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2-5
2-5 Complex
Complex Numbers
Numbers and
and Roots
Roots
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 2Algebra
Algebra22
Holt
McDougal
2-5
Complex Numbers and Roots
Warm Up
Simplify each expression.
2.
1.
3.
Find the zeros of the function by
completing the square.
4. f(x) = x2 – 18x + 16
Rewrite the function in vertex form.
5. f(x) =
x2
+ 8x – 24
Holt McDougal Algebra 2
𝒇 𝒙 = 𝒙+𝟒
𝟐
− 𝟒𝟎
2-5
Complex Numbers and Roots
Objectives
Define and use imaginary and complex
numbers.
Solve quadratic equations with
complex roots.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Vocabulary
imaginary unit
imaginary number
complex number
real part
imaginary part
complex conjugate
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Essential Question
• How do you find complex zeros of
quadratic functions?
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
You can see in the graph of f(x) = x2 + 1 below
that f has no real zeros. If you solve the
corresponding equation 0 = x2 + 1, you find
that x =
,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
imaginary unit i is defined
as
. You can use the imaginary
unit to write the square root of
any negative number.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 1A: Simplifying Square Roots of Negative
Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 1B: Simplifying Square Roots of Negative
Numbers
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
4 6i  4i 6
Holt McDougal Algebra 2
Express in terms of i.
2-5
Complex Numbers and Roots
Check It Out! Example 1a
Express the number in terms of i.
Factor out –1.
Product Property.
Product Property.
Simplify.
Express in terms of i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Check It Out! Example 1b
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Check It Out! Example 1c
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 2A: Solving a Quadratic Equation with
Imaginary Solutions
Solve the equation.
Take square roots.
Express in terms of i.
Check
x2 = –144
(12i)2 –144
144i 2 –144
144(–1) –144 
Holt McDougal Algebra 2
x2 =
(–12i)2
144i 2
144(–1)
–144
–144
–144
–144 
2-5
Complex Numbers and Roots
Example 2B: Solving a Quadratic Equation with
Imaginary Solutions
Solve the equation.
5x2 + 90 = 0
Add –90 to both sides.
Divide both sides by 5.
Take square roots.
Express in terms of i.
Check
5x2 + 90 = 0
0
5(18)i 2 +90 0
90(–1) +90 0 
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Check It Out! Example 2a
Solve the equation.
x2 = –36
Take square roots.
Express in terms of i.
Check
x2 = –36
(6i)2
36i 2
36(–1)
Holt McDougal Algebra 2
–36
–36
–36 
x2 = –36
(–6i)2 –36
36i 2 –36
36(–1) –36 
2-5
Complex Numbers and Roots
Check It Out! Example 2b
Solve the equation.
x2 + 48 = 0
x2 = –48
Add –48 to both sides.
Take square roots.
Express in terms of i.
Check
x2 + 48 = 0
+ 48
(48)i 2 + 48
48(–1) + 48
Holt McDougal Algebra 2
0
0
0
2-5
Complex Numbers and Roots
Check It Out! Example 2c
Solve the equation.
9x2 + 25 = 0
9x2 = –25
Add –25 to both sides.
Divide both sides by 9.
Take square roots.
Express in terms of i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
A complex number is a
number that can be written
in the form a + bi, where a
and b are real numbers and
i=
. The set of real
numbers is a subset of the
set of complex numbers C.
Every complex number has a real part a and an
imaginary part b.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Real numbers are complex numbers where b = 0.
4
Examples: 4, −10, , 0.5, and 15.
3
Imaginary numbers are complex numbers where
a = 0 and b ≠ 0. These are sometimes called
pure imaginary numbers.
2
Examples: 𝑖, 4i, − 𝑖, 𝑖 6.
5
Two complex numbers are equal if and only if their
real parts are equal and their imaginary parts are
equal.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 3: Equating Two Complex Numbers
Find the values of x and y that make the equation
4x + 10i = 2 – (4y)i true .
Real parts
4x + 10i = 2 – (4y)i
Imaginary parts
4x = 2
Equate the
real parts.
Solve for x.
Holt McDougal Algebra 2
10 = –4y Equate the
imaginary parts.
Solve for y.
2-5
Complex Numbers and Roots
Check It Out! Example 3a
Find the values of x and y that make each
equation true.
2x – 6i = –8 + (20y)i
Real parts
2x – 6i = –8 + (20y)i
Imaginary parts
2x = –8
Equate the
real parts.
x = –4
Solve for x.
Holt McDougal Algebra 2
–6 = 20y
Equate the
imaginary parts.
Solve for y.
2-5
Complex Numbers and Roots
Check It Out! Example 3b
Find the values of x and y that make each
equation true.
–8 + (6y)i = 5x – i 6
Real parts
–8 + (6y)i = 5x – i 6
Imaginary parts
–8 = 5x
Equate the real
parts.
Solve for x.
Holt McDougal Algebra 2
Equate the
imaginary parts.
Solve for y.
2-5
Complex Numbers and Roots
Example 4A: Finding Complex Zeros of Quadratic
Functions
Find the zeros of the function.
f(x) = x2 + 10x + 26
x2 + 10x + 26 = 0
Set equal to 0.
x2 + 10x +
Rewrite.
= –26 +
x2 + 10x + 25 = –26 + 25
(x + 5)2 = –1
Add
to both sides.
Factor.
Take square roots.
Simplify.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 4B: Finding Complex Zeros of Quadratic
Functions
Find the zeros of the function.
g(x) = x2 + 4x + 12
x2 + 4x + 12 = 0
Set equal to 0.
x2 + 4x +
Rewrite.
= –12 +
x2 + 4x + 4 = –12 + 4
(x + 2)2 = –8
Add
to both sides.
Factor.
Take square roots.
Simplify.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Check It Out! Example 4a
Find the zeros of the function.
f(x) = x2 + 4x + 13
x2 + 4x + 13 = 0
Set equal to 0.
x2 + 4x +
Rewrite.
= –13 +
x2 + 4x + 4 = –13 + 4
(x + 2)2 = –9
Add
to both sides.
Factor.
Take square roots.
x = –2 ± 3i
Holt McDougal Algebra 2
Simplify.
2-5
Complex Numbers and Roots
Check It Out! Example 4b
Find the zeros of the function.
g(x) = x2 – 8x + 18
x2 – 8x + 18 = 0
Set equal to 0.
x2 – 8x +
Rewrite.
= –18 +
x2 – 8x + 16 = –18 + 16
Add
to both sides.
Factor.
Take square roots.
Simplify.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
The solutions
and
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.
Helpful Hint
When given one complex root, you can always
find the other by finding its conjugate.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Example 5: Finding Complex Zeros of Quadratic
Functions
Find each complex conjugate.
B. 6i
A. 8 + 5i
8 + 5i
8 – 5i
Write as a + bi.
Find a – bi.
Holt McDougal Algebra 2
0 + 6i
0 – 6i
–6i
Write as a + bi.
Find a – bi.
Simplify.
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Complex Numbers and Roots
Check It Out! Example 5
Find each complex conjugate.
B.
A. 9 – i
9 + (–i)
9 – (–i)
9+i
C. –8i
0 + (–8)i
0 – (–8)i
8i
Holt McDougal Algebra 2
Write as a + bi.
Write as a + bi.
Find a – bi.
Find a – bi.
Simplify.
Write as a + bi.
Find a – bi.
Simplify.
2-5
Complex Numbers and Roots
Lesson Quiz
1. Express
in terms of i.
Solve each equation.
2. 3x2 + 96 = 0
3. x2 + 8x +20 = 0
4. Find the values of x and y that make the
equation 3x +8i = 12 – (12y)i true.
5. Find the complex conjugate of
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
Essential Question
• How do you find complex zeros of
quadratic functions?
• Find the zeros of the related equation by
completing the square and taking the
square root of each side. Simplify the
radical of the negative number by
replacing with the imaginary unit i.
Holt McDougal Algebra 2
2-5
Complex Numbers and Roots
• What’s a mathematician’s favorite
dessert?
•
−1ce cream
Holt McDougal Algebra 2