Download 5-5 Complex Numbers and Roots

5-5
Complex Numbers and Roots
Solve the equation: x2 + 16 = 0
x2 = -16
Isolate the variable
𝑥2 =
Square root each side
−16
There is not a real solution to the square root of a
negative number.
Instead, a new set of numbers are required.
Imaginary numbers are defined as the square root
of negative numbers.
The imaginary unit: i =
Holt Algebra 2
−1
5-5
Complex Numbers and Roots
X-intercepts are defined as the solutions to
quadratic functions.
Parabolas that do not have x-intercepts have
imaginary solutions.
Holt Algebra 2
5-5
Complex Numbers and Roots
For any positive real number b,
2
2
b  b  1  bi
where i is the imaginary unit and bi
is called the pure imaginary
number.
Holt Algebra 2
5-5
i0
i1
i2
i3
i4
i5
i6
i7
Complex Numbers and Roots
Cycle of i
=1
=i
2
= −1 =-1
3
= −1 = −1
=1
=i
= -1
=-i
Holt Algebra 2
2
−1 = -i
5-5
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 Algebra 2
5-5
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 Algebra 2
Express in terms of i.
5-5
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 Algebra 2
5-5
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 Algebra 2
5-5
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 Algebra 2
x2 =
(–12i)2
144i 2
144(–1)
–144
–144
–144
–144 
5-5
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 Algebra 2
5-5
Complex Numbers and Roots
Complex numbers are numbers written in the
form: a + bi
Every complex number has a real part a and an
imaginary part b.
Two complex numbers are equal if and only if their real
parts are equal and their imaginary parts are equal.
Holt Algebra 2
5-5
Complex Numbers and Roots
Find the values of x and y that make the equation
4x + 10i = 2 – (4y)i true .
Real parts
4x + 10i = 2 – (4y)i
4x = 2
Imaginary parts
Equate the
10 = –4y Equate the
imaginary parts.
real parts.
Solve for x.
Holt Algebra 2
Solve for y.
5-5
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 Algebra 2
5-5
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 Algebra 2
5-5
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 Algebra 2
Simplify.
5-5
Complex Numbers and Roots
HW pg. 353
#’s 18-25, 36, 46-51
Holt Algebra 2