Complex Numbers

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

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. 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).

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 Simplify:

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

Complex Numbers A Complex Number is a combination of: Complex Number: 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

(2 + 3i) + (1 – 6i) You try: (5 – 2i) – (–4 – i)

Multiplying (3 + 2i)(1 – 4i) (2 + 7i)(2 – 7i)

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

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

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 Negative Types of Solution(s): 2 real solutions 1 real repeated solution 2 COMPLEX Solutions

Homework Operations on Complex Numbers and the Quadratic Formula