# Quadratic Equations and Functions

## Solve Applications of Quadratic Equations


### Solve Applications Modeled by Quadratic Equations

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications.
Let’s first summarize the methods we now have to solve quadratic equations.
As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even and odd integers, by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is n + 2. The next one would be n + 2 + 2 or n + 4. This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
Some applications of odd or even consecutive integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
We will use the formula for the area of a triangle to solve the next example.
Recall that when we solve geometric applications, it is helpful to draw the figure.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
We will use the formula for the area of a rectangle to solve the next example.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
The height of a projectile shot upward from the ground is modeled by a quadratic equation. The initial velocity, v0, propels the object up until gravity causes the object to fall back down.
We can use this formula to find how many seconds it will take for a firework to reach a specific height.
We have solved uniform motion problems using the formula D = rt in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula D = rt assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula 
Some uniform motion problems are also modeled by quadratic equations.
Work applications can also be modeled by quadratic equations. We will set them up using the same methods we used when we solved them with rational equations.We’ll use a similar scenario now.

### Key Concepts

1. Methods to Solve Quadratic Equations


2. How to use a Problem-Solving Strategy.


3. Area of a Triangle


4. Area of a Rectangle


5. Pythagorean Theorem


6. Projectile motion



### Practice Makes Pefect

Solve Applications Modeled by Quadratic Equations
In the following exercises, solve using any method.
In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.

### Writing Exercises


### Self Check

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