Warm UP

1. Warm UP • How many real roots does the following quadratic have? 3x² - 4x + 9 = 0 (-4)² -4(3)(9) -92 NO REAL ROOTS

2. The Quadratic Formula • We already know how to find the number of x-intercepts a quadratic has……right? • The Discriminant • b² - 4ac For example 5x² + 3x + 4 + 0 a. 5 b. 3 c. 4 (3)² - 4(5)(4) = - 71 NONE Since the discriminant is negative, how many real roots does the quadratic have?

4. The Quadratic Formula • The discriminate will tell you how many x-intercepts the quadratic has • The quadratic formula will tell you what they are!

5. The Quadratic Formula • Intro: We already know the standard form of a quadratic equation is: y = ax2 + bx + c • The constants are: a , b, c • The variables are: y, x

6. What it means • The ROOTS (or solutions) of a polynomial are its x-intercepts • Recall: The x-intercepts occur where y = 0.

7. The Easy Way • Example: Find the roots: y = x2 + x - 6 • Solution: Factoring: y = (x + 3)(x - 2) • 0 = (x + 3)(x - 2) • The roots are: • x = -3; x = 2

8. The Challenge • But what about NASTYtrinomials that don’t factor?

9. The Formula • After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor!

10. Example

11. Solution!

12. Check your answer! Plug in your answers for x. If you’re right, you’ll get y = 0.

13. Your turn

14. Tricks of the Trade Remember: All the terms must be on one side BEFORE you use the quadratic formula. • Example: Solve 3m2 - 8 = 10m • Solution:3m2 - 10m - 8 = 0 • a = 3, b = -10, c = -8

15. Your turn! • Solve: 3x2 = 7 - 2x • Solution: 3x2 + 2x - 7 = 0 • a = 3, b = 2, c = -7