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790 Chapter 13 Solving Quadratic Equations and Inequalities

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Problem 1 Get Your Free T-Shirts!

The Perris Pandas baseball team has a new promotional activity to encourage fans to attend games: launching free T-shirts! They can launch a T-shirt in the air with an initial velocity of 91 feet per second from 5 1 __

2 feet off the ground (the height of the team mascot)�

A T-shirt’s height can be modeled with the quadratic function h(t) 5 216t2 1 91t 1 5�5, where t is the time in seconds and h(t) is the height of the launched T-shirt in feet� They want to know how long it will take for a T-shirt to land back on the ground after being launched�

1. What methods can you use to determine how long it will take for a T-shirt to reach the ground?

When you encounter a quadratic equation or function that is difficult to factor, you can always use the Quadratic Formula to determine the roots or zeros�

For a quadratic equation of the form ax2 1 bx 1 c 5 0, the solutions can be calculated using the

QuadraticFormulax 5 2b 6 √_________

b2 2 4ac _______________ 2a

�

Can you create a sketch

of the function to get an idea of the path of

the T-shirt?

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13.1 The Quadratic Formula 791

Consider the function f(x) 5 24x2 2 40x 2 99�

First, determine the values of a, b, and c�

a 5 24, b 5 240, c 5 299

Next, substitute the values into the Quadratic Formula�

x 5 2(240) 6 √

__________________ (240)2 24(24)(299) ____________________________

2(24)

Then, simplify to determine the zeros of the function�

x 5 40 6 √____________

1600 2 1584 ___________________ 28

x 5 40 6 √___

16 _________ 28

x 5 40 6 4 _______ 28

x 5 40 1 4 _______ 28

or x 5 40 2 4 _______ 28

x 5 44 ___ 28

or x 5 36 ___ 28

x 5 25�5 or x 5 24�5

The zeros of the function f(x) 5 24x2 2 40x 2 99 are 25�5 and 24�5�

Let’s go back to the Get Your Free T-Shirts! problem situation�

2. Use the Quadratic Formula to determine how long it will take for a T-shirt to land back on the ground after being launched�

a. Identify the values of a, b, and c for the function h(t) 5 216t2 1 91t 1 5�5�

b. Determine the zero(s) for the given function�

Before using the Quadratic Formula, be sure to write the quadratic function in standard form: ax2 1 bx 1 c�

Be sure to use parentheses

when substituting so you keep track of the correct

positive and negative signs.

Do you think your

solutions will need to be exact or approximate?

What is the context of the problem?

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796 Chapter 13 Solving Quadratic Equations and Inequalities

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4. Use the Quadratic Formula to determine the zeros for each function� Round your solutions to the nearest hundredth�

a. f(x) 5 x2 2 7x 1 11 b. h(x) 5 3x2 211x 2 2

Sketch a graph to

support your reasoning.

5. Let’s think about the zeros of quadratic functions�

a. How many zeros did each quadratic function in this lesson have?

b. Do all quadratic functions have two zeros? Explain why or why not�

c. Could a quadratic function have no zeros? Explain why or why not�

d. Could a quadratic function can have more than two zeros? Explain why or why not�

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