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Quadratic Equations,
Quadratic Functions, and
Complex Numbers
Copyright © Cengage Learning. All rights reserved.
9
Section
9.5
Complex Numbers
Copyright © Cengage Learning. All rights reserved.
Objectives
1. Simplify powers of i.
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22. Simplify square roots containing negative
radicands.
33. Perform operations with complex numbers.
44. Rationalize a denominator, expressing the
answer in a + bi form.
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Objectives
5 Solve a quadratic equation that has complexnumber solutions.
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Complex Numbers
The solutions of some quadratic equations are not real
numbers.
For example, we solve the equation x2 + 2x + 5 = 0 and
obtain the following solutions
or
Each of these solutions involves
, which is not a real
number. Thus, the solutions of this equation are not real
numbers.
As we will see, the solutions of this equation are from a set
called the set of complex numbers.
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1.
Simplify powers of i
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Simplify powers of i
The imaginary number
letter i. Since
is usually denoted by the
it follows that
i 2 = –1
The powers of i produce an interesting pattern:
i=
i2 =
i5 = i4  i = 1  i = i
=i
= –1
i 6 = i 4  i 2 = 1(–1) = –1
i 3 = i 2  i = –1  i = –i
i 7 = i 4  i 3 = 1(–i) = –i
i 4 = i 2  i 2 = (–1)(–1) = 1
i 8 = i 4  i 4 = (1)(1) = 1
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Simplify powers of i
The pattern continues: i, –1, –i, 1, . . . .
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2.
Simplify square roots
containing negative radicands
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Simplify square roots containing negative radicands
If we assume that multiplication of imaginary numbers is
commutative and associative, then
(2i)2 = 22i 2
= 4(–1)
= –4
Since (2i)2 = –4, it follows that 2i is a square root of –4, and
we write
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Simplify square roots containing negative radicands
This result could have been obtained by the following
process:
Likewise, we have
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Simplify square roots containing negative radicands
In general, we have the following rules.
Properties of Radicals
If at least one of a and b is a nonnegative real number,
then
and
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3.
Perform operations with
complex numbers
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Perform operations with complex numbers
Imaginary numbers such as
,
, and
form a
subset of a broader set of numbers called complex
numbers.
Complex Numbers
A complex number is any number that can be written in
the form a + bi where a and b are real numbers, and
.
The number a is called the real part, and the number b is
called the imaginary part of the complex number a + bi.
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Perform operations with complex numbers
If b = 0, the complex number a + bi is the real number a. If
b  0 and a = 0, the complex number 0 + bi (or just bi) is an
imaginary number.
Figure 9-13 shows the relationship of the real numbers to
the imaginary and complex numbers.
Figure 9-13
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Perform operations with complex numbers
Equality of Complex Numbers
The complex numbers a + bi and c + di are equal if and
only if
a = c and b = d
Here are several examples of equal complex numbers.
, because
, because
and
and
.
.
x + yi = 4 + 7i if and only if x = 4 and y = 7.
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Perform operations with complex numbers
Addition and Subtraction of Complex Numbers
Complex numbers are added and subtracted as if they
were binomials:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) – (c + di) = (a – c) + (b – d)i
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Example
Perform each operation:
a. (8 + 4i) + (12 + 8i)
b. (7 – 4i) + (9 + 2i)
c. (–6 + i) – (3 – 4i)
d. (2 – 4i) – (–4 + 3i)
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Example – Solution
We will use the rules for addition and subtraction of
complex numbers.
a. (8 + 4i) + (12 + 8i) = 8 + 4i + 12 + 8i
= 20 + 12i
b. (7 – 4i) + (9 + 2i) = 7 – 4i + 9 + 2i
= 16 – 2i
c. (–6 + i) – (3 – 4i) = –6 + i – 3 + 4i
= –9 + 5i
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Example – Solution
cont’d
d. (2 – 4i) – (–4 + 3i) = 2 – 4i + 4 – 3i
= 6 – 7i
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Perform operations with complex numbers
To multiply a complex number by an imaginary number, we
use the distributive property to remove parentheses and
then simplify.
Multiplication of Complex Numbers
Complex numbers are multiplied as if they were binomials,
with i 2 = –1:
(a + bi) (c + di) = ac + adi + bci + bdi 2
= (ac – bd) + (ad + bc)i
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4. Rationalize a denominator,
expressing the answer in a + bi form
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Rationalize a denominator, expressing the answer in a + bi form
The fraction
is not in simplest form because the
denominator contains a radical:
To simplify the fraction, we must rationalize the
denominator just as we did when we simplified rational
expressions.
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Example
Rationalize each denominator and write the result in
a + bi form.
a.
b.
Solution:
We can multiply each numerator and denominator by i and
simplify.
a.
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Example – Solution
cont’d
b.
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Rationalize a denominator, expressing the answer in a + bi form
To rationalize the denominators of fractions such as
,
, and
, we must multiply the numerator and
denominator by the complex conjugate of the denominator.
Complex Conjugates
The complex numbers a + bi and a – bi are called complex
conjugates of each other.
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Rationalize a denominator, expressing the answer in a + bi form
For example,
• 3 + 4i and 3 – 4i are complex conjugates.
• 5 – 7i and 5 + 7i are complex conjugates.
• 8 + 17i and 8 – 17i are complex conjugates.
In general, the product of the complex number a + bi and
its complex conjugate a – bi is the real number a2 + b2.
(a + bi)(a – bi) = a2 – abi + abi – b2i 2
= a2 – b2(–1)
= a2 + b2
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5.
Solve a quadratic equation that
has complex-number solutions
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Solve a quadratic equation that has complex number solutions
The solutions of many quadratic equations are not real
numbers.
For example, the solutions of the equation x2 + x + 1 = 0
are not real numbers. We will show that this is true in the
next example.
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Example
Solve: x2 + x + 1 = 0
Solution:
We will solve the equation by using the quadratic formula
with a = 1, b = 1, and c = 1:
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Example – Solution
cont’d
Expressed in a + bi form, the solutions are
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