# Factoring

## Quadratic Equations

We have already solved linear equations, equations of the form . In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. Listed below are some examples of quadratic equations:
The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get .
The general form of a quadratic equation is .
To solve quadratic equations we need methods different than the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.

### Solve Quadratic Equations Using the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.
We will now use the Zero Product Property, to solve a quadratic equation.
We usually will do a little more work than we did in this last example to solve the linear equations that result from using the Zero Product Property.
Notice when we checked the solutions that each of them made just one factor equal to zero. But the product was zero for both solutions.
It may appear that there is only one factor in the next example. Remember, however, that  means .

### Solve Quadratic Equations by Factoring

Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we must be sure to start with the quadratic equation in standard form, . Then we can factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?
The left side in the next example is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.
The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree more than two by using the Zero Product Property, just like we solved quadratic equations.
When we factor the quadratic equation in the next example we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.

### Solve Applications Modeled by Quadratic Equations

The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.
We will start with a number problem to get practice translating words into a quadratic equation.
Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give 132.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In an earlier chapter, we used the Pythagorean Theorem . It gave the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.

### Key Concepts

1. Zero Product Property If , then either  or  or both. See .
2. Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See .



3. Use a problem solving strategy to solve word problems See .




### Section Exercises


### Practice Makes Perfect

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Chapter 7 Review Exercises


### 7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
Factor by Grouping
In the following exercises, factor by grouping.

### 7.2 Factor Trinomials of the form 

Factor Trinomials of the Form 
In the following exercises, factor each trinomial of the form .
Factor Trinomials of the Form 
In the following examples, factor each trinomial of the form .

### 7.3 Factoring Trinomials of the form 

Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Factor Trinomials of the Form 
In the following exercises, factor completely.
Factor Trinomials Using the “ac” Method
In the following exercises, factor.
Factor Trinomials with a GCF Using the “ac” Method
In the following exercises, factor.

### 7.4 Factoring Special Products

Factor Perfect Square Trinomials
In the following exercises, factor.
Factor Differences of Squares
In the following exercises, factor.
Factor Sums and Differences of Cubes
In the following exercises, factor.

### 7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.

### 7.6 Quadratic Equations

Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.

### Practice Test

In the following exercises, find the Greatest Common Factor in each expression.
In the following exercises, factor completely.
In the following exercises, solve.