5-9 Complex

TEKS FOCUS VOCABULARY TEKS (7)(A) Add, subtract, • of a complex – • Imaginary number – any number of and multiply complex The absolute value of a the form a + bi, where a and b are real numbers. is its distance from the origin on the numbers and b ≠ 0 complex number plane. TEKS (1)(F) Analyze • – The imaginary unit i mathematical relationships • Complex conjugates – number pairs of is the complex number whose to connect and communicate the form a + bi and a - bi is -1. mathematical ideas. • Complex number – Complex numbers • Pure imaginary number – If a = 0 and Additional TEKS (1)(D), are the real numbers and imaginary b ≠ 0, the number a + bi is a pure (4)(F), (7)(B) numbers. imaginary number. • Complex number plane – The complex number plane is identical to the • Analyze – closely examine objects, ideas, coordinate plane, except each ordered or relationships to learn more about pair (a, b) represents the complex their nature number a + bi.

ESSENTIAL UNDERSTANDING • The complex numbers are based on a number whose • You can define operations on the of complex numbers square is -1. so that when you restrict the operations to the subset of • Every quadratic equation has complex number solutions real numbers, you get the familiar operations on the real (that sometimes are real numbers). numbers.

Key Concept of a Negative

Algebra Example Note For any positive number a, -5 = i 5 ( -5)2 = (i 5)2 = i2( 5)2 -a = -1 # a 1 1 1 = -1# 5 = -51 (not 5) 1 = 1-1 # a = i a. 1 1 1

Key Concept Complex Numbers ( You can write a complex number in a ϩ bi Complex Numbers (a ؉ bi the form a + bi, where a and b are real Imaginary Numbers numbers. Real Imaginary Real (a ϩ bi, b  0) If b = 0, the number a + bi is a real part part Numbers number. (a ϩ 0i) Pure Imaginary If a = 0 and b  0, the number a + bi is a Numbers pure imaginary number. (0 ϩ bi, b  0)

202 Lesson 5-9 Complex Numbers Key Concept Complex Number Plane

In the complex number plane, the point (a, b) represents the complex number a + bi. To graph a complex number, imaginary axis locate the real part on the horizontal axis and the imaginary 3 real axis part on the vertical axis. 2 4 The absolute value of a complex number is its distance Ϫ1i ؊2i from the origin in the . Ϫ2i a + bi = a2 + b2 3 ؊ 2i 2 0 0 ͉3 Ϫ 2i͉ ϭ ͱ13

Problem 1 TEKS Process Standard (1)(E)

Simplifying a Number Using i How do you write 18 by using the imaginary unit i? Is 18 a real 1 number?1 -18 = -1 # 18 No. There is no real 1 1 1 18 Property of Square Roots number that when = - # 1 1 multiplied by itself gives = i 18 Definition of i 18. You must use the # - 1 imaginary unit i to write = i # 3 2 Simplify. -18. 1 1 = 3i 2 1 Problem 2 TEKS Process Standard (1)(D) Graphing in the Complex Number Plane What are the graph and absolute value of each number?

A 5 3i  8i imaginary axis -5 + 3i = (-5)2 + 32 6i 234 6 units up 0 0 = B 6i 1 ؊5 ؉ 3i 4i Where is a pure imaginary number in 6i = 0 + 6i 5 units left, the complex plane? 2i 02 62 3 units up The real part of a pure 0 0 = 0 +0 real axis 2 imaginary number is 0. = 36 Ϫ6 Ϫ4 Ϫ2 2 4 The number must be on 1 the imaginary axis. = 6 Ϫ2i

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Adding and Subtracting Complex Numbers What is each sum or difference? How is adding complex numbers A (4  3i)  (4  3i) similar to 4 ( 4) ( 3i) 3i Use the commutative and associative properties. adding algebraic + - + - + expressions? 0 + 0 = 0 4 - 3i and -4 + 3i are additive inverses. Adding the real parts and imaginary parts B (5  3i)  (2  4i) separately is like adding 5 - 3i + 2 - 4i To subtract, add the opposite. like terms. 5 + 2 - 3i - 4i Use the commutative and associative properties. 7 - 7i Simplify.

Problem 4 Multiplying Complex Numbers What is each product?

A (3i)(5  2i) -15i + 6i2 Distributive Property 2 How do you multiply -15i + 6(-1) Substitute -1 for i . two binomials? -6 - 15i Simplify. Multiply each term of one binomial by B (4  3i)(1  2i) C (6  i)(6  i) each term of the other 4 8i 3i 6i2 36 6i 6i i2 binomial. - - - - + - - -4 - 8i - 3i - 6(-1) Substitute 36 + 6i - 6i - (-1) 2 2 - 11i Ϫ1 for i . 37

Problem 5 Dividing Complex Numbers What is each quotient? What is the goal? 9  12i 2  3i Write the quotient in the A B 3i 1  4i form a + bi. Multiply numerator 9 + 12i -3i 2 + 3i 1 + 4i 3i # -3i and denominator 1 - 4i # 1 + 4i by the complex 2 2 -27i - 36i conjugate of the 2 + 8i + 3i + 12i 2 2 -9i denominator. 1 + 4i - 4i - 16i -27i - 36(-1) 2 + 8i + 3i + 12(-1)

-9(-1) Substitute 1 + 4i - 4i - 16(-1) 2 36 - 27i Ϫ1 for i . -10 + 11i 9 17 10 11 4 - 3i -17 + 17i

204 Lesson 5-9 Complex Numbers Problem 6 TEKS Process Standard (1)(F) Factoring Using Complex Conjugates What is the factored form of 2x2  32? Is the expression 2 factorable using real 2x + 32 numbers? 2(x2 + 16) Factor out the GCF. No. Look for factors using 2 2 2 complex numbers. 2(x + 4i )(x - 4i ) Use a + b = (a + bi )(a - bi ) to factor (x + 16).

Check

2(x2 + 4xi - 4xi - 16i2) Multiply the binomials. 2(x2 - 16(-1)) i 2 = -1 2(x2 + 16) Simplify within the binomial. 2x2 + 32 Multiply.

Problem 7

Finding Imaginary Solutions What are the solutions of 2x2  3x  5  0?

Use the Quadratic Formula with a 2, b 3, and b b2  4ac = = - x  t c = 5. 22a

(3) t (3)2  4(2)(5) Simplify.  22(2)

3 9  40  t 24 3 31  t 24 3 31  t i 4 24

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H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O

Simplify each number by using the imaginary number i.

1. -4 2. -7 3. -15 4. -50 For additional support when Plot each1 complex number1 and find its absolute1 value. 1 completing your homework, go to PearsonTEXAS.com. 5. 2i 6. 5 + 12i 7. 2 - 2i 8. 3 - 6i Simplify each expression.

9. (2 + 4i) + (4 - i) 10. (-3 - 5i) + (4 - 2i) 11. (7 + 9i) + (-5i) 12. (12 + 5i) - (2 - i) 13. (-6 - 7i) - (1 + 3i) 14. (8 + i)(2 + 7i) 15. (-6 - 5i)(1 + 3i) 16. (-6i)2 17. (9 + 4i)2

Write each quotient as a complex number. 18. 3 - 2i 19. -2i 20. 4 - 3i 5i 1 + i -1 - 4i 21. i + 2 22. 4 23. 3 + 2i i - 2 2 - 3i (1 + i)2 Find the factored forms of each expression. Check your answer.

24. x2 + 25 25. x2 + 1 26. 3s2 + 75 2 1 2 2 27. x + 4 28. 4b + 1 29. -9x - 100

Find all solutions to each quadratic equation.

30. x2 + 2x + 3 = 0 31. -3x2 + x - 3 = 0 32. 2x2 - 4x + 7 = 0 33. x2 - 2x + 2 = 0 34. x2 + 5 = 4x 35. 2x(x - 3) = -5

36. a. Name the complex number represented by each point F imaginary axis on the graph at the right. 4i b. Find the additive inverse of each number. D B 2i c. Find the complex conjugate of each number.

d. Find the absolute value of each number. A Ϫ3 3 real C axis 37. Connect Mathematical Ideas (1)(F) In the complex Ϫ2i number plane, what geometric figure describes the E Ϫ4i complex numbers with absolute value 10? 38. Solve (x + 3i)(x - 3i) = 34. Simplify each expression.

39. (8i)(4i)(-9i) 40. (2 + -1) + (-3 + -16) 41. (8 - -1) - (-3 + -16) 42. 2i(5 -13i) 1 43. -5(11+ 2i) + 3i(3 - 41i) 44. (3 + -4)(4 + -1) 1 1

206 Lesson 5-9 Complex Numbers 45. Analyze Mathematical Relationships (1)(F) In the equation x2 - 6x + c = 0, find values of c that will give: a. two real solutions b. two imaginary solutions c. one real solution

46. A student wrote the numbers 1, 5, 1 + 3i, and 4 + 3i to represent the vertices of a quadrilateral in the complex number plane. What type of quadrilateral has these vertices? 1 The of a complex number z is z where z  0. Find the multiplicative inverse, or reciprocal, of each complex number. Then use complex conjugates to simplify. Check each answer by multiplying it by the original number.

47. 2 + 5i 48. 8 - 12i 49. a + bi Find the sum and product of the solutions of each equation.

50. x2 - 2x + 3 = 0 51. 5x2 + 2x + 1 = 0 52. -2x2 + 3x - 3 = 0 2 b For ax  bx  c  0, the sum of the solutions is a and the product of the c solutions is a. Find a quadratic equation for each pair of solutions. Assume a  1. 53. -6i and 6i 54. 2 + 5i and 2 - 5i 55. 4 - 3i and 4 + 3i Two complex numbers a  bi and c  di are equal when a  c and b  d. Solve each equation for x and y.

56. 2x + 3yi = -14 + 9i 57. 3x + 19i = 16 - 8yi 58. -14 - 3i = 2x + yi

59. Show that the product of any complex number a + bi and its complex conjugate is a real number. 60. For what real values of x and y is (x + yi)2 an imaginary number? 61. Explain Mathematical Ideas (1)(G) True or false: The conjugate of the additive inverse of a complex number is equal to the additive inverse of the conjugate of that complex number. Explain your answer.

TEXAS Test Practice

62. How can you rewrite the expression (8 - 5i)2 in the form a + bi? A. 39 + 80i B. 39 - 80i C. 69 + 80i D. 69 - 80i 63. How many solutions does the quadratic equation 4x2 - 12x + 9 = 0 have? F. two real solutions H. two imaginary solutions G. one real solution J. one imaginary solution

64. What are the solutions of 3x2 - 2x - 4 = 0? 1 13 1 i 11 1 13 1 i 11 A. { B. { C. - { D. - { 31 31 3 1 3 1 65. Using factoring, what are all four solutions to x4 - 16 = 0? Show your work.

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