# Quadratic Equations

## Solve Applications Modeled by Quadratic Equations


### Solve Applications of the Quadratic Formula

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. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even integers and consecutive 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 . The next one would be  or . 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 consecutive odd integers or consecutive even 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 geometry 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.
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 upwards is modeled by a quadratic equation. The initial velocity, , propels the object up until gravity causes the object to fall back down.
We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.

### Key Concepts

1. Area of a Triangle For a triangle with base, , and height, , the area, , is given by the formula: 


2. Pythagorean Theorem In any right triangle, where  and  are the lengths of the legs, and  is the length of the hypothenuse, 


3. Projectile motion The height in feet, , of an object shot upwards into the air with initial velocity, , after  seconds can be modeled by the formula:


### Practice Makes Perfect

Solve Applications of the Quadratic Formula
In the following exercises, solve by using methods of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?