If i = 1 ,what are the values of the following? Hint: When you square a square root, all you.
Download ReportTranscript If i = 1 ,what are the values of the following? Hint: When you square a square root, all you.
Slide 1
If i = 1 ,what are the values of the following? Hint: When you
square a square root, all you do is take off the square root.
1. i2
2. i4
3. i6
Slide 2
5.5 Complex Numbers
Slide 3
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.
Slide 4
Slide 5
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of
i.
Slide 6
Express the number in terms of i.
Factor out –1.
4 6i 4i 6
Product
Property.
Simplify
.
Express in terms of
i.
Slide 7
Whiteboards
Express the number in terms of i.
Slide 8
Try this on your own.
Express the number in terms of i.
Slide 9
Whiteboards
Express the number in terms of i.
Slide 10
Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
This is just like solving x2 = 144. You take the
Square root of both sides, making sure you
consider both the positive and negative roots.
Check
x2 = –144
(12i)2 –144
144i 2 –144
–144
144(–1)
x2 = –144
(–12i)2
–144
144i 2 –144
144(–1)
–144
Slide 11
Example 2:
Solve the equation.
5x2 + 90 = 0
What happens here?
What happens?
Solve by taking square roots.
Simplify.
Check
5x2 + 90 = 0
0
5(18)i 2 +90 0
90(–1) +90 0
Slide 12
Now you try!
Solve the equation.
x2 = –36
Slide 13
Your turn
Solve the equation.
x2 + 48 = 0
Slide 14
One more
Solve the equation.
9x2 + 25 = 0
Slide 15
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.
Slide 16
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. For example, 7 can be written as 7 = 7 + 0i.
(7 is the real part a, and 0i is the complex part b).
Two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are equal.
Slide 17
• In the complex number 3x – 5yi, what is
the real part?
• What is the complex part?
Slide 18
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
Set the
real parts
equal to
each other.
Solve for
x.
10 = –4y
Set the
imaginary parts
equal to each
other.
Solve for
y.
Slide 19
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
x = –4
Equate the
real parts.
Solve for
x.
–6 = 20y
Equate the
imaginary
parts.
Solve for
y.
Slide 20
Your turn
Find the values of x and y that make each equation
true.
–8 + (6y)i = 5x – i
6
Slide 21
Find the zeros of the function.
f(x) = x2 + 10x + 26
x2 + 10x + 26 = 0
Set equal to 0.
x2 + 10x +
Rewrite.
Add
to both
sides.
= –26 +
x2 + 10x + 25 = –26 + 25
(x + 5)2 = –1
Factor.
Take square roots.
Simplify.
Slide 22
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.
Slide 23
Example 4a
Find the zeros of the function.
f(x) = x2 + 4x + 13
x2
+ 4x + 13 = 0
x2 + 4x +
= –13 +
x2 + 4x + 4 = –13 + 4
(x + 2)2 = –9
x = –2 ± 3i
Set equal to
0.
Rewrite.
Complete the square.
Factor
.
Take square
roots.
Simplify.
Slide 24
Your turn
Find the zeros of the function.
g(x) = x2 – 8x + 18
Slide 25
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.
Slide 26
Example 5: Finding Complex Zeros of Quadratic Functions
Find each complex conjugate.
A. 8 + 5i
B. 6i
8 + 5i
Write as a +
bi.
8 – 5i
Find a – bi.
0 + 6i
0 – 6i
–6i
Write as a +
bi.
Find a – bi.
Simplify.
Reminder: ONLY the imaginary parts of complex
conjugates are opposites.
Slide 27
Your turn
Find each complex conjugate.
A. 9 – i
C. –8i
B.
If i = 1 ,what are the values of the following? Hint: When you
square a square root, all you do is take off the square root.
1. i2
2. i4
3. i6
Slide 2
5.5 Complex Numbers
Slide 3
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.
Slide 4
Slide 5
Express the number in terms of i.
Factor out –1.
Product Property.
Simplify.
Multiply.
Express in terms of
i.
Slide 6
Express the number in terms of i.
Factor out –1.
4 6i 4i 6
Product
Property.
Simplify
.
Express in terms of
i.
Slide 7
Whiteboards
Express the number in terms of i.
Slide 8
Try this on your own.
Express the number in terms of i.
Slide 9
Whiteboards
Express the number in terms of i.
Slide 10
Solving a Quadratic Equation with Imaginary Solutions
Solve the equation.
This is just like solving x2 = 144. You take the
Square root of both sides, making sure you
consider both the positive and negative roots.
Check
x2 = –144
(12i)2 –144
144i 2 –144
–144
144(–1)
x2 = –144
(–12i)2
–144
144i 2 –144
144(–1)
–144
Slide 11
Example 2:
Solve the equation.
5x2 + 90 = 0
What happens here?
What happens?
Solve by taking square roots.
Simplify.
Check
5x2 + 90 = 0
0
5(18)i 2 +90 0
90(–1) +90 0
Slide 12
Now you try!
Solve the equation.
x2 = –36
Slide 13
Your turn
Solve the equation.
x2 + 48 = 0
Slide 14
One more
Solve the equation.
9x2 + 25 = 0
Slide 15
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.
Slide 16
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. For example, 7 can be written as 7 = 7 + 0i.
(7 is the real part a, and 0i is the complex part b).
Two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are equal.
Slide 17
• In the complex number 3x – 5yi, what is
the real part?
• What is the complex part?
Slide 18
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
Set the
real parts
equal to
each other.
Solve for
x.
10 = –4y
Set the
imaginary parts
equal to each
other.
Solve for
y.
Slide 19
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
x = –4
Equate the
real parts.
Solve for
x.
–6 = 20y
Equate the
imaginary
parts.
Solve for
y.
Slide 20
Your turn
Find the values of x and y that make each equation
true.
–8 + (6y)i = 5x – i
6
Slide 21
Find the zeros of the function.
f(x) = x2 + 10x + 26
x2 + 10x + 26 = 0
Set equal to 0.
x2 + 10x +
Rewrite.
Add
to both
sides.
= –26 +
x2 + 10x + 25 = –26 + 25
(x + 5)2 = –1
Factor.
Take square roots.
Simplify.
Slide 22
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.
Slide 23
Example 4a
Find the zeros of the function.
f(x) = x2 + 4x + 13
x2
+ 4x + 13 = 0
x2 + 4x +
= –13 +
x2 + 4x + 4 = –13 + 4
(x + 2)2 = –9
x = –2 ± 3i
Set equal to
0.
Rewrite.
Complete the square.
Factor
.
Take square
roots.
Simplify.
Slide 24
Your turn
Find the zeros of the function.
g(x) = x2 – 8x + 18
Slide 25
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.
Slide 26
Example 5: Finding Complex Zeros of Quadratic Functions
Find each complex conjugate.
A. 8 + 5i
B. 6i
8 + 5i
Write as a +
bi.
8 – 5i
Find a – bi.
0 + 6i
0 – 6i
–6i
Write as a +
bi.
Find a – bi.
Simplify.
Reminder: ONLY the imaginary parts of complex
conjugates are opposites.
Slide 27
Your turn
Find each complex conjugate.
A. 9 – i
C. –8i
B.