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
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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
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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
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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
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
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
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