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Page 300. Complex Solutions. Complex Numbers. Complex Numbers are numbers that are associated with a second real number with an ordered pair a + b i Conjugate is the complex number’s opposite sign Example: 2 + 3 i ‘s conjugate is 2 – 3 i

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Page 300

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### Page 300

4.5 - Complex Numbers

4.5 - Complex Numbers

## Complex Solutions

4.5 - Complex Numbers

### Complex Numbers

Complex Numbersare numbers that are associated with a second real number with an ordered pair

a + b i

Conjugate is the complex number’s opposite sign

Example: 2 + 3i ‘s conjugate is 2 – 3i

Remember: NO IMAGINARY NUMBERSin the denominator

Imaginary Number

Real Number

4.5 - Complex Numbers

### Example 1

Find the values of x and y that make the equation 4x + 10i= 2 – (4y)itrue.

Real parts

4x + 10i = 2 – (4y)i

Imaginary parts

4.5 - Complex Numbers

### Example 2

Find the values of x and y that make the equation 2x – 6i= –8 + (20y)itrue.

4.5 - Complex Numbers

Find the values of x and y that make the equation

true.

4.5 - Complex Numbers

### Imaginary Numbers

Rules with Imaginaries:

i1 = i

i2 = –1

Answers CAN have an i but CAN NOT have two i’s

Two i’s make –1

4.5 - Complex Numbers

### Example 3

Solve the equation, 5x2 + 90 = 0

4.5 - Complex Numbers

Solve the equation, 9x2 + 25 = 0

4.5 - Complex Numbers

### Example 4

Solve the equation, 3x2 – 2x + 5 = 0

4.5 - Complex Numbers

Solve the equation, 2x2 + x + 3 = 0

4.5 - Complex Numbers

### Cube Roots

4.5 - Complex Numbers

### Sum and Difference of Cube Roots

• Take the GCF Out

• Put the equation into perfect cubes

• Put the bases of the cubes into parentheses

• Refer to the bases in the parentheses and follow the format of S1OMAS2

S1 – Square the first base

O – Take Opposite of the original problem’s sign

M – Multiply the first and second base (leave the signs alone)

S2 – Square the second base

4.5 - Complex Numbers

### Example 5

Solve the equation, x3 + 27 = 0

4.5 - Complex Numbers

### Example 6

Solve the equation, x3 – 27 = 0

4.5 - Complex Numbers

### Example 7

Solve the equation, x3 – 1 = 0

4.5 - Complex Numbers

Solve the equation, x3 + 64 = 0

4.5 - Complex Numbers

### Review

Rationalize

What is the problem with the bottom?

We try to get rid of the radicals on the bottom so the bottom will be even

With the CONJUGATE

4.5 - Complex Numbers

### Example 7

Rationalize

What is the problem with the bottom?

We try to get rid of the radicals on the bottom so the bottom will be even

With the CONJUGATE

4.5 - Complex Numbers

### Example 8

Rationalize

4.5 - Complex Numbers

### Example 9

Rationalize

4.5 - Complex Numbers