1. Solving Quadratic Equations

2. Review of Solving Quadratic Equations ax 2 +bx +c = 0 A quadratic equation is a 2nd degree equation whose graph is a parabola.

3. Solve with Perfect Square Binomial 1.Get Perfect Squares on Both Sides of Equation. 2.Take Square Root of Perfect Squares 3.Solve Positive Number & Neg. Number 4.Check Answers by Putting into Original Eq.

4. Solve with Perfect Square Binomial 1.Get Perfect Squares Binomial. 2.Take Square Root of Both Sides of Eq 3.Simplify 4.Check Answers by Putting into Original Eq.

5. Solve Quadratic Eq. By Factoring 1.Set Eq = 0 2.Factor 3.Set Each Factor = 0 4.Check Answers by Putting into Original Eq.

6. What If You Can’t Factor The Equation? The “X” factor procedure doesn’t work which means it can not be factored -36 4

7. You can solve this equation by Completing The Square

8. Review of a Trinomial Square (x+5) 2 = x 2 + 10x + 25 When you square a binomial you get a TRINOMIAL SQUARE

9. What is Completing The Square? x 2 + 10x + WHAT WILL MAKE A TRINOMIAL SQUARE? ?

10. Now we will use these Algebra tiles to show Completing the Square 1 x x2x2 -x-x 2

11. x 2 + 10x + ? It takes 25 ones to complete this square Note that 25 is half of b squared ( 10 / 2 ) 2 = 5 2 = 25 b

12. 2 2 Remember this to Complete The Square b

14. Finding Half of b Squared b

15. Add to Complete the Square b This is now a trinomial square

16. x 2 +6x+9 is a Trinomial Square x 2 +6x+9 x+3 (x+3) 2 =x 2 +6x+9

17. More Examples of Half of b Squared

18. More Examples of Half of b Squared

19. More Examples of Half of b Squared

20. Complete The Square and State The Binomial Square

23. Solve This Quad. Eq.

24. If the “X” factor doesn’t work, you can’t solve by Factoring -31 4

25. Solve by Completing The Square

26. 1.First Put c on One Side of equation.

27. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a Skip this step when a=1

28. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ADD HALF OF b SQUARED ( b / 2 ) 2 =( 4 / 2 ) 2 =4 b

29. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square

30. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides

31. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

32. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide equation by a 3.Complete The Square ( b / 2 ) 2 = 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

33. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 =( 8 / 2 ) 2 =4 2 = 16 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

34. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

35. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 =( 2 / 2 ) 2 =1 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

36. Solve by Completing The Square

37. 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 6 / 2 ) 2 =3 2 =9 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

38. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 9 / 2 ) 2 = 81 / 4 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

39. Solve by Completing The Square 1.First Put c on One Side of equation. 2.Divide by a 3.Complete The Square ( b / 2 ) 2 = ( 2 / 2 ) 2 =1 4.Write Trinomial Square as a Binomial Square 5.Square Root both sides 6.Solve for x (Note 2 Answers)

40. Solve Quadratic Equations With The Quadratic Formula So far we have solved quadratic equations with FACTORING (Solves Equations with Integer Solutions) & COMPLETING THE SQUARE (Solves All quadratic equations) THE QUADRATIC FORMULA ALSO SOLVES ALL QUADRATIC EQUATIONS

41. QUADRATIC EQUATION

42. - b 2 a + 2 X=

43. Solve with Quadratic Equation

44. 1.Put Equation into Std. Form (ax 2 +bx+c=0) a=1bc

45. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. a=1bc

46. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical a=1bc

47. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

48. Solve with Quadratic Equation

49. 1.Put Equation into Std. Form (ax 2 +bx+c=0) a=1bc

50. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. a=1bc

51. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical a=1bc

52. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

53. Solve by quadratic equation

54. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

55. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x a=1bc

56. Solve with Quadratic Equation 1.Put Equation into Std. Form (ax 2 +bx+c=0) 2.Plug a, b & c into the Quad. Eq. 3.Simplify Radical 4.Solve for x abc

57. The Discriminant The Discriminant equals b 2 -4ac The Discriminant is what’s under the RADICAL

58. The Discriminant The Discriminant equals b 2 -4ac 2 Solutions if positive 1 Solution if 0 0 Solutions if negative

59. The Discriminant tells you how many solutions or roots x This quadratic has two roots so the Discriminant must be positive y

60. The Discriminant tells you how many solutions or roots This quadratic has one roots so the Discriminant must be zero X

61. The Discriminant tells you how many solutions or roots This quadratic has no roots so the Discriminant must be negative X

62. Find The Number of Solutions (Roots) 1.Identify a, b & c. 2.Put values into the discriminant b 2 -4ac 3.If Pos. 2 Solutions If Zero 1 Solution If Neg. 0 Solutions The discriminant is neg. so there are no solutions

63. Determine The Number of Solutions or Roots

65. Singing the Quadratic Equation (To the melody of Pop Goes The Weasel) Minus B plus or minus the square root B squared minus 4AC All over 2A, That’s the quadratic equation. It’s the greatest equation of all time. It even has a wonderful rhyme. If you don’t know it, It would be a crime That’s the quadratic equation. Math is the most taught subject in the world So you’d better believe me Bone up to pass the SAT You won’t end up a weasel.

66. SOLVING QUADRATIC EQUATIONS

67. What Method To Use SquareRoot when there is a perfect square on one side of eq. Use “X” Factor when trinomial is factorable Use quadratic formula or complete the square if not factorable