Section 7.5 Complex Numbers and Solving Quadratic Equations with Complex Solutions.

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


9. 6 6 10. 81


11. 81 81 12. 81 81


13. 100

25


14. 6 5 6


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


19.


3 10i i

20.


2 7 9i i

21.


4 3 5i i

22.


7 6 3 8i i

23.


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.


2

3 4i

28. 5 3

2 7

i

i


29. 6 8

1

i

i


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


33. 2 7x


34. 21 4x


35. 23 49x


36. 22 3 49x


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


40. 2 10 25x x


41. 3 4 5x x


42.


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



(b) 2 2 1 0x x



(c) 2 1 0x

Section 7.5 Complex Numbers and Solving Quadratic Equations with Complex Solutions.

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