671

LESSON

24 LEARNING OBJECTIVES

➤ Today I am: comparing two methods for fi nding the x-intercepts.

➤ So that I can: determine when to use each method.

➤ I’ll know I have it when I can: identify the advantages and disadvantages of

each method.

Finding x-Intercepts Again?

Opening Discussion

The quadratic function, y 5 x2 2 6x 1 8, can be written as y 5 (x 2 2)(x 2 4) and as y 5 (x 2 3)2 2 1. Deshi and Ame wanted to fi nd the x-intercepts of this function. Their work is shown below.

1. Read over each method and then discuss which method you think is the easiest to use.

Deshi’s Method: Using Factored Form Ame’s Method: Using Vertex Form

y 5 (x 2 2)(x 2 4)

0 5 (x 2 2)(x 2 4)

(x 2 2) 5 0 or (x 2 4) 5 0

x 5 2 or x 5 4

y 5 (x 2 3)2 2 1

0 5 (x 2 3)2 2 1

1 5 (x 2 3)2

61 5 x 2 3

x 5 61 1 3

x 5 1 1 3 5 4 or x 5 21 1 3 5 2

y o

rti staff

672 Module 4 Quadratic Functions

Practice Exercises

For each quadratic function, use the form given to determine the x-intercepts. Your partner will use the other form. Then check that you are getting the same x-intercepts. Be sure to switch methods for each problem.

2. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts

y 5 (x 1 1)(x 2 3) y 5 (x 2 1)2 2 4

3. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts

y 5 2(x 2 4)(x 1 2) y 5 2(x 2 1)2 2 18

y o

O cxtlKX 3 O CX 124

4 41 0 X 3 0f 2TI X 3 4 CX l12 X 1I 11

1121 2 3 1 2 1

O 2CX 4 X 12 0 24 15 18118 118

X 4 2 18 221 022T9 cxTI3 X 1113 X

11 4 2

Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 673

4. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts

y x x= − +13

( 7)( 1) y x= − −13

( 3) 163

2

5. A. Using Factored Form to Find x-Intercepts B. Using Vertex Form to Find x-Intercepts

y 5 (2x 2 1)(2x 1 1) y 5 4x2 2 1

O f CX 7 Cx 11 0 15 35 1Z 3 Cx 35

7 I F6 cx1 4 X 3314 X

7 I

0 2 1 2 1 1 I I

2 1 0 2 1 0 I 1 42

54x L x I I

2 2

It _x

674 Module 4 Quadratic Functions

Discussion

6. Avery prefers the vertex form of quadratic functions. He found the x-intercepts for y 5 (x 2 5)2 2 12 in the following way.

A. Write in each step Avery took.

y 5 (x 2 5)2 2 12

O 5 (x 2 5)2 2 12

12 5 (x 2 5)2

± = −x12 5

= ±x 5 12

= + = −x or x5 2 3 5 2 3

B. What is different about the equation Avery worked with as compared to the ones you used in Exercises 1–5?

7. Kenja really likes the vertex form but she ran into a problem with y 2 10 5 (x 2 1)2. What problem did Kenja face? What are the x-intercepts? What does this mean?

8. Neema wondered how to write the equation of the graph at the right in factored form. What should Neema do to get an equation of this function? Is there a factored form?

Let y oAdd 12square root bothsidesAdd 5Simplify T2

b

Theyshouldbe zero O lo ex D2Flo xThere's no Xintercept

Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 675

9. Write an equation that has no x-intercepts. Exchange equations with your partner and prove there are no x-intercepts for your partner’s equation.

10. Marine thought she made a mistake when finding the x-intercepts of the equation y 5 2(x 2 3)(x 2 3) and got only one x-intercept of 3. Explain what Marine’s graph would look like and why this isn’t a mistake.

11. Write the advantages and disadvantages to each method.

Advantages Disadvantages

Usi

ng F

acto

red

Form

to

Fin

d x-

Inte

rcep

tsU

sing

Ver

tex

Form

to

Fin

d x-

Inte

rcep

ts

12. Are there any advantages to using the standard form to get the x-intercepts? Explain your thinking.

Or n

3

FastEasy to find Factorintercepted short 4 long

NosquarerootCI specialcases

Good if the Have to completethe

answer is irrational Square

Fl T2 longerremember toput

676 Module 4 Quadratic Functions

Practice Exercises—Finding x-Intercepts

For each equation below, determine the x-intercepts, if there are any. You may use any method.

13. y 5 (x 2 5)(3x 1 2) 14. y 5 (4x 1 1)(2x 2 1) 15. y 5 23(x 1 4)(2x 1 3)

16. y 5 2(x 2 3)2 2 50 17. y 5 23(x 1 4)2 1 12 18. y 1 10 5 14

(x 2 1)2

19. y 5 2x2 1 4x 1 4 20. y 5 x2 1 10x 1 9 21. y 2 11 5 x2 2 6x

O X 5 3 2

X 5 0X 5

3 12 0

39 2

23

0 3 x145 12 0 10 14 X 112 3CX14123 3

T T 10 14CxD2

4 Cx 1412 x 14 IT4o cx4 4 44412 X Ianto X lG 2 II

112To X12510,1 air

0 2 2 4 4completethesquarecannotfactor

0 2622 1 2foggy

4

0 2 X1172 22 22 2 X115FT cxNo real x intercepts

Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 677

Practice Exercises—Finding the Vertex

For each equation below, determine the vertex. You may use any method.

22. y 5 (x 2 5)(3x 1 2) 23. y 5 (4x 1 1)(2x 2 1) 24. y 5 23(x 1 4)(2x 1 3)

25. y 5 2(x 2 3)2 2 50 26. y 5 23(x 1 4)2 1 12 27. y 1 10 5 14

(x 2 1)2

28. y 5 2x2 1 4x 1 4 29. y 5 x2 1 10x 1 9 30. y 2 11 5 x2 2 6x

Lesson Summary

When a quadratic equation is not conducive to factoring, we can solve by completing

the square.

Completing the square can be used to find solutions that are irrational, something

very difficult to do by factoring.

O XI GX 12

O x 761.9

12 9

O x 1332 21

IT CX 3

IDT XI3

Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 679

NAME: PERIOD: DATE:

Homework Problem Set

1. Solve the equation for b: 2b2 2 9b 5 3b2 2 4b 2 14.

2. Solve for x. 12 5 x2 1 6x

3. Solve for x. 4x2 2 40x 1 93 5 0

680 Module 4 Quadratic Functions

Solve each equation by completing the square.

4. x2 2 2x 5 12

5. 12

r2 2 6r 5 2 (Hint: Consider multiplying every term by 2.)

6. 2p2 1 8p 5 26

7. Challenge 2y2 1 3y 2 5 5 4

Unit 10 Completing the Square and the Quadratic Formula Lesson 24 Finding x-Intercepts Again? 681

Solve each equation. Use any method.

8. p2 2 2p 5 8

9. 2q2 1 8q 5 4

10. 13

m2 1 2m 1 8 5 5

11. 24x2 5 24x 1 11

12. Challenge Rewrite the expression by completing the square: 12

b2 2 4b 1 13.

682 Module 4 Quadratic Functions

Determine the x-intercepts of each quadratic function, if there are any.

13. y 5 (2x 2 1)(x 1 2) 14. y 5 x(4x 1 1) 15. y 5 (x 2 7)(2x 2 5)

16. y 5 (x 2 4)2 2 1 17. y 5 2(x 1 3)2 2 2 18. y 1 16 5 (x 2 2)2

19. y 5 3x2 2 7x 20. y 5 x2 2 5x 2 24 21. y 5 25x2 2 1