1. Real and Complex Roots 20 October 2010

2. What is a root again?

3. Review: Algebraic Definition of Root(s) The value or values of x for which the equation equals 0.

4. Review: How we solve for the root(s) algebraically Step 1a: If given an equation, convert the equation into standard form. Step 1b: If given an equation in the form y = replace y with 0. Step 1c: If given an expression, set the expression equal to 0. Step 2: Solve for x.

5. Graphical Definition of Root(s) The value or values of x at which the equation intersects (crosses) the x-axis.

6. How we identify the root(s) of an equation graphically: Step 1: Convert the equation into the form y = Step 2: Graph the equation. Step 3: Identify the values of x where the graph crosses the x- axis.

7. Example: y = x 2 - 4

8. Your Turn: For the problems below, solve the root(s) of the following equations. Confirm your answers graphically. 1. y = x 2 – 3x + 22. -10 = x 2 + 2x – 9

9. y = x 2 + 4 Complex Roots

10. Your Turn: For the problems below, solve the root(s) of the following equations. Confirm your answers graphically. 1. y = x 2 – 3x + 22. y = x 2 + 36 3. y = x 2 + 6x + 94. y = x 2 + 2x + 25

11. Classifying Roots, Part I A quadratic equation can have one of the following three types of roots: 2 unique, complex roots 2 unique, real roots 1 unique, real root (This includes cases where there is only one value for x or the same value for x repeats twice.) We use the discriminant to classify roots.

12. Your Friend, the Discriminant In the Quadratic Formula, the discriminant is the value of the radical.

13. Classifying Roots, Part II If the discriminant is positive, then there are 2 unique, real roots. If the discriminant is zero, then there is 1 unique, real root. If the discriminant is negative, then there are 2 unique, complex roots. The roots are complex conjugates of each other.

14. Find Your Group: 1. Each of you will be given a card with a quadratic equation on it. Don’t write on the card! 2. Calculate the value of your descriminant. (Don’t forget to simplify!) 3. Find the other members of your group that have the same value for the discriminant. 4. Sit together in a group. 5. There will be groups of 3 and groups of 4.

15. With Your Group, Classify the roots (2 real, 1 real, or 2 complex) using the discriminant: 1. y = 4x 2 – 12x + 42. y = -x 2 + 25 3. y = 12x 2 – 1444. y = x 2 + 196 5. y = 4x 2 + 646. y = x 2 + 10x + 25