1. Complex Numbers

2. Warm Up SOLVE the following polynomials by factoring: 2x2 + 7x + 3 = 0
Solve the following quadratics using the Quadratic Formula:
2x2 – 7x – 3 = 0
8x2 + 6x + 5 = 0

3. 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

4. Square Roots of Negative Numbers
Up until now, you've been told that you can't take the square root of a negative number.
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).

5. 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.”

6. DEFINITION
The imaginary number, i, is defined to be:
So,

7. Using Imaginary Numbers
Simplify:

8. Using Imaginary Numbers
Simplify:
Tip: Treat i like a variable. But remember, i2 = -1

9. Complex Numbers A Complex Number is a combination of: Complex Number:
A Real Number (1, 12.38, , ¾, √2, 1998)
And an Imaginary Number (i)
Complex Number:

10. Special Examples
Either “part” can be 0

11. Real
Imaginary
You try:
(2 + 3i) + (1 – 6i)
(5 – 2i) – (–4 – i)

13. Round Robin Practice Do #1 on your own paper. Then pass.
Check your neighbor’s work, then do #2 on their paper. Pass. Repeat

14. Remember using the Quadratic Formula in Math 2???
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

15. Quadratics with Complex Solutions
Now that you know about complex numbers, you can find solutions to ALL quadratics.
Example: x2 – 10x + 34 = 0

16. You Try
Solve 3x2 – 4x + 10 = 0

17. Quadratics with Complex Solutions
Use the discriminant to determine the type and number of solutions.
If the discriminant (b2 – 4ac) is…
Discriminant:
Positive
Zero
Negative
Types of Solution(s):
2 real solutions
1 real solution
2 COMPLEX Solutions

18. Homework
Operations on Complex Numbers and the Quadratic Formula