1. Imaginary and Complex Numbers (5.4) Roots when the Discriminant is negative

2. First, a little POD Find the vertex and roots of x 2 - 10x + 34 = 0.

3. First, a little POD x 2 - 10x + 34 = 0 What else are the roots called? What are the real solutions to the POD? What does that mean about the x-intercepts of the parabola y = x 2 - 10x + 34?

4. First, a little POD x 2 - 10x + 34 = 0 What does that mean about the x-intercepts of the parabola y = x 2 - 10x + 34?

5. Imaginary Numbers When we have the square root of negative numbers, we are dealing with imaginary numbers (as you know). The basics:i = √-1. i 2 = -1 i 3 = -i i 4 = 1 So, i 5 = what? See the pattern? What does i 55 equal?

6. Imaginary numbers What do the roots of the POD look like using imaginary number notation?

7. Complex numbers Complex numbers are a combination of real and imaginary numbers in the form a + bi, where a is the real component and bi is the imaginary component. (What happens if a = 0? If b = 0? See why all numbers are complex numbers?)

8. Complex numbers If a + bi is one complex number, then a - bi is its complex conjugate. What is the complex conjugate of each of the following? 6 - 7i5 + 3i -46i m - 9i-q + ri

9. Complex numbers Complex conjugates have a special relationship. To find it, just FOIL them. (6 - 7i)(6 + 7i) = (5 + 3i)(5 - 3i) = (m - 9i)(m + 9i) = (-q + ri)(-q - ri) = What happens to the i-terms?

10. Complex numbers Graphing complex numbers is straightforward, but we do it on a special coordinate plane. Measure the real component along the horizontal axis and the imaginary along the vertical axis. Graph each of the following: 6 - 7i-8i 5 + 3i6 -2 - 4i-4 0-9 + i