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Transcript
```Warm-Up #53
11/4/16
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
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
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
```
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