B Apply

45. 1. Name the complex number represented by each point on the graph below.
    2. Find the additive inverse of each number.
    3. Find the complex conjugate of each number.
    4. Find the absolute value of each number.

    
    Image Long Description

46. Think About a Plan In the complex number plane, what geometric figure describes the complex numbers with absolute value 10?
    • What does the absolute value of a complex number represent?
    • How can you use the complex number plane to solve this problem?
47. Solve ( x + 3 i ) ( x − 3 i ) = 34 . open x plus 3 i close open x minus 3 i close equals 34 .

Simplify each expression.

48. ( 8 i ) ( 4 i ) ( − 9 i ) open 8 i close open 4 i close open negative 9 i close
49. ( 2 + − 1 ) + ( − 3 + − 16 ) open . 2 plus , square root of negative 1 end root . close . plus . open . negative 3 plus , square root of negative 16 end root . close
50. ( 4 + − 9 ) + ( 6 − − 49 ) open . 4 plus , square root of negative 9 end root . close . plus . open . 6 minus , square root of negative 49 end root . close
51. ( 10 + − 9 ) − ( 2 + − 25 ) open . 10 plus , square root of negative 9 end root . close . minus . open . 2 plus , square root of negative 25 end root . close
52. ( 8 − − 1 ) − ( − 3 + − 16 ) open . 8 minus , square root of negative 1 end root . close . minus . open . negative 3 plus , square root of negative 16 end root . close
53. 2 i ( 5 − 3 i ) 2 i open 5 minus 3 i close
54. − 5 ( 1 + 2 i ) + 3 i ( 3 − 4 i ) negative 5 open 1 plus 2 i close plus 3 i open 3 minus 4 i close
55. ( 3 + − 4 ) ( 4 + − 1 ) open . 3 plus , square root of negative 4 end root . close . open . 4 plus , square root of negative 1 end root . close
56. Open-Ended In the equation x 2 − 6 x + c = 0 , x squared , minus 6 x plus c equals 0 comma  find values of c that will give:
    1. two real solutions
    2. two imaginary solutions
    3. one real solution
57. A student wrote the numbers 1, 5, 1 + 3 i, and 4 + 3 i 4 plus 3 i  to represent the vertices of a quadrilateral in the complex number plane. What type of quadrilateral has these vertices?

The multiplicative inverse of a complex number z is 1 z 1 over z  where z ≠ 0 . bold italic z not equal to 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.

58. 2 + 5 i
59. 8 − 12 i 8 minus 12 i
60. a + bi

Find the sum and product of the roots of each equation.

61. x 2 − 2 x + 3 = 0 x squared , minus 2 x plus 3 equals 0
62. 5 x 2 + 2 x + 1 = 0 5 x squared , plus 2 x plus 1 equals 0
63. − 2 x 2 + 3 x − 3 = 0 negative 2 , x squared , plus 3 x minus 3 equals 0

For a x 2 + b x + c = 0 , bold italic eh , bold italic x squared , plus bold italic b bold italic x plus bold italic c equals 0 comma  the sum of the roots is − b a negative , b over eh  and the product of the roots is c a . c over eh , .

Find a quadratic equation for each pair of roots. Assume a = 1.

64. − 6 negative 6 i and 6 i
65. 2 + 5 i 2 plus 5 i  and 2 − 5 i 2 minus 5 i
66. 4 − 3 i 4 minus 3 i  and 4 + 3 i 4 plus 3 i

Two complex numbers a + bi and c + di are equal when a = c and b = d. Solve each equation for x and y.

67. 2 x + 3 y i = − 14 + 9 i 2 x plus 3 y i equals negative 14 plus 9 i
68. 3 x + 19 i = 16 − 8 y i 3 x plus 19 i equals 16 minus 8 y i
69. − 14 − 3 i = 2 x + y i negative 14 minus 3 i equals 2 x plus y i

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