the quadratic equation , as follows:

Graph of where and the line where
	Intersects at two distinct points	Touches at one point (tangent)	Does not intersect