Table of Contents |
Quadratic equations can be used to represent the path of objects as they go up in the air, and come back down due to the force of gravity.
EXAMPLE
Imagine a ball being thrown straight up into the air. For the first few seconds, the height of the ball will be increasing, but eventually, it will reach a maximum height and then begin to fall back down, eventually hitting the group. In relation to time, the graph might look something like this:We can use this equation to answer some questions about the height of the ball, and how long it is in the air.
EXAMPLE
Suppose we want to figure out when the ball is more than 70 feet in the air. We will use a quadratic inequality to represent this situation.Write as an equation | |
Set equal to zero by subtracting 70 from both sides | |
Our equation |
Factor out -16 | |
Factor the quadratic expression | |
Set each factor equal to zero | |
Simplify each factor | |
Our solutions |
Identify coefficients b, and c | |
Use quadratic formula | |
Plug in | |
Square 80 and multiply 4, -16, and -64 | |
Simplify discriminant | |
Evaluate square root | |
Evaluate denominator | |
Separate into two fractions, one with addition and one with subtraction | |
Evaluate numerators of both fractions | |
Simplify | |
, and | Our solutions |
Test Value | Result | Interval | |
---|---|---|---|
0 | This inequality is FALSE, so this interval is NOT part of the solution. | ||
2 | This inequality is TRUE, so this interval is part of the solution. | ||
5 | This inequality is FALSE, so this interval is NOT part of the solution. |
Let's take a look at an example where a quadratic inequality has no solutions.
EXAMPLE
We would like to know when the ball is above 120 feet. To answer this question, we follow the same steps as in our previous example, but with a different inequality to represent this new scenario.Write as an equation | |
Set equal to zero by subtracting 120 from both sides | |
Our equation |
Plug in | |
Square 80 and multiply 4, -16, and -114 | |
Evaluate discriminant | |
Non-real solution |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License