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Complex Numbers
Warm Up
• SOLVE the following polynomials by factoring:1. 2x2 + 7x + 3 = 02. 3x2 – 6x = 0
• Solve the following quadratics using the Quadratic Formula:
3. 2x2 – 7x – 3 = 04. 8x2 + 6x + 5 = 0
Today’s Objectives
• Students will be introduced to Complex Numbers
• Students will add, subtract, and multiply complex numbers
• Students will solve quadratics with complex solutions
Solve
X2 + 5 = 0
Square Roots of Negative Numbers
• Up until now, you've been told that you can't take the square root of a negative number.o
• That's because you had no numbers which were negative after you'd squared them (so you couldn't "go backwards" by taking the square root).o
Imaginary Numbers
• You actually can take the square root of a negative number, but it involves using a new number to do it.
• When this new number was invented, nobody believed that any "real world" use would be found for it, so this new number was called "i", standing for "imaginary.”
DEFINITION
• The imaginary number, i, is defined to be:
• So,
Using Imaginary Numbers
1. Simplify:
2. Simplify:
3. Simplify:
4. Simplify:
3i
5i
3i√2
-i√6
Using Imaginary Numbers
5. Simplify:
6. Simplify:
7. Simplify:
8. Simplify:
Tip: Treat i like a variable. But remember, i2 = -1
5i
11i
12i2 = -12
-6i3 = -6 * -i = 6i
Complex Numbers
• A Complex Number is a combination of:– A Real Number (1, 12.38, -0.8625, ¾, √2, 1998)– And an Imaginary Number (i)
• Complex Number:
Special Examples
Either “part” can be 0
1. (2 + 3i) + (1 – 6i)
You try:
(5 – 2i) – (–4 – i)
3 – 3i
9 - i
Multiplying
(3 + 2i)(1 – 4i)
(2 + 7i)(2 – 7i)
3 -12i +2i -8i2
3 – 10i +811 – 10i
4 -14i +14i – 49i2
4 + 49 = 53
Round Robin Practice• Do #1 on your own paper. Then pass.• Check your neighbor’s work, then do #2 on
their paper. Pass. Repeat1. 2. 3. 4. 5. 6.
Remember using the Quadratic Formula in Math 2???
• Quadratic Formula:
• This was another way to find our zeros/solutions/x-intercepts
• When the part under the square root was negative (the discriminant), what did we write as our answer?– No REAL Solution
Video example
• Quadratic Formula with complex solutions
Quadratics with Complex Solutions
• Now that you know about complex numbers, you can find solutions to ALL quadratics.
• Example: x2 – 10x + 34 = 0
You Try
• Solve 3x2 – 4x + 10 = 0
Quadratics with Complex Solutions
• Graph to tell the types of solutions
• If the discriminant (b2 – 4ac) is…
Discriminant: Positive Zero NegativeTypes of Solution(s): 2 real solutions 1 real repeated
solution2 COMPLEX Solutions
Homework
• Operations on Complex Numbers and the Quadratic Formula
