Solving Quadratic Equations by Graphing
Objective Help you understand that the
solutions of a quadratic equation occur where the graph of the
corresponding function intersects the xaxis.
In this lesson, you should be able to use the graphing
techniques you already learned to help you solve or approximate
solutions to quadratic equations. Let's begin by stating a
definition.
Quadratic Equations
Quadratic Equation
A quadratic equation is an equation of the form f ( x ) = 0,
where f ( x ) = ax^{ 2} + bx + c is a quadratic function.
The goal in solving a quadratic equation is to find what x
values make the y value of the quadratic function y = f ( x )
equal to zero. The y value of the function will be zero where the
graph intersects the xaxis. Geometrically, this is because a
solution of an equation f ( x ) = 0 occurs when the graph of the
function y = f ( x ) intersects the line y = 0. But the line y =
0 is the xaxis.
Recall that the graph of a quadratic function is a parabola.
So the solutions of a quadratic equation occur where the parabola
representing the graph of the quadratic intersects the xaxis.
Try to draw some parabolas like the following, and observe
that a parabola can either
• not intersect the xaxis,
• intersect the x axis in exactly one point, or
• intersect the x axis in two points.
1. There may be no real solutions. This will
occur if the parabola does not intersect the xaxis. Either the
parabola opens upwards and the vertex (a minimum) lies above the
x axis, or the parabola opens downwards and the vertex (a
maximum) lies below the xaxis.
2. There may be one solution. This occurs
when the parabola intersects the x axis in exactly one point.
This happens when the vertex of the parabola lies on the x axis.
3. There may be two solutions. This occurs
when the parabola intersects the x axis in two points. Either
the parabola opens upward and the vertex lies below the x axis,
or the parabola opens downward and the vertex lies above the x
axis.
Keep in mind that a quadratic equation may have no solutions,
one solution, or two solutions, because a parabola can intersect
the x axis in zero, one, or two points.
