How are the real solutions of a quadratic equation related to the graph of the quadratic function?

They are the points where the graph crosses the x-axis. That is, where

ax^2+bx+c = 0

the solutions to the quadratic equation correspond with the x-intercepts of the corresponding function

e.g.
the solutions of
x^2 - 7x + 12 = 0
are x = 4 or x = 3

and the graph of y = x^2 - 7x + 12 has x-intercepts of 3 and 4

The real solutions of a quadratic equation are directly related to the graph of the quadratic function. Here are the key relationships:

1. x-intercepts: The x-intercepts of the graph represent the real solutions of the quadratic equation. These are the points where the graph intersects the x-axis. The x-coordinate of the x-intercepts gives the value(s) of x that satisfy the quadratic equation.

2. Vertex: The vertex of the graph, represented as (h, k), is a significant point on the graph. If the quadratic equation has real solutions, the x-coordinate of the vertex (h) will be the midpoint between the x-intercepts.

3. Discriminant: The discriminant of the quadratic equation, given by the expression b^2 - 4ac, determines the nature of the real solutions and the shape of the graph.

- If the discriminant is positive, the equation has two distinct real solutions, and the graph will intersect the x-axis at two points.

- If the discriminant is zero, the equation has a repeated real solution, and the graph will touch the x-axis at one point (a tangent).

- If the discriminant is negative, the equation has no real solutions, and the graph of the quadratic function will not intersect the x-axis (it will be entirely above or below).

Overall, the real solutions of the quadratic equation can be visualized and understood by analyzing the x-intercepts, vertex, and discriminant of the corresponding quadratic graph.

The real solutions of a quadratic equation are related to the graph of the quadratic function in the following way:

1. Intersection with the x-axis: The x-coordinates of the points where the graph of the quadratic function intersects the x-axis represent the real solutions of the quadratic equation. These are the values of x for which the function equals zero. If the quadratic function has two distinct x-intercepts, then the corresponding quadratic equation will have two real solutions. If the quadratic function has only one x-intercept (i.e., it touches the x-axis at one point), then the corresponding quadratic equation will have one real solution. And if the quadratic function does not intersect the x-axis (i.e., it is either always positive or always negative), then the corresponding quadratic equation will have no real solutions.

2. Vertex: The vertex of the parabolic graph represents the minimum or maximum point of the quadratic function, depending on whether the quadratic coefficient is positive or negative. If the vertex lies above the x-axis, then the corresponding quadratic equation has no real solutions. If the vertex lies on the x-axis (i.e., the function touches the x-axis at its lowest or highest point), then the corresponding quadratic equation has exactly one real solution. Finally, if the vertex lies below the x-axis, then the corresponding quadratic equation has two real solutions.

In summary, the number and nature of the real solutions of a quadratic equation can be determined by analyzing the graph of the corresponding quadratic function, particularly the x-intercepts and the vertex.