Name:_________________________________________  Class:_________________________________________  Hour:_________________________________________

Algebra  2B  What  You  Need  to  Know  5.4  –  5.6,  5.9  Test  A2.5.4.1  Be  able  to  solve  quadratic  equations  using  the  Square  Root  Property.  Solve  each  equation  by  using  the  square  root  property  

1.        2.      3.     A2.5.4.2  Be  able  to  use  the  “completing  the  square”  method  to  turn  quadratic  expressions  into  perfect  square  trinomials.    

 4.       5.        6.                    A2.5.4.3  Be  able  to  solve  quadratic  equations  by  “completing  the  square.”  For  quadratic  expressions  of  the  form  ax2  +  bx  +  c  where  a  =  1  

 7.       8.       9.                            

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Ready to Go On? 365

SECTION 5A

Quiz for Lessons 5-1 Through 5-6

5-1 Using Transformations to Graph Quadratic FunctionsUsing the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.

1. g (x) = (x + 2) 2 - 4 2. g (x) = -4 (x - 1) 2 3. g (x) = 1 _ 2

x 2 + 1

Use the description to write each quadratic function in vertex form.

4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.

5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.

5-2 Properties of Quadratic Functions in Standard FormFor each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

6. f (x) = x 2 - 4x + 3 7. g (x) = - x 2 + 2x - 1 8. h (x) = x 2 - 6x

9. A football kick is modeled by the function h (x) = -0.0075 x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and FactoringFind the roots of each equation by factoring.

10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4 x 2 + 8x = 0

5-4 Completing the SquareSolve each equation by completing the square.

13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8

Write each function in vertex form, and identify its vertex.

16. f (x) = x 2 + 24x + 138 17. g (x) = x 2 - 12x + 39 18. h (x) = 5 x 2 - 20x + 9

5-5 Complex Numbers and RootsSolve each equation.

19. 3 x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0

5-6 The Quadratic FormulaFind the zeros of each function by using the Quadratic Formula.

22. f (x) = (x + 6) 2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2 x 2 - 5x + 3

25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4 t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?

a207se_c05rg1_0365.indd 365a207se_c05rg1_0365.indd 365 9/29/05 1:45:49 PM9/29/05 1:45:49 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

a207se_c05l04_0341_0348.indd 345a207se_c05l04_0341_0348.indd 345 9/28/05 6:11:14 PM9/28/05 6:11:14 PM

Ready to Go On? 365

SECTION 5A

Quiz for Lessons 5-1 Through 5-6

5-1 Using Transformations to Graph Quadratic FunctionsUsing the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.

1. g (x) = (x + 2) 2 - 4 2. g (x) = -4 (x - 1) 2 3. g (x) = 1 _ 2

x 2 + 1

Use the description to write each quadratic function in vertex form.

4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.

5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.

5-2 Properties of Quadratic Functions in Standard FormFor each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

6. f (x) = x 2 - 4x + 3 7. g (x) = - x 2 + 2x - 1 8. h (x) = x 2 - 6x

9. A football kick is modeled by the function h (x) = -0.0075 x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and FactoringFind the roots of each equation by factoring.

10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4 x 2 + 8x = 0

5-4 Completing the SquareSolve each equation by completing the square.

13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8

Write each function in vertex form, and identify its vertex.

16. f (x) = x 2 + 24x + 138 17. g (x) = x 2 - 12x + 39 18. h (x) = 5 x 2 - 20x + 9

5-5 Complex Numbers and RootsSolve each equation.

19. 3 x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0

5-6 The Quadratic FormulaFind the zeros of each function by using the Quadratic Formula.

22. f (x) = (x + 6) 2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2 x 2 - 5x + 3

25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4 t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?

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Ready to Go On? 365

SECTION 5A

Quiz for Lessons 5-1 Through 5-6

5-1 Using Transformations to Graph Quadratic FunctionsUsing the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.

1. g (x) = (x + 2) 2 - 4 2. g (x) = -4 (x - 1) 2 3. g (x) = 1 _ 2

x 2 + 1

Use the description to write each quadratic function in vertex form.

4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.

5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.

5-2 Properties of Quadratic Functions in Standard FormFor each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

6. f (x) = x 2 - 4x + 3 7. g (x) = - x 2 + 2x - 1 8. h (x) = x 2 - 6x

9. A football kick is modeled by the function h (x) = -0.0075 x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and FactoringFind the roots of each equation by factoring.

10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4 x 2 + 8x = 0

5-4 Completing the SquareSolve each equation by completing the square.

13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8

Write each function in vertex form, and identify its vertex.

16. f (x) = x 2 + 24x + 138 17. g (x) = x 2 - 12x + 39 18. h (x) = 5 x 2 - 20x + 9

5-5 Complex Numbers and RootsSolve each equation.

19. 3 x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0

5-6 The Quadratic FormulaFind the zeros of each function by using the Quadratic Formula.

22. f (x) = (x + 6) 2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2 x 2 - 5x + 3

25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4 t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?

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Name:_________________________________________  Class:_________________________________________  Hour:_________________________________________

A2.5.4.4  Be  able  to  write  a  quadratic  equation  in  vertex  form  and  identify  its  vertex.      

10.     11.                    12.                        A2.5.5.1  Be  able  to  simplify  square  roots  of  negative  numbers.  

 

13.                  14.                          15.       16.                          A2.5.5.2  Be  able  to  solve  quadratic  equations  with  complex  roots.    

 18.         19.       20.                          

Ready to Go On? 365

SECTION 5A

Quiz for Lessons 5-1 Through 5-6

5-1 Using Transformations to Graph Quadratic FunctionsUsing the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.

1. g (x) = (x + 2) 2 - 4 2. g (x) = -4 (x - 1) 2 3. g (x) = 1 _ 2

x 2 + 1

Use the description to write each quadratic function in vertex form.

4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.

5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.

5-2 Properties of Quadratic Functions in Standard FormFor each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

6. f (x) = x 2 - 4x + 3 7. g (x) = - x 2 + 2x - 1 8. h (x) = x 2 - 6x

9. A football kick is modeled by the function h (x) = -0.0075 x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and FactoringFind the roots of each equation by factoring.

10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4 x 2 + 8x = 0

5-4 Completing the SquareSolve each equation by completing the square.

13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8

Write each function in vertex form, and identify its vertex.

16. f (x) = x 2 + 24x + 138 17. g (x) = x 2 - 12x + 39 18. h (x) = 5 x 2 - 20x + 9

5-5 Complex Numbers and RootsSolve each equation.

19. 3 x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0

5-6 The Quadratic FormulaFind the zeros of each function by using the Quadratic Formula.

22. f (x) = (x + 6) 2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2 x 2 - 5x + 3

25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4 t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?

a207se_c05rg1_0365.indd 365a207se_c05rg1_0365.indd 365 9/29/05 1:45:49 PM9/29/05 1:45:49 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

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Ready to Go On? 365

SECTION 5A

Quiz for Lessons 5-1 Through 5-6

5-1 Using Transformations to Graph Quadratic FunctionsUsing the graph of f (x) = x 2 as a guide, describe the transformations, and then graph each function.

1. g (x) = (x + 2) 2 - 4 2. g (x) = -4 (x - 1) 2 3. g (x) = 1 _ 2

x 2 + 1

Use the description to write each quadratic function in vertex form.

4. f (x) = x 2 is vertically stretched by a factor of 9 and translated 2 units left to create g.

5. f (x) = x 2 is reflected across the x-axis and translated 4 units up to create g.

5-2 Properties of Quadratic Functions in Standard FormFor each function, (a) determine whether the graph opens upward or downward, (b) find the axis of symmetry, (c) find the vertex, (d) find the y-intercept, and (e) graph the function.

6. f (x) = x 2 - 4x + 3 7. g (x) = - x 2 + 2x - 1 8. h (x) = x 2 - 6x

9. A football kick is modeled by the function h (x) = -0.0075 x 2 + 0.5x + 5, where h is the height of the ball in feet and x is the horizontal distance in feet that the ball travels. Find the maximum height of the ball to the nearest foot.

5-3 Solving Quadratic Equations by Graphing and FactoringFind the roots of each equation by factoring.

10. x 2 - 100 = 0 11. x 2 + 5x = 24 12. 4 x 2 + 8x = 0

5-4 Completing the SquareSolve each equation by completing the square.

13. x 2 - 6x = 40 14. x 2 + 18x = 15 15. x 2 + 14x = 8

Write each function in vertex form, and identify its vertex.

16. f (x) = x 2 + 24x + 138 17. g (x) = x 2 - 12x + 39 18. h (x) = 5 x 2 - 20x + 9

5-5 Complex Numbers and RootsSolve each equation.

19. 3 x 2 = -48 20. x 2 - 20x = -125 21. x 2 - 8x + 30 = 0

5-6 The Quadratic FormulaFind the zeros of each function by using the Quadratic Formula.

22. f (x) = (x + 6) 2 + 2 23. g (x) = x 2 + 7x + 15 24. h (x) = 2 x 2 - 5x + 3

25. A bicyclist is riding at a speed of 18 mi/h when she starts down a long hill. The distance d she travels in feet can be modeled by d (t) = 4 t 2 + 18t, where t is the time in seconds. How long will it take her to reach the bottom of a 400-foot-long hill?

a207se_c05rg1_0365.indd 365a207se_c05rg1_0365.indd 365 9/29/05 1:45:49 PM9/29/05 1:45:49 PM

Skills Practice p. S12Application Practice p. S36

Extra Practice

5- 4 Completing the Square 345

ExercisesExercisesKEYWORD: MB7 Parent

KEYWORD: MB7 5-4

5-4

GUIDED PRACTICE 1. Vocabulary What must you add to the expression x 2 + bx to complete the square?

SEE EXAMPLE 1 p. 341

Solve each equation.

2. (x - 2) 2 = 16 3. x 2 - 10x + 25 = 16 4. x 2 - 2x + 1 = 3

SEE EXAMPLE 2 p. 342

Complete the square for each expression. Write the resulting expression as a binomial squared.

5. x 2 + 14x + 6. x 2 - 12x + 7. x 2 - 9x +

SEE EXAMPLE 3 p. 343

Solve each equation by completing the square.

8. x 2 - 6x = -4 9. x 2 + 8 = 6x 10. 2 x 2 - 20x = 8

11. x 2 = 24 - 4x 12. 10x + x 2 = 42 13. 2 x 2 + 8x - 15 = 0

SEE EXAMPLE 4 p. 344

Write each function in vertex form, and identify its vertex.

14. f (x) = x 2 + 6x - 3 15. g (x) = x 2 - 10x + 11 16. h (x) = 3 x 2 - 24x + 53

17. f (x) = x 2 + 8x - 10 18. g (x) = x 2 - 3x + 16 19. h (x) = 3 x 2 - 12x - 4

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

20–22 1 23–25 2 26–31 3 32–37 4

Independent Practice Solve each equation.

20. (x + 2) 2 = 36 21. x 2 - 6x + 9 = 100 22. (x - 3) 2 = 5

Complete the square for each expression. Write the resulting expression as a binomial squared.

23. x 2 - 18x + 24. x 2 + 10x + 25. x 2 - 1 _ 2

x +

Solve each equation by completing the square.

26. x 2 + 2x = 7 27. x 2 - 4x = -1 28. 2 x 2 - 8x = 22

29. 8x = x 2 + 12 30. x 2 + 3x - 5 = 0 31. 3 x 2 + 6x = 1

Write each function in vertex form, and identify its vertex.

32. f (x) = x 2 - 4x + 13 33. g (x) = x 2 + 14x + 71 34. h (x) = 9 x 2 + 18x - 3

35. f (x) = x 2 + 4x - 7 36. g (x) = x 2 - 16x + 2 37. h (x) = 2 x 2 + 6x + 25

38. Engineering The height h above the roadway of the main cable of the Golden Gate Bridge can be modeled by the function h (x) = 1 ____ 9000 x 2 - 7 __ 15 x + 500, where x is the distance in feet from the left tower.

x

h

a. Complete the square, and write the function in vertex form. b. What is the vertex, and what does it represent? c. Multi-Step The left and right towers have the same height. What is the

distance in feet between them?

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Name:_________________________________________  Class:_________________________________________  Hour:_________________________________________

A2.5.5.3  Be  able  to  find  complex  zeros  of  quadratic  functions.    

21.              22.              23.          A2.5.6.1  Be  able  to  classify  roots  using  the  discriminant.    

 24.       25.                  26.                  A2.5.6.2  Be  able  to  solve  quadratic  equations  using  the  Quadratic  Formula.  

 27.         28.                            

Name:_________________________________________  Class:_________________________________________  Hour:_________________________________________

A2.5.9.1  Be  able  to  add  and  subtract  complex  numbers    

29.     30.          31.              A2.5.9.2  Be  able  to  multiply  complex  numbers  

 32.       33.         34.                  A2.5.9.3  Be  able  to  evaluate  powers  of  i  Simplify  

35.             36.         37.                A2.5.9.4  Be  able  to  divide  complex  numbers  Simplify  38.             39.           40.  

                   

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

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386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

5- 9 Operations with Complex Numbers 387

Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit self-similarity. As a result, scientists can use fractals to model these objects.

Fractals

Multiply. Write the result in the form a + bi.

55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i)

58. (5 + 6i) 2 59. (7 - 5i) (-3 + 9i) 60. -4 (8 + 12i)

Simplify.

61. i27 62. -i11 63. 5 i10

64. 2 - 3i _ i 65. 5 - 2i _

3 + i 66. 3 _

-1 - 5i

67. 19 + 9i_5 + i

68. 8 + 4i_7 + i

69. 6 + 3i_2 - 2i

Write the complex number represented by each point on the graph.

70. A

71. B

72. C

73. D

74. E

Find the absolute value of each complex number.

75. 3 - i 76. 7i 77. -2 - 6i

78. -1 - 8i 79. 0 80. 5 + 4i

81. 3 _ 2

- 1 _ 2

i 82. 5 - i √ " 3 83. 2 √ " 2 - i √ " 3

84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = ( Z n ) 2 + c, where c is a constant.

a. Find Z2 using Z2 = (Z1)2 + 0.25. Let Z1 = 0.5 + 0.6i.

b. Find Z 3 using Z 3 = ( Z 2 ) 2 + 0.25. Use Z 2 that you obtained in part a.

c. Find Z 4 using Z4 = (Z3)2 + 0.25. Use Z3 that you obtained in part b.

Simplify. Write the result in the form a + bi.

85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i)

88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 90. (6 - 5i)2

91. (-2 - 3i) 2 92. (5 + 7i) (5 - 7i) 93. (2 - i) (2 + i) (2 - i)

94. 3 - i11 95. i52 - i48 96. i35 - i24 + i18

97. 12 + i _ i 98. 18 - 3i _

i 99. 4 + 2i _

6 + i

100. 1 + i_-2 + 4i

101. 4_2 - 3i

102. 6_ √ " 2 - i

a207se_c05l09_0382_0389.indd 387a207se_c05l09_0382_0389.indd 387 5/5/06 3:02:46 PM5/5/06 3:02:46 PM

Ready to Go On? 391

SECTION 5B

Quiz for Lessons 5-7 Through 5-9

5-7 Solving Quadratic InequalitiesGraph each inequality.

1. y > - x 2 + 6x 2. y ≤ - x 2 - x + 2

Solve each inequality by using tables or graphs.

3. x 2 - 4x + 1 > 6 4. 2x 2 + 2x -10 ≤ 2

Solve each inequality by using algebra.

5. x 2 + 4x - 7 ≥ 5 6. x 2 - 8x < 0

7. The function p (r) = -1000 r 2 + 6400r - 4400 models the monthly profit p of a small DVD-rental store, where r is the rental price of a DVD. For what range of rental prices does the store earn a monthly profit of at least $5000?

5-8 Curve Fitting with Quadratic ModelsDetermine whether each data set could represent a quadratic function. Explain.

8. x 5 6 7 8 9

y 13 11 7 1 -7

9. x -4 -2 0 2 4

y 10 8 4 8 10

Write a quadratic function that fits each set of points.

10. (0, 4) , (2, 0) , and (3, 1) 11. (1, 3) , (2, 5) , and (4, 3)

For Exercises 12–14, use the table of maximum load allowances for various heights of spruce columns.

12. Find a quadratic regression equation to model the maximum load given the height.

13. Use your model to predict the maximum load allowed for a 6.5 ft spruce column.

14. Use your model to predict the maximum load allowed for an 8 ft spruce column.

5-9 Operations with Complex NumbersFind each absolute value.

15. -6i 16. 3 + 4i 17. 2 - i

Perform each indicated operation, and write the result in the form a + bi.

18. (3 - 5i) - (6 - i) 19. (-6 + 4i) + (7 - 2i)

20. 3i (4 + i) 21. (3 + i) (5 - i)

22. (1 - 4i) (1 + 4i) 23. 3 i 15

24. 2 - 7i _ -i

25. 3 - i _ 4 - 2i

Maximum Load Allowance No. 1 Common Spruce

Height of Column (ft)

Maximum Load (lb)

4 7280

5 7100

6 6650

7 5960

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5- 9 Operations with Complex Numbers 387

Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit self-similarity. As a result, scientists can use fractals to model these objects.

Fractals

Multiply. Write the result in the form a + bi.

55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i)

58. (5 + 6i) 2 59. (7 - 5i) (-3 + 9i) 60. -4 (8 + 12i)

Simplify.

61. i27 62. -i11 63. 5 i10

64. 2 - 3i _ i 65. 5 - 2i _

3 + i 66. 3 _

-1 - 5i

67. 19 + 9i_5 + i

68. 8 + 4i_7 + i

69. 6 + 3i_2 - 2i

Write the complex number represented by each point on the graph.

70. A

71. B

72. C

73. D

74. E

Find the absolute value of each complex number.

75. 3 - i 76. 7i 77. -2 - 6i

78. -1 - 8i 79. 0 80. 5 + 4i

81. 3 _ 2

- 1 _ 2

i 82. 5 - i √ " 3 83. 2 √ " 2 - i √ " 3

84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = ( Z n ) 2 + c, where c is a constant.

a. Find Z2 using Z2 = (Z1)2 + 0.25. Let Z1 = 0.5 + 0.6i.

b. Find Z 3 using Z 3 = ( Z 2 ) 2 + 0.25. Use Z 2 that you obtained in part a.

c. Find Z 4 using Z4 = (Z3)2 + 0.25. Use Z3 that you obtained in part b.

Simplify. Write the result in the form a + bi.

85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i)

88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 90. (6 - 5i)2

91. (-2 - 3i) 2 92. (5 + 7i) (5 - 7i) 93. (2 - i) (2 + i) (2 - i)

94. 3 - i11 95. i52 - i48 96. i35 - i24 + i18

97. 12 + i _ i 98. 18 - 3i _

i 99. 4 + 2i _

6 + i

100. 1 + i_-2 + 4i

101. 4_2 - 3i

102. 6_ √ " 2 - i

a207se_c05l09_0382_0389.indd 387a207se_c05l09_0382_0389.indd 387 5/5/06 3:02:46 PM5/5/06 3:02:46 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

a207se_c05l09_0382_0389.indd 386a207se_c05l09_0382_0389.indd 386 9/28/05 6:24:59 PM9/28/05 6:24:59 PM

5- 9 Operations with Complex Numbers 387

Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit self-similarity. As a result, scientists can use fractals to model these objects.

Fractals

Multiply. Write the result in the form a + bi.

55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i)

58. (5 + 6i) 2 59. (7 - 5i) (-3 + 9i) 60. -4 (8 + 12i)

Simplify.

61. i27 62. -i11 63. 5 i10

64. 2 - 3i _ i 65. 5 - 2i _

3 + i 66. 3 _

-1 - 5i

67. 19 + 9i_5 + i

68. 8 + 4i_7 + i

69. 6 + 3i_2 - 2i

Write the complex number represented by each point on the graph.

70. A

71. B

72. C

73. D

74. E

Find the absolute value of each complex number.

75. 3 - i 76. 7i 77. -2 - 6i

78. -1 - 8i 79. 0 80. 5 + 4i

81. 3 _ 2

- 1 _ 2

i 82. 5 - i √ " 3 83. 2 √ " 2 - i √ " 3

84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = ( Z n ) 2 + c, where c is a constant.

a. Find Z2 using Z2 = (Z1)2 + 0.25. Let Z1 = 0.5 + 0.6i.

b. Find Z 3 using Z 3 = ( Z 2 ) 2 + 0.25. Use Z 2 that you obtained in part a.

c. Find Z 4 using Z4 = (Z3)2 + 0.25. Use Z3 that you obtained in part b.

Simplify. Write the result in the form a + bi.

85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i)

88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 90. (6 - 5i)2

91. (-2 - 3i) 2 92. (5 + 7i) (5 - 7i) 93. (2 - i) (2 + i) (2 - i)

94. 3 - i11 95. i52 - i48 96. i35 - i24 + i18

97. 12 + i _ i 98. 18 - 3i _

i 99. 4 + 2i _

6 + i

100. 1 + i_-2 + 4i

101. 4_2 - 3i

102. 6_ √ " 2 - i

a207se_c05l09_0382_0389.indd 387a207se_c05l09_0382_0389.indd 387 5/5/06 3:02:46 PM5/5/06 3:02:46 PM

386 Chapter 5 Quadratic Functions

ExercisesExercises

GUIDED PRACTICE 1. Vocabulary In the complex number plane, the horizontal axis represents

−−− ? numbers, and the vertical axis represents −−− ? numbers. (real, irrational, or imaginary)

SEE EXAMPLE 1 p. 382

Graph each complex number.

2. 4 3. -i 4. 3 + 2i 5. -2 - 3i

SEE EXAMPLE 2 p. 383

Find each absolute value.

6. 4 - 5i 7. -33.3 8. -9i

9. 5 + 12i 10. -1 + i 11. 15i

SEE EXAMPLE 3 p. 383

Add or subtract. Write the result in the form a + bi.

12. (2 + 5i) + (-2 + 5i) 13. (-1 - 8i) + (4 + 3i) 14. (1 - 3i) - (7 + i)

15. (4 - 8i) + (-13 + 23i) 16. (6 + 17i) - (18 - 9i) 17. (-30 + i) - (-2 + 20i)

SEE EXAMPLE 4 p. 384

Find each sum by graphing on the complex plane.

18. (3 + 4i) + (-2 - 4i) 19. (-2 - 5i) + (-1 + 4i) 20. (-4 - 4i) + (4 + 2i)

SEE EXAMPLE 5 p. 384

Multiply. Write the result in the form a + bi.

21. (1 - 2i) (1 + 2i) 22. 3i (5 + 2i) 23. (9 + i) (4 - i)

24. (6 + 8i) (5 - 4i) 25. (3 + i) 2 26. (-4 - 5i) (2 + 10i )

SEE EXAMPLE 6 p. 385

Simplify.

27. - i 9 28. 2 i 15 29. i 30

SEE EXAMPLE 7 p. 385

30. 5 - 4i _ i 31. 11 - 5i _

2 - 4i 32. 8 + 2i _

5 + i

33. 17 _ 4 + i

34. 45 - 3i _ 7 - 8i

35. -3 - 12i _ 6i

PRACTICE AND PROBLEM SOLVING

For See Exercises Example

36–39 1 40–45 2 46–51 3 52–54 4 55–60 5 61–63 6 64–69 7

Independent Practice Graph each complex number.

36. -3 37. -2.5i 38. 1 + i 39. 4 - 3i

Find each absolute value.

40. 2 + 3i 41. -18 42. 4 _ 5

i 43. 6 - 8i 44. -0.5i 45. 10 - 4i

Add or subtract. Write the result in the form a + bi.

46. (8 - 9i) - (-2 - i) 47. 4i - (11 - 3i) 48. (4 - 2i) + (-9 - 5i)

49. (13 + 6i) + (15 + 35i) 50. (3 - i) - (-3 + i) 51. -16 + (12 + 9i)

Find each sum by graphing on the complex plane.

52. (4 + i) + (-3i) 53. (5 + 4i) + (-1 + 2i) 54. (-3 - 3i) + (4 - 3i)

5-9

KEYWORD: MB7 Parent

KEYWORD: MB7 5-9

Skills Practice p. S13Application Practice p. S36

Extra Practice

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5- 9 Operations with Complex Numbers 387

Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit self-similarity. As a result, scientists can use fractals to model these objects.

Fractals

Multiply. Write the result in the form a + bi.

55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i)

58. (5 + 6i) 2 59. (7 - 5i) (-3 + 9i) 60. -4 (8 + 12i)

Simplify.

61. i27 62. -i11 63. 5 i10

64. 2 - 3i _ i 65. 5 - 2i _

3 + i 66. 3 _

-1 - 5i

67. 19 + 9i_5 + i

68. 8 + 4i_7 + i

69. 6 + 3i_2 - 2i

Write the complex number represented by each point on the graph.

70. A

71. B

72. C

73. D

74. E

Find the absolute value of each complex number.

75. 3 - i 76. 7i 77. -2 - 6i

78. -1 - 8i 79. 0 80. 5 + 4i

81. 3 _ 2

- 1 _ 2

i 82. 5 - i √ " 3 83. 2 √ " 2 - i √ " 3

84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = ( Z n ) 2 + c, where c is a constant.

a. Find Z2 using Z2 = (Z1)2 + 0.25. Let Z1 = 0.5 + 0.6i.

b. Find Z 3 using Z 3 = ( Z 2 ) 2 + 0.25. Use Z 2 that you obtained in part a.

c. Find Z 4 using Z4 = (Z3)2 + 0.25. Use Z3 that you obtained in part b.

Simplify. Write the result in the form a + bi.

85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i)

88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 90. (6 - 5i)2

91. (-2 - 3i) 2 92. (5 + 7i) (5 - 7i) 93. (2 - i) (2 + i) (2 - i)

94. 3 - i11 95. i52 - i48 96. i35 - i24 + i18

97. 12 + i _ i 98. 18 - 3i _

i 99. 4 + 2i _

6 + i

100. 1 + i_-2 + 4i

101. 4_2 - 3i

102. 6_ √ " 2 - i

a207se_c05l09_0382_0389.indd 387a207se_c05l09_0382_0389.indd 387 5/5/06 3:02:46 PM5/5/06 3:02:46 PM

5- 9 Operations with Complex Numbers 387

Fractals are self-similar, which means that smaller parts of a fractal are similar to the fractal as a whole. Many objects in nature, such as the veins of leaves and snow crystals, also exhibit self-similarity. As a result, scientists can use fractals to model these objects.

Fractals

Multiply. Write the result in the form a + bi.

55. -12i (-1 + 4i) 56. (3 - 5i)(2 + 9i) 57. (7 + 2i)(7 - 2i)

58. (5 + 6i) 2 59. (7 - 5i) (-3 + 9i) 60. -4 (8 + 12i)

Simplify.

61. i27 62. -i11 63. 5 i10

64. 2 - 3i _ i 65. 5 - 2i _

3 + i 66. 3 _

-1 - 5i

67. 19 + 9i_5 + i

68. 8 + 4i_7 + i

69. 6 + 3i_2 - 2i

Write the complex number represented by each point on the graph.

70. A

71. B

72. C

73. D

74. E

Find the absolute value of each complex number.

75. 3 - i 76. 7i 77. -2 - 6i

78. -1 - 8i 79. 0 80. 5 + 4i

81. 3 _ 2

- 1 _ 2

i 82. 5 - i √ " 3 83. 2 √ " 2 - i √ " 3

84. Fractals Fractals are patterns produced using complex numbers and the repetition of a mathematical formula. Substitute the first number into the formula. Then take the result, put it back into the formula, and so on. Each complex number produced by the formula can be used to assign a color to a pixel on a computer screen. The result is an image such as the one at right. Many common fractals are based on the Julia Set, whose formula is Z n + 1 = ( Z n ) 2 + c, where c is a constant.

a. Find Z2 using Z2 = (Z1)2 + 0.25. Let Z1 = 0.5 + 0.6i.

b. Find Z 3 using Z 3 = ( Z 2 ) 2 + 0.25. Use Z 2 that you obtained in part a.

c. Find Z 4 using Z4 = (Z3)2 + 0.25. Use Z3 that you obtained in part b.

Simplify. Write the result in the form a + bi.

85. (3.5 + 5.2i) + (6 - 2.3i) 86. 6i - (4 + 5i) 87. (-2.3 + i) - (7.4 - 0.3i)

88. (-8 - 11i) + (-1 + i) 89. i(4 + i) 90. (6 - 5i)2

91. (-2 - 3i) 2 92. (5 + 7i) (5 - 7i) 93. (2 - i) (2 + i) (2 - i)

94. 3 - i11 95. i52 - i48 96. i35 - i24 + i18

97. 12 + i _ i 98. 18 - 3i _

i 99. 4 + 2i _

6 + i

100. 1 + i_-2 + 4i

101. 4_2 - 3i

102. 6_ √ " 2 - i

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