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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

Page 300

4.5 - Complex Numbers


Page 300

4.5 - Complex Numbers


Complex solutions

Complex Solutions

4.5 - Complex Numbers


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

SIMPLIFY RADICALS FIRST then OPERATE

Imaginary Number

Real Number

4.5 - Complex Numbers


Example 1

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

Example 2

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

4.5 - Complex Numbers


Your turn

Your Turn

Find the values of x and y that make the equation

true.

4.5 - Complex Numbers


Imaginary 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

Example 3

Solve the equation, 5x2 + 90 = 0

4.5 - Complex Numbers


Your turn1

Your Turn

Solve the equation, 9x2 + 25 = 0

4.5 - Complex Numbers


Example 4

Example 4

Solve the equation, 3x2 – 2x + 5 = 0

4.5 - Complex Numbers


Your turn2

Your Turn

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

4.5 - Complex Numbers


Cube roots

Cube Roots

4.5 - Complex Numbers


Sum and difference of cube roots

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)

A – Put addition sign

S2 – Square the second base

4.5 - Complex Numbers


Example 5

Example 5

Solve the equation, x3 + 27 = 0

4.5 - Complex Numbers


Example 6

Example 6

Solve the equation, x3 – 27 = 0

4.5 - Complex Numbers


Example 7

Example 7

Solve the equation, x3 – 1 = 0

4.5 - Complex Numbers


Your turn3

Your Turn

Solve the equation, x3 + 64 = 0

4.5 - Complex Numbers


Review

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 71

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

Example 8

Rationalize

4.5 - Complex Numbers


Example 9

Example 9

Rationalize

4.5 - Complex Numbers


Your tur n

Your Turn

Rationalize

4.5 - Complex Numbers


Assignment

Assignment

Page 300

23-31 odd, 55-67 odd

4.5 - Complex Numbers


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