Download Complex Numbers - Leon County Schools

Warm-Up #53
11/4/16
Solve the following quadratic equation.
1. x2 + 4x = 3
Complex Numbers
Determine the number of real solutions.
Section 4-8
11/4/16
2. 2x2 - 3x + 7 = 0
3. x2 = 6x + 5
EQ: What are complex numbers? Why are they based
on a number whose square is -1?
Finding Square Roots
Complex Numbers
Solve each quadratic function by finding square roots.
The complex numbers are based on a number called the
imaginary unit i, whose square is -1.
1. 7x2 - 63 = 0
x=±3
two real solutions
and
2
2. 5x + 125 = 0
x2 = -25
no real solutions → imaginary
solutions
Example
5x2 + 125 = 0
Simplify a Number Using i
Complex Numbers
Write the following using the imaginary unit i.
Complex numbers can be written in the form a + bi, where
a and b are real numbers.
1.
3.
2.
4.
If b = 0, the number a + bi is a
real number.
If a = 0 and b ≠ 0, the number a + bi is a pure imaginary
number.
Warm-Up #54
11/7/16
Solve each quadratic equation.
Graphing Complex Numbers
3. 2x2 - 3x + 5 = 0
In the complex number plane, the point (a, b) represents
a + bi.
Example
2 + 3i
The absolute value of a
complex number is its
distance from the origin in
the complex plane.
Graphing Complex Numbers
Adding & Subtracting Complex Numbers
Graph each number and find its distance from the origin.
We can also add and subtract complex numbers.
1. -5 + 3i
To do so, we combine the real parts and the imaginary
parts separately, similar to combining like-terms.
1. 3x2 + 24 = 0
2. x2 + 4x = -5
2. 6i
3. 5 - i
1. (7 - 2i) + (-3 + i)
3. (5 - 3i) - (-2 + 4i)
2. (4 - 3i) + ( -4 + 3i)
4. (8 + 6i) - (8 - 6i)
Multiplying Complex Numbers
Dividing Complex Numbers
What is each product?
We can use complex conjugates to simplify quotients of
complex numbers.
Complex Conjugate
1. (3i)(-5 + 2i)
2. (4 + 3i)(-1 - 2i)
3. (6 + 2i)(6 - 2i)
What is each quotient?
Complex Conjugate
(a + bi)(a - bi) = a2 + b2
1.
2.
(a + bi)(a - bi)
3.
Finding Imaginary Solutions
We can solve any quadratic equation now using complex
numbers.
1. x2 + 72 = 0
2. 3x2 - x + 2 = 0
3. x2 - 4x + 5 = 0