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Quadratic Inequalities That Cannot Be Factored

There is another method, called the test point method, that can be used instead of the sign graph to solve inequalities. The following example shows how to solve a quadratic inequality that involves a prime polynomial.


Example 1

Solving a quadratic inequality using the quadratic formula

Solve x2 - 4x - 6 > 0 and graph the solution set.


The quadratic polynomial is prime, but we can solve x2 - 4x - 6 = 0 by the quadratic formula:

As in the previous examples, the solutions to the equation divide the number line into the intervals (-, 2 - ), (2 -, 2 +), and (2 +, ) on which the quadratic polynomial has either a positive or negative value. To determine which, we select an arbitrary test point in each interval. Because 2 + 5.2 and 2 - ≈ -1.2, we choose a test point that is less than -1.2, one between -1.2 and 5.2, and one that is greater than 5.2. We have selected -2, 0, and 7 for test points, as shown in the figure below.

Now evaluate x2 - 4x - 6 at each test point.

Test point Value of x2 - 4x - 6 at the test point Sign of x2 - 4x - 6 in interval of test point
-2 6 Positive
0 -6 Negative
7 15 Positive

Because x2 - 4x - 6 is positive at the test points -2 and 7, it is positive at every point in the intervals containing those test points. So the solution set to the inequality x2 - 4x - 6 > 0 is

and its graph is shown in the figure below.

The test point method used in Example 1 can be used also on inequalities that do factor. We summarize the strategy for solving inequalities using test points below.


Strategy for Solving Quadratic Inequalities

1. Rewrite the inequality with 0 on the right.

2. Solve the quadratic equation that results from replacing the inequality symbol with the equals symbol.

3. Locate the solutions to the quadratic equation on a number line.

4. Select a test point in each interval determined by the solutions to the quadratic equation.

5. Test each point in the original quadratic inequality to determine which intervals satisfy the inequality.

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