Section 7.5

Complex Numbers and Solving Quadratic Equations with Complex Solutions

The Imaginary Number i

The imaginary number i is defined as 1i . So:

2i 3i 4i

Objective 1: Express complex numbers in standard form.

Notice that any power of i can be simplified to i, −1, −i, or 1. Simplify each power of i.

1. 2. 3.

4. 5. 6.

5i 17i 22i

60i 107i 801i

If a and b are real numbers and , then:

Algebraic Form Numerical Example

is a complex number with a ____________ term a and an ____________ term bi.

has a real term ______ and an imaginary term ______.

1i

a bi 2 7i

Complex Numbers

Simplify each expression and write the result in the standard a bi form.

7. 36 8. 6

Simplify each expression and write the result in the standard a bi form.

9. 6 6 10. 81

Simplify each expression and write the result in the standard a bi form.

11. 81 81 12. 81 81

Simplify each expression and write the result in the standard a bi form.

13. 100

25

Simplify each expression and write the result in the standard a bi form.

14. 6 5 6

Simplify each expression and write the result in the standard a bi form.

15. 36 25 49

Objective 2: Add, subtract, multiply, and divide complex numbers.

Addition of Binomials

5 7 3 4 5 3 7 4

8 3

x y x y x x y y

x y

5 7 3 4 5 3 7 4

8 3

i i i i

i

Addition of Complex Numbers

The arithmetic of complex numbers is very similar to the arithmetic of binomials

16. 3 7 9 8i i

Simplify each expression and write the result in the standard form. a bi

17.

Simplify each expression and write the result in the standard form.

4 7 3 2i i

a bi

18. 6 7 3i

Simplify each expression and write the result in the standard a bi form.

19.

Simplify each expression and write the result in the standard a bi form.

3 10i i

20.

Simplify each expression and write the result in the standard a bi form.

2 7 9i i

21.

Simplify each expression and write the result in the standard a bi form.

4 3 5i i

22.

Simplify each expression and write the result in the standard a bi form.

7 6 3 8i i

23.

Simplify each expression and write the result in the standard a bi form.

22 5i

Complex Conjugates: The conjugate of a bi is .a bi

Write the conjugate of each expression. Then multiply the expression by its conjugate.

Expression Conjugate

Product

24. 5 2i

Complex Conjugates: The conjugate of a bi is .a bi

Write the conjugate of each expression. Then multiply the expression by its conjugate.

Expression Conjugate

Product

25. 8i

Dividing Complex Numbers - The fact that the product of a complex number and its conjugate is always a real number plays a key role in the division of complex numbers as outlined in the following box.

Steps for Dividing Complex Numbers

Step 1. Write the division problem as a fraction.

Step 2. Multiply both the numerator and the denominator by the conjugate of the denominator.

Step 3. Simplify the result, and express it in standard a bi form.

Example 3 1 2i i

26.

Perform the indicated operations and express the result in a bi form.

50 1 3i

27.

Perform the indicated operations and express the result in a bi form.

2

3 4i

28. 5 3

2 7

i

i

Perform the indicated operations and express the result in a bi form.

29. 6 8

1

i

i

Perform the indicated operations and express the result in a bi form.

30. Determine whether or not is a solution of the equation 2 6 13 0.x x

2 5x i

31. Determine whether or not is a solution of the equation

3 2x i 2 6 13 0.x x

Objective 3: Solve a quadratic equation with imaginary solutions

Recall solving quadratic equations by extraction of roots from Section 7.1: The solutions of 2x k are x kand x k

Solve each quadratic equation by extraction of roots.

32. 2 16x

Solve each quadratic equation by extraction of roots.

33. 2 7x

Solve each quadratic equation by extraction of roots.

34. 21 4x

Solve each quadratic equation by extraction of roots.

35. 23 49x

Solve each quadratic equation by extraction of roots.

36. 22 3 49x

Solve each quadratic equation by extraction of roots.

37. 23 2 10x

Recall solving quadratic equations by the Quadratic Formula from Section 7.3: The solutions of the quadratic equation

2 0ax bx c with real coefficients a, b, and c, when 0,a

are 2 4

.2

b b acx

a

Use the quadratic formula to solve each quadratic equation.

38. 2 2 6 0x x

39. 23 2 4x x

Use the quadratic formula to solve each quadratic equation.

40. 2 10 25x x

Use the quadratic formula to solve each quadratic equation.

41. 3 4 5x x

Use the quadratic formula to solve each quadratic equation.

42.

Use the quadratic formula to solve each quadratic equation.

22 2 4 0x x x (Hint: Use the zero factor principle.)

Construct a quadratic equation in x that has the given solutions.

43. and 5x i 5x i

Construct a quadratic equation in x that has the given solutions.

44. and 3x i 3x i

45. Determine the discriminant of each of these quadratic equations and then determine the solution of each equation.

Equation Discriminant Solution

(a) 2 1 0x

45. Determine the discriminant of each of these quadratic equations and then determine the solution of each equation.

Equation Discriminant Solution

(b) 2 2 1 0x x

45. Determine the discriminant of each of these quadratic equations and then determine the solution of each equation.

Equation Discriminant Solution

(c) 2 1 0x