Download 5-5 Complex Numbers and Roots

5-5
Complex Numbers
and Roots
TEKS 2A.2.B Foundations for functions: use complex numbers to
describe the solutions of quadraic equations.
Solve quadratic equations
with complex roots.
Vocabulary
imaginary unit
imaginary number
complex number
real part
imaginary part
complex conjugate
Also 2A.2.A, 2A.5.E,
2A.6.A, 2A.6.B, 2A.8.A,
2A.8.D
Andrew Toos/CartoonResource.com
Why learn this?
Complex numbers can be used to
describe the zeros of quadratic
functions that have no real
zeros. (See Example 4.)
Objectives
Define and use imaginary
and complex numbers.
You can see in the graph of
f (x) = x 2 + 1 below that f has no real
zeros. If you solve the corresponding
equation 0 = x 2 + 1, you find that
x = ± √
-1 , which has no real solutions.
However, you can find solutions if you define
the square root of negative numbers, which is
why imaginary numbers were invented. The
-1 . You can use
imaginary unit i is defined as √
the imaginary unit to write the square root of any
negative number.
Þ
v­Ý®ÊÊÝÓÊÊ£
Ý
œÊ݇ˆ˜ÌiÀVi«ÌÃ
Imaginary Numbers
WORDS
An imaginary number is the square
root of a negative number.
Imaginary numbers can be written
in the form bi, where b is a real
number and i is the imaginary unit.
NUMBERS
√
-1 = i
If b is a positive
real number,
= i √
√-2
= √
then √-b
-1 √
2 = i √2
b
√
-4 = √
-1 √
4 = 2i
The square of an imaginary number
is the original negative number.
EXAMPLE
1
ALGEBRA
and √
-b 2 = bi.
2
( √
-1 ) = i 2 = -1
2
( √
-b ) = -b
Simplifying Square Roots of Negative Numbers
Express each number in terms of i.
A 3 √
-16
B - √
-75
(16)(-1) Factor out -1.
3 √
(75)(-1)
- √
Factor out -1.
3 √
16 √
-1 Product Property
- √
75 √
-1
Product Property
3 · 4 √
-1
Simplify.
- √
25 √
3 √
-1
Product Property
12 √
-1
Multiply.
-5 √
3 √
-1
Simplify.
12i
Express in terms
of i.
-5 √
3 i = -5i √
3 Express in terms
Express each number in terms of i.
1a. √
-12
1b. 2 √
-36
350
of i.
1 √
1c. -_
-63
3
Chapter 5 Quadratic Functions
a207se_c05l05_0350_0355.indd 350
12/14/05 1:52:32 PM
EXAMPLE
2
Solving a Quadratic Equation with Imaginary Solutions
Solve each equation.
A x 2 = -81
B 3x 2 + 75 = 0
-81 Take square
x = ± √
x = ±9i
3x 2 = -75
roots.
Express in
terms of i.
x 2 = -25 Divide both sides by 3.
x = ± √
-25 Take square roots.
Check
x = ±5i Express in terms of i.
Check −−−−−−−−−−−−
3x 2 + 75 = 0
2
3(±5i) + 75 0
3(25)i 2 + 75 0
75(-1) + 75 0 ✔
x 2 = -81
x 2 = -81
−−−−−−−−−
−−−−−−−−−
2
(9i) -81
(-9i) 2 -81
81i 2 -81
81i 2 -81
81(-1) -81 ✔ 81(-1) -81 ✔
Solve each equation.
2a. x 2 = -36
2b. x 2 + 48 = 0
A complex number is a number
that can be written in the form
a + bi, where a and b are real
numbers and i = √
-1 . The set of
real numbers is a subset of the set
of complex numbers .
2c. 9x 2 + 25 = 0
œ“«iÝÊ Õ“LiÀÃÊο
ÎÊÊLj ÊÊÊÊÎÊÊÊÚÚ
ÊÓÊÊÊˆÊ ÊÊÊÊ{Êʈ
Î
,i>
ՓLiÀÃÊώ
£
ÚÚ
ÊÊÊÊÊÊÊ£°ÇÎÊÊÊÊäÊÊÊÊû
Ó
Ü
е
™°ÊÈÊÊÊÊÊÊÊȖÓÊ
Ê
Every complex number has a real
part a and an imaginary part b.
Real part
Add -75 to both sides.
“>}ˆ˜>ÀÞ
ՓLiÀÃ
ˆÊÊÊÊΈÊÊÊÊxˆ
е
ÊȖÇÊ
еÊ
Imaginary part
Real numbers are complex numbers where b = 0. Imaginary numbers are
complex numbers where a = 0 and b ≠ 0. These are sometimes called pure
imaginary numbers.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal.
EXAMPLE
3
Equating Two Complex Numbers
Find the values of x and y that make the equation 3x - 5i = 6 - (10y)i true.
Real parts
3x - 5i = 6 - (10y)i
Imaginary parts
3x = 6
x=2
Equate the real parts.
Solve for x.
-5 = -(10y) Equate the imaginary parts.
1 =y
_
Solve for y.
2
Find the values of x and y that make each equation true.
3a. 2x - 6i = -8 + (20y)i
3b. -8 + (6y)i = 5x - i √
6
5- 5 Complex Numbers and Roots
a207se_c05l05_0350_0355.indd 351
351
12/14/05 1:52:41 PM
EXAMPLE
4
Finding Complex Zeros of Quadratic Functions
Find the zeros of each function.
A f (x) = x 2 - 2x + 5
B
x 2 - 2x + 5 = 0
x 2 - 2x +
x 2 + 10x + 35 = 0
Set equal to 0.
= -5 +
x 2 + 10x +
Rewrite.
()
g (x) = x 2 + 10x + 35
2
= -35 +
x - 2x + 1 = -5 + 1
b
Add __
.
x + 10x + 25 = -35 + 25
(x - 1) = -4
Factor.
(x + 5) 2 = -10
2
2
2
x - 1 = ± √
-4
2
x + 5 = ± √
-10
Take square roots.
x = 1 ± 2i
x = -5 ± i √
10
Simplify.
Find the zeros of each function.
4a. f (x) = x 2 + 4x + 13
4b. g(x) = x 2 - 8x + 18
When given one
complex root, you
can always find the
other by finding its
conjugate.
EXAMPLE
The solutions -5 + i √
10 and -5 - i √
10 in Example 4B are related. These
solutions are a complex conjugate pair. Their real parts are equal and their
imaginary parts are opposites. The complex conjugate of any complex number
a + bi is the complex number a - bi.
If a quadratic equation with real coefficients has nonreal roots, those roots are
complex conjugates.
5
Finding Complex Conjugates
Find each complex conjugate.
A 2i - 15
-15 + 2i
-15 - 2i
B
Write as a + bi.
Find a - bi.
Find each complex conjugate.
5a. 9 - i
5b. i + √
3
-4i
0 + (-4)i
0 - (-4)i
4i
Write as a + bi.
Find a - bi.
Simplify.
5c. -8i
THINK AND DISCUSS
1. Given that one solution of a quadratic equation is 3 + i, explain how to
determine the other solution.
2. Describe a number of the form a + bi in which a ≠ 0 and b = 0.
Then describe a number in which a = 0 and b ≠ 0. Are both numbers
complex? Explain.
3. GET ORGANIZED Copy and complete
the graphic organizer. In each box or
oval, give a definition and examples of
each type of number.
352
œ“«iÝÊ Õ“LiÀÃ
,i>Ê
ՓLiÀÃ
“>}ˆ˜>ÀÞÊ
ՓLiÀÃ
Chapter 5 Quadratic Functions
a207se_c05l05_0350_0355.indd 352
12/14/05 1:52:44 PM