The real solutions of a quadratic equation are related to the graph of the quadratic function in the following ways:
1. x-intercepts: The x-values of the real solutions of the quadratic equation correspond to the x-intercepts of the quadratic function's graph. These are the points where the function crosses the x-axis.
2. Vertex: The vertex of the quadratic function represents the minimum or maximum point of the parabolic curve. The x-coordinate of the vertex can be found using the formula x = -b/2a, where the quadratic equation is in the form ax^2 + bx + c = 0.
a. If the quadratic equation has two real solutions, the vertex lies exactly in the middle between these two solutions on the x-axis.
b. If the quadratic equation has one real solution (a double root), the vertex coincides with this solution on the x-axis.
3. Discriminant: The discriminant of the quadratic equation, which is given by the expression b^2 - 4ac, provides information about the nature of the solutions.
a. If the discriminant is positive (b^2 - 4ac > 0), the quadratic equation has two distinct real solutions. On the graph, this means the parabola intersects the x-axis at two different points.
b. If the discriminant is zero (b^2 - 4ac = 0), the quadratic equation has one real solution (a double root). On the graph, this means the parabola touches the x-axis at exactly one point.
c. If the discriminant is negative (b^2 - 4ac < 0), the quadratic equation has no real solutions (only complex solutions). On the graph, this means the parabola does not intersect the x-axis, i.e., it lies entirely above or below the x-axis.