1. Algebra 2: Unit 5 Continued
Factoring Quadratic expression

2. Factors
Factors are numbers or expressions that you multiply to get another number or expression. Ex. 3 and 4 are factors of 12 because 3x4 = 12

3. Factors
What are the following expressions factors of? 1. 4 and 5? 2. 5 and (x + 10) 3. 4 and (2x + 3) 4. (x + 3) and (x - 4) 5. (x + 2) and (x + 4) 6. (x – 4) and (x – 5)

4. GCF
One way to factor an expression is to factor out a GCF or a GREATEST COMMON FACTOR. EX: 4x2 + 20x – 12 EX: 9n2 – 24n

5. Try Some!
Factor:
9x2 +3x – 18
7p2 + 21
4w2 + 2w

6. Factors of Quadratic Expressions
When you multiply 2 binomials:
(x + a)(x + b) = x2 + (a +b)x + (ab)
This only works when the coefficient for x2 is 1.

7. Finding Factors of Quadratic Expressions
When a = 1: x2 + bx + c Step 1. Determine the signs of the factors Step 2. Find 2 numbers that’s product is c, and who’s sum is b.

8. Sign table! 2nd sign + Same - Different 1st sign + or - (x+ )(x+ )
Question
2nd sign +
Same -
Different
1st sign
+ or -
Answer
(x+ )(x+ )
(x - )(x - )
(x + )(x - ) OR
(x - )(x + )

9. Examples
Factor: 1. X2 + 5x x2 – 10x x2 – 6x – x2 + 4x – 45

10. Examples
Factor: 1. X2 + 6x x2 – 13x x2 – 5x – x2 – 16

11. Slide Factor Divide Reduce More Factoring! When a does NOT equal 1.
Steps
Slide
Factor
Divide
Reduce

12. Example!
Factor: 1. 3x2 – 16x + 5

13. Example!
Factor: 2. 2x2 + 11x + 12

14. Example!
Factor: 3. 2x2 + 7x – 9

15. Try Some!
Factor 1. 5t2 + 28t m2 – 11m + 15

16. March 20th Warm Up
Find the Vertex, Axis of Symmetry, X-intercept, and Y-intercept for each:
y = x2 + 8x + 9
y= 2(x – 3)2 + 5

17. Quadratic Equations

18. Quadratic Equation
Standard Form of Quadratic Function: y = ax2 + bx + c Standard Form of Quadratic Equation: 0 = ax2 + bx + c

19. Solutions
A SOLUTION to a quadratic equation is a value for x, that will make 0 = ax2 + bx + c true. A quadratic equation always have 2 solutions.

20. 5 ways to solve
There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

21. SOLVING BY FACTORING

22. Solve by factoring; 2x2 – 11x = -15

23. Solve by factoring; x2 + 7x = 18

24. Factoring
Solve by factoring; 1. 2x2 + 4x = x2 – 8x = 0 3. x2 – 9x + 18 = 0

25. Solving by Finding Square Roots
For any real number x; X2 = n x = Example: x2 = 25

26. Solve
Solve by finding the square root; 5x2 – 180 = 0

27. Solve
Solve by finding the square root; 4x2 – 25 = 0

28. Try Some!
Solve by finding the Square Root: 1. x2 – 25 = 0 2. x2 – 15 = x2 – 14 = (x – 4)2 = 25

29. Quadratic Equations
Solving by Graphing

30. Warm Up March 21st
A model for a company’s revenue is R = -15p p + 12,000 where p is the price in dollars of the company’s product. What PRICE will maximize the Revenue? What is the maximum revenue? Convert to vertex form: y = 2x2 + 6x - 8

31. 5 ways to solve
There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

32. Solving by Graphing
For a quadratic function, y = ax2 +bx + c, a zero of the function, or where a function crosses the x-axis, is a solution of the equations ax2 + bx + c = 0

33. Examples
Solve x2 – 5x + 2 = 0

34. Examples
Solve x2 + 6x + 4 = 0

35. Examples
Solve 3x2 + 5x – 12 = 8

36. Examples
Solve x2 = -2x + 7

37. Complex Numbers

38. Quick Review
Simplifying Radicals If the number has a perfect square factor, you can bring out the perfect square. EX:

39. Try Some

40. Try this:
Solve the following quadratic equations by finding the square root:
4x = 0
What happens?

41. Complex Numbers

42. Imaginary Number: i
The Imaginary number This can be used to find the root of any negative number. EX

43. Properties of i
This pattern repeats!!

44. Graphing Complex Number

45. Absolute Values

46. Operations with Complex Numbers
The Imaginary unit, i, can be treated as a variable Adding Complex Number EX: (8 + 3i) + ( i)

47. Try Some!
7 – (3 + 2i)
(4 – 6i) + 3i

48. Operations with Complex Numbers
Multiplying Complex Numbers: Example: (5i)(-4i) Example: (2 + 3i)(-3 + 5i)

49. Try Some!
(6 – 5i)(4 – 3i)
(4 – 9i)(4 + 3i)

50. Now we can SOLVE THIS!
Solve 4x = 0

51. Absolute Values

52. Completing the Square

53. Warm Up
Factor each Expressions

54. 5 ways to solve
There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

55. Solving a Perfect Square Trinomial
We can solve a Perfect Square Trinomial using square roots.
A Perfect Square Trinomial is one with two of the same factors!
X2 + 10x + 25 = 36

56. Solving a Perfect Square Trinomial
X2 – 14x + 49 = 81

57. What if it’s not a Perfect Square Trinomials?!
If an equation is NOT a perfect square Trinomial, we can use a method called COMPLETING THE SQUARE.

58. Completing the Square
Using the formula for completing the square, turn each trinomial into a perfect square trinomial.

59. Solving by Competing the Square
Solve by completing the square: X2 + 6x + 8 = 0

60. Solving by Competing the Square
Solve by completing the square: X2 – 12x + 5 = 0

61. Solving by Competing the Square
Solve by completing the square: X2 – 8x + 36 = 0

62. Solving Quadratic Equations

63. Warm Up Write in Vertex Form: y = 2x2 + 6x – 8 Simplify |2i + 4|
Simplify (3i – 2)(5i + 3)

64. Solve by Factoring
2x2 – x = 3 x2 + 6x + 8 = 0

65. Solve by Finding the Square Root
5x2 = x = 0

66. Solve by Graphing
X2 + 5x + 3 = 0 3x2 – 5x – 4 = 0

67. Solve by Completing the Square
X2 – 3x = 28 x2 + 6x – 41 = 0

68. 5 ways to solve
There are 5 ways to solve quadratic equations: Factoring Finding the Square Root Graphing Completing the Square Quadratic Formula

69. Quadratic Formula
The Quadratic Formulas is our final way to Solve! It works when all else fails!

70. Examples
2x2 + 6x + 1 = 0

71. Examples
X2 – 4x + 3 = 0 3x2 + 2x – 1 = 0 X2 = 3x – 1 8x2 – 2x – 3 = 0

72. Discriminant

73. Discriminant

74. Discriminant
IF the Discriminant is POSITIVE then there are 2 REAL solutions IF the Discriminant is ZERO then there is ONE REAL solution IF the Discriminant is NEGATIVE then there are 2 IMAGINARY solutions.

75. Using the Discriminant
The weekly revenue for a company is: R = -3p2 + 60p , where p is the price of the company’s product. Use the discriminant to find whether there is a price the company can sell their product to reach a maximum revenue of $1500?