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Use of Parentheses or Brackets (The Distributive Law)
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Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
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Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
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Rise and Run
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Writing the Terms of a Polynomial in Descending Order
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Comparing Integers on a Number Line
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The Quadratic Formula
Writing a Quadratic with Given Solutions
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Solving Systems of Equations Using Substitution
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Product of a Sum and a Difference
Solving First Degree Inequalities
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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.

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 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.

 
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