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Math 102: College Mathematics14 chapters | 108 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn what simple steps you can take to solve quadratic inequalities using two binomials. See how it only requires a few steps to solve quadratic inequalities using this method.

On the surface, solving quadratic inequalities using two binomials sounds like a dry and boring topic. And, it might be, but knowing how to do this is useful in the real world. Say you work at a circus and need to figure out when one ball is underneath another ball when you throw each ball into the air; you can use what you learn in this video lesson to find the answer.

To begin, let's review some important terms. A **quadratic inequality** is an inequality where, after moving all your terms to one side, you end up with a quadratic on the one side, like this:

- -
*x*^2 + 5*x*+6 < 0

What do we know about **quadratic expressions**? We know that, in general, a quadratic has three terms in it: a term with a squared variable, a term with a single variable, and a term with only a number. The squared variable term is the part that makes the expression a quadratic.

We also know that a quadratic expression can also have less than three terms. As long as it has the squared variable term, then it is a quadratic. Because we have two sides to our inequality, our original problem can have terms on both sides. As long as you end up with a quadratic after moving all your terms to one side, you are working with a quadratic inequality. The inequality below, for example, is also a quadratic inequality.

We will use this inequality to show the two easy parts to solving quadratic inequalities.

If our first quadratic expression tells us the path of the first ball, and the second quadratic expression tells us the path of our second ball, then solving the inequality will tell us when our first ball is underneath the second ball.

So, now let's see how to go about solving this inequality.

The first part of the solution is to move everything to the left side. We will use algebra skills to do this. We will subtract and add the terms on the right side to both sides of the equations as needed to clear the right side. We look at what we have on the right side. We have a -6 and a -*x*^2 term. To move the -*x*^2 term, we add it to both sides and combine like terms. To move the -6 term, we add it to both sides as well and combine like terms. Like terms are terms with the same variable and same exponent. After doing this, we end up with a zero on the right side. On the left we have -x^2 + 5*x* + 6.

We have moved all our terms to one side, and we now have one combined quadratic to solve for. After this first part, we are ready for part two, which is solving the combined quadratic for zero and getting our answer.

To finish solving our quadratic inequality, we need to solve the combined quadratic for zero by temporarily changing the inequality sign into an equal sign. We can use what we know about factoring quadratics to find our zeroes. Our quadratic factors into this:

- (-
*x*- 1) (*x*- 6) = 0

After factoring, to find our zeroes, we set each factor equal to zero, and solve for the variable. We set the first factor, -*x* - 1, equal to zero to find the first zero. Solving for *x*, we add 1 to both sides to get -*x* = 1. We then multiply by -1 on both sides to get *x* = -1 for our first zero. We set our second factor, *x* - 6, equal to zero to find the second zero. We then add 6 to both sides to get *x* = 6 as our next zero. Our zeroes are located where *x* is -1 and 6.

From the zeroes that we have just found, we will now label a number of ranges based on the number of zeroes. In our case, we have a range where *x* < -1, another range where *x* is > -1 and *x* < 6, and a third where *x* > 6.

Our correct answer is one of these ranges. To figure out which range is the correct answer, we can graph our combined quadratic or think about the properties of quadratics. What we want to do is to find the range or ranges where our combined quadratic fits the inequality. For our problem, we need to find the parts where the quadratic is less than zero.

Let's think about the properties of quadratics. We know that if the *x*^2 term is negative, the quadratic is a parabola that opens downward. If the parabola opens downward, then the parts below zero are the parts that are outside the two zeroes or the area outside the middle area. So, our answer is the ranges of *x* < -1 and *x* > 6. Our answer has two ranges, and that's okay. Our answer tells us that when *x* < -1 or *x* > 6, our first ball is under our second ball.

We have used less than and greater than symbols since our inequality problem uses a less than symbol instead of a less than or equal to symbol. If it had an inequality sign with an 'or equal' part, then our answer ranges would use the symbol with the 'or equal' part as well.

We can summarize the steps we took to solving quadratic inequalities using two binomials to these:

- Move everything to the left side of the inequality.
- Solve the combined quadratic for zero by factoring.
- Find the ranges based on the zeroes of the quadratic.
- Pick the range or ranges where the inequality is true.

Solving these types of problems is not difficult and is useful for finding out real-world solutions, such as our two balls circus scenario.

- Recognize the properties of quadratic expressions
- Know how to solve for the variable
- List the steps for solving quadratic inequalities using two binomials

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Math 102: College Mathematics14 chapters | 108 lessons

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- Multiplying Binomials Using FOIL and the Area Method 7:26
- Multiplying Binomials Using FOIL & the Area Method: Practice Problems 5:46
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- How to Solve Quadratics That Are Not in Standard Form 6:14
- Solving Quadratic Inequalities Using Two Binomials 5:36
- Go to Graphing and Factoring Quadratic Equations

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