1. M3U3D5 Warm Up Multiply: (-5 + 2i)(6 – 4i) -30 +20i + 12i – 8i 2 -30 + 32i – 8(-1) -30 + 32i + 8 -22 + 32i FOIL

2. Remember the cycle of “i” i 0 = 1 i 2 = -1 i 1 = i i 3 = -i

3. Homework Check:

5. Quadratic Formula, Discriminant, and Zeroes OBJ: Solve quadratic equations with real coefficients that have complex solutions.

6. THE QUADRATIC FORMULA 1.When you solve using completing the square on the general formula you get: 2.This is the quadratic formula! 3.Just identify a, b, and c then substitute into the formula.

7. The Quadratic Formula Derived

9. The quadratic formula is used to solve any quadratic equation. The quadratic formula is: What are a, b, and c? Standard form of a quadratic equation is: The Quadratic Formula

10. Solve the problem by factoring.

11. Solve the same problem using The Quadratic Formula

12. 1. You try.

13. 2. Another example.

14. WHY USE THE QUADRATIC FORMULA? The quadratic formula allows you to solve ANY quadratic equation, even if you cannot factor it. An important piece of the quadratic formula is what’s under the radical: b 2 – 4ac This piece is called the discriminant.

15. WHY IS THE DISCRIMINANT IMPORTANT? The discriminant tells you the number and types of answers (roots) you will get. The discriminant can be +, –, or 0 which actually tells you a lot! Since the discriminant is under a radical, think about what it means if you have a positive or negative number or 0 under the radical.

16. WHAT THE DISCRIMINANT TELLS YOU! Value of the DiscriminantNature of the Solutions Negative2 imaginary solutions Zero1 Real Solution Positive – perfect square2 Reals- Rational Positive – non-perfect square 2 Reals- Irrational

17. Example #1 Find the value of the discriminant and describe the nature of the roots (real, imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula) 1. a=2, b=7, c=-11 Discriminant = Value of discriminant=137 Positive-NON perfect square Nature of the Roots – 2 Reals - Irrational

18. Example #1- continued Solve using the Quadratic Formula

19. Solving Quadratic Equations by the Quadratic Formula Try the following examples. Do your work on your paper and then check your answers.

20. The Quadratic Formula and the Discriminant REMEMBER, the discriminant is the radicand portion of the quadratic formula (b 2 – 4ac). It is used to discriminate among the possible number and type of solutions a quadratic equation will have. b 2 – 4ac Name and Type of Solution Positive Zero Negative Two real solutions One real solutions Two complex, non-real solutions

21. The Quadratic Formula and the Discriminate b 2 – 4ac Name and Type of Solution Positive Zero Negative Two real solutions One real solutions Two complex, non-real solutions Positive Two real solutions

22. The Quadratic Formula and the Discriminate b 2 – 4ac Name and Type of Solution Positive Zero Negative Two real solutions One real solutions Two complex, non-real solutions Negative Two complex, non-real solutions

23. 25x 2 - 60x + 36 = 0 Find the zeros using the Quadratic Formula

24. Exact Solution:

25. Calculator Solution: x = 1.2 Check Intercepts!

26. Use the Quadratic Formula to solve: f(x) = 3x 2 + 2 - 4x 3x 2 - 4x + 2 = 0 Find the zeros using the Quadratic Formula…You try!

27. No real solution. Check Intercepts.

28. A real world application of The Quadratic Formula: Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 20 feet x + 2 x

29. The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 20 feet x + 2 x The Pythagorean Theorem a 2 + b 2 = c 2 (x + 2) 2 + x 2 = 20 2 x 2 + 4x + 4 + x 2 = 400 2x 2 + 4x + 4 = 400 2x 2 + 4x – 396 = 0 2(x 2 + 2x – 198) = 0

30. The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 20 feet x + 2 x The Pythagorean Theorem a 2 + b 2 = c 2 2(x 2 + 2x – 198) = 0

31. The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 20 feet x + 2 x The Pythagorean Theorem a 2 + b 2 = c 2

32. The Quadratic Formula Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 20 feet x + 2 x The Pythagorean Theorem a 2 + b 2 = c 2 28 – 20 =8 ft

33. Classwork U3D5 Packet Pages 1 & 2

34. Homework U3D5 Packet Pages 3 & 4 odds