# Equations and Inequalities

## Quadratic Equations


### Learning Objectives

1. Recognize and use the appropriate method to factor a polynomial completely (IA 6.4.1)

### Objective 1: Recognize and use the appropriate method to factor a polynomial completely  (IA 6.4.1)


The following chart summarizes the factoring methods and outlines a strategy you should use when factoring polynomials.


### Practice Makes Perfect

Recognize and use the appropriate method to factor a polynomial completely. Factor each of the following polynomials completely, if a polynomial does not factor label it as prime.

The computer monitor on the left in  is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.

### Solving Quadratic Equations by Factoring

An equation containing a second-degree polynomial is called a quadratic equation. For example, equations such as 
  and 
  are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.
Often the easiest method of solving a quadratic equation is factoring. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.
If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if 
  then 
  or 
  where a and b are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.
Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression 
  by multiplying the two factors together.
The product is a quadratic expression. Set equal to zero, 
  is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.
The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form, 
  where a, b, and c are real numbers, and 
  The equation 
  is in standard form.
We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.

### Solving Quadratics with a Leading Coefficient of 1

In the quadratic equation 
  the leading coefficient, or the coefficient of 
  is 1. We have one method of factoring quadratic equations in this form.

### Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1

When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:
1. With the quadratic in standard form, 
  multiply 
  
2. Find two numbers whose product equals 
  and whose sum equals 
  
3. Rewrite the equation replacing the 
  term with two terms using the numbers found in step 1 as coefficients of x.
4. Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
5. Factor out the expression in parentheses.
6. Set the expressions equal to zero and solve for the variable.

### Using the Square Root Property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property, in which we isolate the 
  term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the 
  term so that the square root property can be used.

### Completing the Square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.
We will use the example 
  to illustrate each step.
1. Given a quadratic equation that cannot be factored, and with 

2. Multiply the 

3. 
Add 

4. The left side of the equation can now be factored as a perfect square.

5. Use the square root property and solve.

6. The solutions are 

### Using the Quadratic Formula

The fourth method of solving a quadratic equation is by using the quadratic formula, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.
We can derive the quadratic formula by completing the square. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by 
  and obtain a positive a. Given 
  
  we will complete the square as follows:
1. First, move the constant term to the right side of the equal sign:

2. As we want the leading coefficient to equal 1, divide through by 

3. Then, find 

4. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:

5. Now, use the square root property, which gives

6. Finally, add 


### The Discriminant

The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant, or the expression under the radical, 
  The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect.  relates the value of the discriminant to the solutions of a quadratic equation.

### Using the Pythagorean Theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem. It is based on a right triangle, and states the relationship among the lengths of the sides as 
  where 
  and 
  refer to the legs of a right triangle adjacent to the 
  angle, and 
  refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.
We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.
The Pythagorean Theorem is given as
where 
  and 
  refer to the legs of a right triangle adjacent to the 
  angle, and 
  refers to the hypotenuse, as shown in .

### Key Equations


### Key Concepts

1. Many quadratic equations can be solved by factoring when the equation has a leading coefficient of 1 or if the equation is a difference of squares. The zero-product property is then used to find solutions. See , , and .
2. Many quadratic equations with a leading coefficient other than 1 can be solved by factoring using the grouping method. See  and .
3. Another method for solving quadratics is the square root property. The variable is squared. We isolate the squared term and take the square root of both sides of the equation. The solution will yield a positive and negative solution. See  and .
4. Completing the square is a method of solving quadratic equations when the equation cannot be factored. See .
5. A highly dependable method for solving quadratic equations is the quadratic formula, based on the coefficients and the constant term in the equation. See  and .
6. The discriminant is used to indicate the nature of the roots that the quadratic equation will yield: real or complex, rational or irrational, and how many of each. See .
7. The Pythagorean Theorem, among the most famous theorems in history, is used to solve right-triangle problems and has applications in numerous fields. Solving for the length of one side of a right triangle requires solving a quadratic equation. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, solve the quadratic equation by factoring.
For the following exercises, solve the quadratic equation by using the square root property.
For the following exercises, solve the quadratic equation by completing the square. Show each step.
For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.
For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution.

### Technology

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the x-intercepts) by using 2. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.

### Extensions


### Real-World Applications
