Download 9-3 Solving Quadratic Equations

9-3
Solving Quadratic
Equations
Vocabulary
Review
1. Multiple Choice Toby wants to buy concert tickets. Each ticket costs $45.00.
Circle the function rule Toby could use to find the cost c of any number of tickets t.
c 5 45t t 5 45c
45 5 ct c 5 45
2. What is the difference between a relation and a function?
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Vocabulary Builder
root (noun) root
Main Idea: A root is a solution of an equation in the form f (x) 5 0. The related
function y 5 f (x) has a zero, and the graph of y 5 f (x) has an x-intercept, for each
root of the equation.
Example: The equation x 2 2 1 5 0 has two roots. So, the function y 5 x 2 2 1
has two zeros: 21 and 1.
Use Your Vocabulary
Complete each statement with the correct word from the list below. Each word is
used exactly once.
root
roots
zero
zeros
3. The function y 5 4x 2 2 1 has two zeros. The equation 0 5 4x 2 2 1 has two 9.
4. Each 9 of an equation in the form f (x) 5 0 is a solution of the equation.
5. The function y 5 2x 2 5 has one 9.
6. One of the two 9 of the function y 5 x 2 2 4 is 2.
Chapter 9
266 Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
Related Words: x-intercept, zero, solution
Key Concept Standard Form of a Quadratic Equation
A quadratic equation is an equation that can be written in the form ax 2 1 bx 1 c 5 0,
where a 2 0. This form is called the standard form of a quadratic equation.
Examples
9x 2 1 81 5 0
x 2 1 4x 1 16 5 0
7. Cross out the quadratic equations that are NOT in standard form.
x 2 2 81 5 0
12x 2 5 6x 1 20
8x 2 2 10 5 25x 8. Draw a line from each quadratic equation in Column A to an equivalent equation in
Column B.
Column A
Column B
x 2 1 3x 1 18 5 5x 1 11
2x 2 1 x 1 1 5 0
3x 2 1 x 5 x 2 2 1
9x 2 1 30x 1 25 5 0
(3x 1 5)2 5 0
x 2 2 2x 1 7 5 0
You can solve quadratic equations by a variety of methods. One way to solve a quadratic
equation ax 2 1 bx 1 c 5 0 is to graph the related quadratic function y 5 ax 2 1 bx 1 c.
The solutions of the equation are the x-intercepts of the related function.
Problem 1 Solving by Graphing
Got It? What are the solutions of the equation
y
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
x 2 2 16 5 0? Use a graph of the related function.
16
9. Graph the related function, y 5 x 2 2 16.
8
x
–8
–4
O
4
8
–8
–16
10. Which part of the graph gives you the solutions of the equation? Circle your answer.
vertex
x-intercept(s)
y-intercept(s)
axis of symmetry HSM11_A1MC_0903_T91319
11. Complete each sentence.
The equation has solutions.
The solutions are and , or 4 .
267
Lesson 9-3
Problem 2 Solving Using Square Roots
Got It? What are the solutions of m2 2 36 5 0?
12. Follow the steps to find the solutions of m2 2 36 5 0.
1
2
Write the original equation.
Isolate m 2 on one side of the equation.
m2 5
Find the square roots of each side and simplify.
3
m56
"
56
Got It? What are the solutions of 3x 2 1 15 5 0?
13. Complete the steps to find the solutions of 3x 2 1 15 5 0.
3x 2 1 15 5 0
Write the original equation.
3x 2HSM11_A1MC_0903_T91320
5 Isolate the x2 term on one side.
x 2 5 x54
Å Divide each side by 3.
Find the square roots of each side.
14. Does 3x 2 1 15 5 0 have a real-number solution? Explain.
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Problem 3 Choosing a Reasonable Solution
Got It? An aquarium is designing a new exhibit to
showcase tropical fish. The exhibit will include a tank
that is a rectangular prism with a length < that is twice
the width w. The height of the tank is 4 ft. The volume
of the tank is 500 ft 3 . What is the width of the tank to
the nearest tenth of a foot?
Chapter 9
268 ℓ ft
4 ft
w
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
15. Complete the reasoning model below.
Think
Write
I need to use the equation for the volume
V=
of a rectangular prism.
I will substitute 500 for V, 2w for 𝓵𝓵, and
500 = (2w) ?
4 for h.
500 =
I need to simplify.
Finally, I need to isolate w and find
±
the square roots of each side. I will
use a calculator.
√
8
±
16. The width of the tank is about ?
? w2
=w
≈w
ft.
Lesson Check HSM11_A1MC_0903_T91321
• Do you UNDERSTAND?
Compare and Contrast When is it easier to solve a quadratic equation of the form
ax2 1 c 5 0 using square roots than to solve it using a graph?
17. Solve each equation below using square roots.
Copyright © by Pearson Education, Inc. or its affiliates. All Rights Reserved.
5x 2 2 50 5 0
5x 2 2 125 5 0
5x 2 5 5x 2 5 x 2 5 x 2 5 x 5 4 x 5 4 18. The equation from Exercise 17 that is easy to solve using a graph is .
19. When is it easier to solve an equation of the form ax 2 1 c 5 0 using square roots?
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Math Success
Check off the vocabulary words that you understand.
standard form of a quadratic equation
root of an equation
zero of a function
Rate how well you can solve a quadratic equation.
Need to
review
0
2
4
6
8
Now I
get it!
10
269
Lesson 9-3