Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Simplify each number by using the imaginary number i .

8. − 4 square root of negative 4 end root
9. − 7 square root of negative 7 end root
10. − 15 square root of negative 15 end root
11. − 81 square root of negative 81 end root
12. − 50 square root of negative 50 end root

See Problem 2.

Plot each complex number and find its absolute value.

13. 2 i
14. 5 + 12 i
15. 2 − 2 i 2 minus 2 i
16. 1 − 4 i 1 minus 4 i
17. 3 − 6 i 3 minus 6 i

See Problems 3 and 4.

Simplify each expression.

18. ( 2 + 4 i ) + ( 4 − i ) open 2 plus 4 i close plus open 4 minus i close
19. ( − 3 − 5 i ) + ( 4 − 2 i ) open negative 3 minus 5 i close plus open 4 minus 2 i close
20. ( 7 + 9 i ) + ( − 5 i ) open 7 plus 9 i close plus open negative 5 i close
21. ( 12 + 5 i ) − ( 2 − i ) open 12 plus 5 i close minus open 2 minus i close
22. ( − 6 − 7 i ) − ( 1 + 3 i ) open negative 6 minus 7 i close minus open 1 plus 3 i close
23. ( 8 + i ) ( 2 + 7 i ) open 8 plus i close open 2 plus 7 i close
24. ( − 6 − 5 i ) ( 1 + 3 i ) open negative 6 minus 5 i close open 1 plus 3 i close
25. ( − 6 i ) 2 open negative 6 i , close squared
26. ( 9 + 4 i ) 2 open 9 plus 4 i , close squared

See Problem 5.

Write each quotient as a complex number.

27. 3 − 2 i 5 i fraction 3 minus 2 i , over 5 i end fraction
28. − 2 i 1 + i fraction negative 2 i , over 1 plus i end fraction
29. 4 − 3 i − 1 − 4 i fraction 4 minus 3 i , over negative 1 minus 4 i end fraction
30. i + 2 i − 2 fraction i plus 2 , over i minus 2 end fraction
31. 4 2 − 3 i fraction 4 , over 2 minus 3 i end fraction
32. 3 + 2 i ( 1 + i ) 2 fraction 3 plus 2 i , over open , 1 plus i , close squared end fraction

See Problem 6.

Solve each equation. Check your answer.

33. x 2 + 25 = 0 x squared , plus 25 equals 0
34. 2 x 2 + 1 = 0 2 x squared , plus 1 equals 0
35. 3 s 2 + 2 = − 62 3 s squared , plus 2 equals negative 62
36. x 2 = − 7 x squared , equals negative 7
37. x 2 + 36 = 0 x squared , plus 36 equals 0
38. − 5 x 2 − 3 = 0 negative 5 , x squared , minus 3 equals 0

See Problem 7.

Find all solutions to each quadratic equation.

39. x 2 + 2 x + 3 = 0 x squared , plus 2 x plus 3 equals 0
40. − 3 x 2 + x − 3 = 0 negative 3 , x squared , plus x minus 3 equals 0
41. 2 x 2 − 4 x + 7 = 0 2 x squared , minus 4 x plus 7 equals 0
42. x 2 − 2 x + 2 = 0 x squared , minus 2 x plus 2 equals 0
43. x 2 + 5 = 4 x x squared , plus 5 equals 4 x
44. 2 x ( x − 3 ) = − 5 2 x open x minus 3 close equals negative 5

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