Quadratic Inequalities

Mathematics Quadratic Inequalities [email protected]

Quadratic Inequalities A quadratic inequality is one that can be written in one of the following standard forms: or or or  

In other words, a quadratic inequality is in standard form when the inequality is set to 0. Just like in a quadratic equation, the degree of the polynomial expression is two.

Solving Quadratic Inequalities Using a Sign Graph of the Factors Note: This method of solving quadratic inequalities only works if the quadratic factors.

Step 1: Write the quadratic inequality in standard form. It is VERY important that one side of the inequality is 0. 0 is our magic number. It is the only number that separates the negatives from the positives. If an expression is greater than 0, then there is no doubt that its sign is positive. Likewise, if it is less than 0, its sign is negative. You can not say this about any other number. Since we are working with inequalities, this idea will come in handy. With this technique we will be looking at the sign of a number to determine if it is a solution or not. Step 2: Solve the quadratic equation, , by factoring to get the boundary point(s). The boundary point(s) will mark off where the quadratic expression is equal to 0. This is like the cross over point. 0 is neither positive or negative.  As mentioned above, this method of solving quadratic inequalities only works if the quadratic factors. If it doesn't factor then you will need to use the test-point method. Step 3: Use the boundary points found in Step 2 to mark off test intervals on the number line and list all of the factors found in Step 2. The boundary point(s) on the number will create test intervals. Step 4: Find the sign of every factor in every interval.  You can choose ANY value in an interval to plug into each factor. Whatever the sign of the factor is with that value gives you the sign you need for that factor in that interval. Make sure that you find the sign of every factor in every interval.  Since the inequality will be set to 0, we are not interested in the actual value that we get when we plug in our test points, but what SIGN (positive or negative) that we get. Step 5: Using the signs found in Step 4, determine the sign of the overall quadratic function in each interval.  Since the inequality will be set to 0, we are not interested in the actual value that we get when we plug in our test points, but what SIGN (positive or negative) that we get.  When you look at the signs of your factors in each interval, keep in mind that they represent a product of the factors that make up your overall quadratic function.  You can determine the sign of the overall quadratic function by using basic multiplication sign rules: o The product of two factors that have the same sign is positive.

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o The product of two factors that have the opposite signs is negative. If the quadratic expression is less than or less than or equal to 0, then we are interested in values that cause the quadratic expression to be negative. If the quadratic expression is greater than or greater than or equal to 0, then we are interested in values that cause our quadratic expression to be positive

“It always seems impossible until its done.” -Nelson Mandela

Mathematics Quadratic Inequalities [email protected]

Step 6: Write the solution set and graph.

“It always seems impossible until its done.” -Nelson Mandela