You have studied how to solve some types of quadratic equations. Solve the following questions. Click “Submit” when finished.

Consider the equation <EQUATION>.

What values of <EQUATION> make this equation true? Use the tool to evaluate <EQUATION> for different values of <EQUATION>. Click Done Exploring when finished.

The only two values that make <EQUATION> equal to zero are <EQUATION> and <EQUATION>.

What values of <EQUATION> make this equation true? Use the tool to evaluate <EQUATION> for different values of <EQUATION>. Which <EQUATION> values make <EQUATION> equal to zero? Click “Done Exploring” when finished.

The only two values that make <EQUATION> equal to zero are <EQUATION> and <EQUATION>.

In general, if the product of two numbers is zero, one of the numbers must be zero. This is called the |b| zero product property |/b| . So, if the product of two factors is zero, one of the two factors is equal to zero. Let’s look at the equation <EQUATION> again. Either <EQUATION> or <EQUATION>. Solve to find that <EQUATION> or <EQUATION>. This confirms the solutions you found earlier.

Use the zero product property to solve this equation for <EQUATION>. Click “Submit” when finished.

You have seen that if the product of two numbers is zero then one of the numbers must be zero. This property is not true for numbers other than zero. For example, if <EQUATION>, there are many possible choices for <EQUATION> and <EQUATION>. <EQUATION> and <EQUATION> could be <EQUATION> and <EQUATION>, <EQUATION> and <EQUATION>, <EQUATION> and <EQUATION>, <EQUATION> and <EQUATION>, and so on. But with zero we know that one of the two numbers must be zero.

Let’s use the zero product principle to solve <EQUATION>. First, we should rewrite <EQUATION> as a product. This means we must factor <EQUATION>. <EQUATION> is the common factor, so we can rewrite the equation as <EQUATION>. Using the zero product principle, we know that either <EQUATION> or <EQUATION>. Therefore, <EQUATION> or <EQUATION>. As always, we should check these solutions by substituting <EQUATION> and <EQUATION> for <EQUATION> in <EQUATION>. True statements result, so the solutions are correct.

Use the zero product principle to solve this equation for <EQUATION>. Click “Submit” when finished.

Consider the parabola <EQUATION>. Let's find the <EQUATION> intercepts of this parabola. Remember that the <EQUATION> intercepts are points on the graph where <EQUATION>. So we can solve <EQUATION> for <EQUATION>. To solve this equation, we can factor and use the zero product principle. <EQUATION> is a common factor, so <EQUATION> can be written as <EQUATION>. So, either <EQUATION> or <EQUATION>. Therefore, either <EQUATION> or <EQUATION>. So the <EQUATION> intercepts of this parabola are <EQUATION> and <EQUATION>.

To graph this equation, we can find a few more pieces of information. We know that the axis of symmetry is midway between the two <EQUATION> intercepts. So, the axis of symmetry must be <EQUATION>. We know that the vertex is on the axis of symmetry. By substituting <EQUATION> into <EQUATION> ,we find that the vertex is <EQUATION>.

Use the zero product principle to find the <EQUATION> intercepts of this equation. Click “Submit” when finished.

Use the graph to solve the equation. Click “Submit” when finished.

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education