SECTION 2.7SECTION 2.7

COMPLEX ZEROS OF A COMPLEX ZEROS OF A QUADRATIC FUNCTION QUADRATIC FUNCTION

SQUARE ROOTS OF NEGATIVE NUMBERSSQUARE ROOTS OF

NEGATIVE NUMBERS

4 Is a value we have dealt Is a value we have dealt with up to now by with up to now by simply saying that it is simply saying that it is not a real number.not a real number.

And, up to now, we have dealt And, up to now, we have dealt with the following equation by with the following equation by simply saying it has no solution:simply saying it has no solution:

xx22 + 4 = 0 + 4 = 0

DEFINITION OF iDEFINITION OF i

1 - i

ii 2 2 = - 1 = - 1

The number The number ii is called an is called an imaginary number. Imaginary imaginary number. Imaginary numbers, along with the real numbers, along with the real numbers, make up a set of numbers, make up a set of numbers known as the numbers known as the complexcomplex numbersnumbers..

COMPLEX NUMBERSCOMPLEX NUMBERS

ImaginaryImaginary RealReal

55 -1-1

1/21/2 .7.7

3

ii 22ii

- 3- 3ii 2/32/3ii

4 + 54 + 5i i -7 + -7 + ii

1/2 + 3/41/2 + 3/4ii

COMPLEX NUMBERSCOMPLEX NUMBERS

All numbers are complex and All numbers are complex and should be thought of in the form:should be thought of in the form:

a + ba + bii

Real Real PartPart

Imaginary PartImaginary Part

COMPLEX NUMBERSCOMPLEX NUMBERS

a + ba + bii

Real Real PartPart

Imaginary PartImaginary Part

When b = 0, the number is a When b = 0, the number is a real number. Otherwise, the real number. Otherwise, the number is imaginary.number is imaginary.

OPERATING ON COMPLEX NUMBERS

OPERATING ON COMPLEX NUMBERS

AdditionAddition::

Example:Example:

(3 + 5i) + ( - 2 + 3i)(3 + 5i) + ( - 2 + 3i)

SubtractionSubtraction::

Example:Example:

(6 + 4i) - ( 3 + 6i)(6 + 4i) - ( 3 + 6i)

OPERATING ON COMPLEX NUMBERS

OPERATING ON COMPLEX NUMBERS

MultiplicationMultiplication::

Example:Example:

(5 + 3i) • (2 + 7i)(5 + 3i) • (2 + 7i)

Example:Example:

(3 + 4i) • ( 3 - 4i)(3 + 4i) • ( 3 - 4i)

CONJUGATESCONJUGATES

2 + 3i = 2 - 3i2 + 3i = 2 - 3i

Multiplying a complex number by Multiplying a complex number by its conjugate always yields a its conjugate always yields a nonnegative real number. nonnegative real number.

THEOREM: If z = a + biTHEOREM: If z = a + bi

z z = az z = a22 + b + b22

Writing the reciprocal of a Writing the reciprocal of a complex number in standard complex number in standard form.form.

Example:Example:

4i 31

Writing the quotient of Writing the quotient of complex numbers in standard complex numbers in standard form.form.

Example:Example:

12i - 5 4i 1

Writing the quotient of Writing the quotient of complex numbers in standard complex numbers in standard form.form.

Example:Example:

3i - 43i - 2

POWERS OF iPOWERS OF i

ii11 = i = i

ii22 = - 1 = - 1

ii33 = - i = - i

ii44 = 1 = 1

ii55 = i = i

and so onand so on

QUADRATIC EQUATIONS WITH A NEGATIVE DISCRIMINANT

QUADRATIC EQUATIONS WITH A NEGATIVE DISCRIMINANT

Quadratic equations with a Quadratic equations with a negative discriminant have no negative discriminant have no real solution. But, if we extend real solution. But, if we extend our number system to the our number system to the complex numbers, quadratic complex numbers, quadratic equations will always have equations will always have solutions because we will then be solutions because we will then be including imaginary numbers.including imaginary numbers.

EXAMPLEEXAMPLE

i 1 2i 4

i 22or 22i 8

EXAMPLEEXAMPLE

Solve the following equations Solve the following equations in the complex number in the complex number system:system:

xx22 = 4 = 4 xx22 = - 9 = - 9

WARNING!WARNING!

36 3- 12-

EXAMPLEEXAMPLE

Solve the following Solve the following equation in the complex equation in the complex number system:number system:

xx22 - 4x + 8 = 0 - 4x + 8 = 0

DISCRIMINANTDISCRIMINANT

If bIf b22 - 4ac > 0 Two unequal real sol’ns - 4ac > 0 Two unequal real sol’ns

If bIf b22 - 4ac = 0 One double real root - 4ac = 0 One double real root

If bIf b22 - 4ac < 0 Two imaginary solutions - 4ac < 0 Two imaginary solutions

EXAMPLE:EXAMPLE:

Without solving, determine Without solving, determine the character of the solution the character of the solution of each equation in the of each equation in the complex number system:complex number system:

3x3x22 + 4x + 5 = 0 + 4x + 5 = 0

2x2x22 + 4x + 1 = 0 + 4x + 1 = 0

9x9x22 - 6x + 1 = 0 - 6x + 1 = 0

CONCLUSION OF SECTION 2.7CONCLUSION OF SECTION 2.7