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Friday 24th of May
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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# 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 x-axis.

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.

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 x-axis. 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 x-axis.

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 x-axis.

Try to draw some parabolas like the following, and observe that a parabola can either

• not intersect the x-axis,

• 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 x-axis. 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 x-axis.

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.