# Algebra

## Quadratic Equations with Two Variables with Applications


### Learning Objectives

After completing this section, you should be able to: 
1. Multiply binomials.
2. Factor trinomials.
3. Solve quadratic equations by graphing.
4. Solve quadratic equations by factoring.
5. Solve quadratic equations using square root method.
6. Solve quadratic equations using the quadratic formula.
7. Solve real world applications modeled by quadratic equations.
In this section, we will discuss quadratic equations. There are several real-world scenarios that can be represented by the graph of a quadratic equation. Think of the Gateway Arch in St. Louis, Missouri. Both ends of the arch are 630 feet apart and the arch is 630 feet tall. You can plot these points on a coordinate system and create a parabola to graph the quadratic equation.

### Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant, or the product of a constant and one or more variables. When it is of the form , where  is a constant and  is a positive whole number, it is called a monomial. Some examples of monomial are 8, , , and .
A monomial or two or more monomials combined by addition or subtraction is a polynomial. Some examples include: , , and . Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term (examples: 14, , , and ). A binomial has exactly two terms (examples: , , , and ), and a trinomial has exactly three terms (examples: , , , and ).
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

### Multiply Binomials

Recall multiplying algebraic expressions from Algebraic Expressions. In this section, we will continue that work and multiply binomials as well. We can use an area model to do multiplication.

### Factoring Trinomials

We’ve just covered how to multiply binomials. Now you will need to “undo” this multiplication—to start with the product and end up with the factors. Let us review an example of multiplying binomials to refresh your memory.
To factor the trinomial means to start with the product, , and end with the factors, . You need to think about where each of the terms in the trinomial came from. The first term came from multiplying the first term in each binomial. So, to get  in the product, each binomial must start with an .
The last term in the trinomial came from multiplying the last term in each binomial. So, the last terms must multiply to 6. What two numbers multiply to 6? The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use? Consider the middle term. It came from adding the outer and inner terms. So the numbers that must have a product of 6 will need a sum of 5.
We’ll test both possibilities and summarize the results in the following table, which will be very helpful when you work with numbers that can be factored in many different ways.
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. We have the factors of . They are .
You can check if the factors are correct by multiplying. Looking back, we started with , which is of the form , where  and . We factored it into two binomials of the form  and .
To get the correct factors, we found two number  and  whose product is  and sum is . With the area model (), start with an empty box and then put in the  term and .
Continue by putting in two terms that add up to  and  ():
Then you find the terms of the binomials on the top and side ():
    
    
    

### Solving Quadratic Equations by Graphing

We have already solved and graphed linear equations in Graphing Linear Equations and Inequalities, equations of the form . In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. The following are some examples of quadratic equations:
The last equation does not 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 , where  and  are real numbers, with . Remember that a solution of an equation is a value of a variable that makes a true statement when substituted into the equation. The solutions of quadratic equations are the values of the variables that make the quadratic equation  true.
To solve quadratic equations, we need methods different than the ones we used in solving linear equations. We will start by solving a quadratic equation from its graph. Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations. Let us look first at graphing the quadratic equation . We will choose integer values of  between  and 2 and find their  values, as shown in the table below.
Notice when we let  and , we got the same value for .
The same thing happened when we let  and . Now, we will plot the points to show the graph of . See .
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this. When  the solution to the quadratic  is 0 because  at .

### Solving Quadratic Equations by Factoring

Another way of solving quadratic equations is by factoring. We will use the Zero Product Property that 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.

### Solving Quadratic Equations Using the Square Root Property

We just solved some quadratic equations by factoring. Let us use factoring to solve the quadratic equation .
The solution is read as “ is equal to positive or negative 3.”
What happens when we have an equation like ? Since 7 is not a perfect square, we cannot solve the equation by factoring. These equations are all of the form . We define the square root of a number in this way: If , then  is a square root of . This leads to the Square Root Property.

### Solving Quadratic Equations Using the Quadratic Formula

This last method we will look at for solving quadratic equations is the quadratic formula. This method works for all quadratic equations, even the quadratic equations we could not factor! To use the quadratic formula, we substitute the values of , , and  into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.

### Solving Real-World Applications Modeled by Quadratic Equations

There are problem solving strategies that will work well for applications that translate to quadratic equations. Here’s a problem-solving strategy to solve word problems:
Step 1: Read the problem. Make sure all the words and ideas are understood.
Step 2: Identify what we are looking for.
Step 3: Name what we are looking for. Choose a variable to represent that quantity.
Step 4: Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
Step 5: Solve the equation using good algebra techniques.
Step 6: Check the answer in the problem and make sure it makes sense.
Step 7: Answer the question with a complete sentence.
Were you surprised by the pair of negative integers that is one of the solutions? In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.

### Check Your Understanding


### Key Terms

1. monomial
2. polynomial
3. binomial
4. trinomial
5. quadratic equation
6. Zero Product Property

### Key Concepts

1. A quadratic equation is an algebraic equation where the highest power (degree) of the equation is two.
2. To solve a quadratic equation is to find the value(s) that when substituted in for the variables, will make the equation equal to zero.
3. There can be two, one, or no solutions to any quadratic equation.
4. There are several methods to solve a quadratic equation. These methods include factoring quadratic equations, graphic quadratic equations, using the square root method, and using the quadratic formula.

### Videos

1.  Factoring with the Box Method (Area Model)
2. Solving Quadratics with the Zero Property
3. Solving Quadratics with the Quadratic Formula