1. Complex Numbers

2. Warm Up SOLVE the following polynomials by factoring: 1.2x 2 + 7x + 3 = 0 2.3x 2 – 6x = 0 Solve the following quadratics using the Quadratic Formula: 3.2x 2 – 7x – 3 = 0 4.8x 2 + 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. Solve X 2 + 5 = 0

5. 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

6. 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.”

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

8. Using Imaginary Numbers 1.Simplify: 2.Simplify: 3.Simplify: 4.Simplify: 3i 5i 3i√2 -i√6

9. Using Imaginary Numbers 5.Simplify: 6.Simplify: 7.Simplify: 8.Simplify: Tip: Treat i like a variable. But remember, i 2 = -1 5i 11i 12i 2 = -12 -6i 3 = -6 * -i = 6i

10. 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:

11. Special Examples Either “part” can be 0

12. Video time! Operations with Complex Numbers

13. 1.(2 + 3i) + (1 – 6i) You try: (5 – 2i) – (–4 – i) 3 – 3i 9 - i

14. Multiplying (3 + 2i)(1 – 4i) (2 + 7i)(2 – 7i) 3 -12i +2i -8i 2 3 – 10i +8 11 – 10i 4 -14i +14i – 49i 2 4 + 49 = 53

15. Round Robin Practice Do #1 on your own paper. Then pass. Check your neighbor’s work, then do #2 on their paper. Pass. Repeat 1. 2. 3. 4. 5. 6.

16. 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

17. Video example Quadratic Formula with complex solutions

18. Quadratics with Complex Solutions Now that you know about complex numbers, you can find solutions to ALL quadratics. Example: x 2 – 10x + 34 = 0

19. You Try Solve 3x 2 – 4x + 10 = 0

20. Quadratics with Complex Solutions Graph to tell the types of solutions If the discriminant (b 2 – 4ac) is… Discriminant:PositiveZeroNegative Types of Solution(s):2 real solutions1 real repeated solution 2 COMPLEX Solutions

21. Homework Operations on Complex Numbers and the Quadratic Formula