Download Section 3.1

CHAPTER 3:
Quadratic Functions and
Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and
Models
3.3 Analyzing Graphs of Quadratic Functions
3.4 Solving Rational Equations and Radical
Equations
3.5 Solving Equations and Inequalities with
Absolute Value
Copyright © 2009 Pearson Education, Inc.
2.2
The Complex Numbers

Perform computations involving complex
numbers.
Copyright © 2009 Pearson Education, Inc.
The Complex-Number System
Some functions have zeros that are not real numbers.
The complex-number system is used to find zeros of
functions that are not real numbers.
When looking at a graph of a function, if the graph
does not cross the x-axis, then it has no x-intercepts,
and thus it has no real-number zeros.
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 4
Example
Express each number in terms of i.
a.
7
d.  64
b.
16
e.
48
c.  13
Solution
a.
7  1 7  1  7
 i 7, or
b.
7i
16  116  1  16
 i  4  4i
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 5
Example (continued)
Solution continued
c.  13   113   1  13
 i 13, or  13i
d.  64   1 64   1  64
 i  8  8i
e.
48  1 48  1  48
 i 16  3
 i4 3
 4i 3, or 4 3i
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 6
Complex Numbers
A complex number is a number of the form a + bi,
where a and b are real numbers. The number a is
said to be the real part of a + bi and the number b
is said to be the imaginary part of a + bi.
Imaginary Number
a + bi, a ≠ 0, b ≠ 0
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Pure Imaginary Number
a + bi, a = 0, b ≠ 0
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Addition and Subtraction
Complex numbers obey the commutative,
associative, and distributive laws.
We add or subtract them as we do binomials.
We collect the real parts and the imaginary parts of
complex numbers just as we collect like terms in
binomials.
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 8
Example
Add or subtract and simplify each of the following.
a. (8 + 6i) + (3 + 2i)
b. (4 + 5i) – (6 – 3i)
Solution
a. (8 + 6i) + (3 + 2i) = (8 + 3) + (6i + 2i)
= 11 + (6 + 2)i = 11 + 8i
b.
(4 + 5i) – (6 – 3i) = (4 – 6) + [5i  (–3i)]
=  2 + 8i
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 9
Multiplication
When
a and
b are real numbers, a  b  ab.
This is not true when
a and
b are not real numbers.
Note: Remember i2 = –1
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 10
Example
Multiply and simplify each of the following.
a.
16  25
b.
1 2i 1 3i 
c.
3  7i 
2
Solution
a.
16  25  1  16  1  25
 i  4 i 5
 i 2  20
 1 20
 20
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 11
Example (continued)
Solution continued
b.
1  2i 1  3i   1  3i  2i  6i 2
 1 3i  2i  6
 5  5i
c.
3  7i 
2
 3  2  3 7i  7i 
2
2
 9  42i  49i 2
 9  42i  49
 40  42i
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 12
Simplifying Powers of i
Recall that 1 raised to an even power is 1, and 1
raised to an odd power is 1.
Simplifying powers of i can then be done by using the
fact that i2 = –1 and expressing the given power of i in
terms of i2.
i  1
i 2  1
Note that powers of i cycle
through i, –1, –i, and 1.
i 3  i 2  i  1i  i
   1  1
 i  i  i   i  1  i  1 i  i
i  i
4
i
5
4
2 2
2
2 2
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2
Slide 3.1 - 13
Conjugates
The conjugate of a complex number a + bi is a  bi.
The numbers a + bi and a  bi are complex conjugates.
Examples:
3 + 7i and 3  7i
14  5i and 14 + 5i
8i and 8i
The product of a complex number and its conjugate is a
real number.
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 14
Multiplying Conjugates - Example
Multiply each of the following.
a. (5 + 7i)(5 – 7i)
b. (8i)(–8i)
Solution
a. (5 + 7i)(5  7i) = 52  (7i)2
= 25  49i2
= 25  49(1)
= 25 + 49
= 74
b. (8i)(–8i) = 64i2
= 64(1)
= 64
Copyright © 2009 Pearson Education, Inc.
Slide 3.1 - 15
Dividing Using Conjugates - Example
Divide 2  5i by 1  6i.
Solution: Write fraction notation. Multiply by 1, using the
conjugate of the denominator to form the symbol for 1.
2  5i 2  5i 1  6i


1  6i 1  6i 1  6i
2  7i  30

1  36
2  5i 1  6i 


1  6i 1  6i 
32  7i

37
2  7i  30i 2

1  36i 2
32 7

 i
37 37
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Slide 3.1 - 16