1. Chapter 4 Quadratics 4.7 Solving Quadratic Equations: The Quadratic Formula

2. Humour Break

3. 4.7 Solving Quadratic Equations: The Quadratic Formula Goals for Today: (1)We can use the quadratic formula to find the zeros of any hard-to-factor quadratic equation in standard form (2)In the quadratic formula, a portion of the formula called the discriminant can be used to tell us whether there are one, two, or no zeros (3)We can find the zeros of a quadratic equation in vertex form using algebra instead of expanding and then factoring

4. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros When we have a quadratic equation in standard form, we learned how to find the zeros by factoring (Chapter 3) When we have a quadratic equation in vertex form, we could expand the equation and put it into standard form & then factor it to find the zeros

5. 4.7 Solving Quadratic Equations: The Quadratic Formula Recall from Chapter 3 that there are times where we couldn’t find the factors of a trinomial because they didn’t factor evenly to integers This didn’t necessarily that there are no factors – but it did mean that there were no factors that we could discover using the butterfly method which is really an organized way of doing guess & check

6. 4.7 Solving Quadratic Equations: The Quadratic Formula At that time I said there is a way...

7. 4.7 Solving Quadratic Equations: The Quadratic Formula Where we have a trinomial in standard form y = ax² + bx + c

8. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 1: Identify what “a”, “b” & “c” are in the trinomial equation Ex.1x² - 4x - 1

9. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 1: Identify what “a”, “b” & “c” are in the trinomial equation Ex.1x² - 4x – 1 a is 1 b is - 4 c is - 1

10. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 2: Plug a, b & c into the quadratic formula Ex.1 a = 1, b = – 4 & c = - 1

11. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 2: Plug a, b & c into the quadratic formula Ex. 1 a = 1, b = – 4 & c = - 1

12. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: Solve... Ex. 1

13. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: Solve... Ex. 1

14. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: (Cont’d... ) Solve... Ex. 1

15. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: (Cont’d... ) Solve... Ex. 1

16. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: (Cont’d... ) Solve... Ex. 1

17. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: (Cont’d... ) Solve... Ex. 1

18. 4.7 Solving Quadratic Equations: The Quadratic Formula Step 3: (Cont’d... ) Solve... Ex. 1

19. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant We can use a portion of the quadratic formula to tell us whether or not a quadratic equation has 1, 2 or no zeros without solving for the zeros

20. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant

21. If b² - 4ac is > 0... there are 2 zeros Ex. y = -2x² + 8x - 1

22. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is > 0... there are 2 zeros Ex. y = -2x² + 8x - 1

23. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is > 0... there are 2 zeros Ex. y = -2x² + 8x – 1 a = -2, b = 8, c = -1 (Identify a, b & c) b² - 4ac (discriminant form.) = 8² - 4(-2)(-1) = 64 – 8 = 56 (+ve so 2 zeros)

24. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is = 0... there is 1 zero Ex. y = -2x² + 8x - 8

25. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is = 0... there is 1 zero Ex. y = -2x² + 8x - 8

26. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is = 0... there is 1 zero Ex. y = -2x² + 8x – 8 a = -2, b = 8, c = -8 (Identify a, b & c) b² - 4ac (discriminant form.) = 8² - 4(-2)(-8) = 64 – 64 = 0 (so 1 zero)

27. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is < 0... there are no zeros Ex. y = 2x² + 8x +15

28. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is < 0... there are no zeros Ex. y = 2x² + 8x +15

29. 4.7 Solving Quadratic Equations: The Quadratic Formula - Discriminant If b² - 4ac is < 0... there are no zeros Ex. y = 2x² + 8x +15 a = 2, b = 8, c = 15 (Identify a, b & c) b² - 4ac (discriminant form.) = 8² - 4(2)(15) = 64 – 120 = -56 (so no zeros)

30. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Ex. y = 2(x – 3)² - 2 Find the zeros...

31. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros y = 2(x – 3)² - 2 y = 2(x² - 6x – 6x + 9) – 2... expand using FOIL y = 2(x² - 6x + 9) – 2 y = 2x² - 12x + 18 – 2 y = 2x² - 12x + 16.... now in standard form y = 2(x² - 6x + 8)... factor our 2 (GCF) y = 2(x – 4)(x – 2)... butterfly chart So, the zeros are 4 and 2

32. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros

33. Ex. y = 2(x – 3)² - 2 Find the zeros... Step 1: Set y = 0 & move the “k” or the “- 2” to the left hand side of the equation

34. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Ex. y = 2(x – 3)² - 2 Find the zeros... Step 1: Set y = to 0 & move the “- 2” to the left hand side (where it becomes a positive 2) 2 = 2(x – 3)²

35. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 2: Divide the left & right-hand side by 2 2 = 2(x – 3)²

36. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 2: Divide the left & right-hand side by 2 2 = 2(x – 3)² -- ---------- 2

37. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 3: Take the square root of both sides 1 = (x – 3)²

38. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 3: Take the square root of both sides* √1 = √(x – 3)² *When you take the square root of the constant, you have to recognize that it could be either + 1 or -1 (i.e. 1 x 1 = 1 & -1 x -1 also = 1)

39. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 4: Use algebra to solve for x +1 = x – 3 or -1 = x – 3

40. 4.7 Solving Quadratic Equations: The Quadratic Formula – Finding The Zeros Step 4: Use algebra to solve for x +1 = x – 3 or -1 = x – 3 1 + 3 = xor-1 + 3 = x x = 4or x = 2

41. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. A digital sensor records the path of Rachel’s soother after she throws it into the air. The equation that models the situation is y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground?

42. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex.

43. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491

44. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491

45. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491

46. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491

47. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491

48. 4.7 Solving Quadratic Equations: The Quadratic Formula – Movement Ex. y = -4.9x² + 20.58x + 0.491 When does the soother hit the ground? a = -4.9, b = 20.58 & c = 0.491 So, the soother hits the ground after 4.22 seconds

49. Homework Tuesday, May 24 th p.403, #4, 5, 6, 11 odd Wednesday, May 25 th p.404, #16, 17, 18 & 19