Module 1( final 2) quadtraic equations and inequalities jq

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Module 1( final 2) quadtraic equations and inequalities jq

  1. 1. Regional Training of MathematicsTeachers for Grade 9 of The K to 12 Enhanced Basic Education Program May 15-19, 2014 Notre Dame of Marbel University Koronadal City
  2. 2. AIRLINE
  3. 3. ASSESSMENT
  4. 4. QUADRATIC
  5. 5. SQUARE ROOT
  6. 6. FACTORING
  7. 7. INEQUALITY
  8. 8. SUGGESTIONS
  9. 9. QUADRILATERALS
  10. 10. PROVING
  11. 11. Grade 9 Mathematics Quarter I – First Grading Period Module 2 – Quadratic Functions Module 1 – Quadratic Equations and Inequalities
  12. 12. Quarter II – Second Grading Period Module 4 – Zero Exponents, Negative Integral Exponents, Rational Exponents and Radicals Module 3 – Variations Grade 9 Mathematics
  13. 13. Quarter III – Third Grading Period Module 6 – Similarity Module 5 – Quadrilaterals Grade 9 Mathematics
  14. 14. Quarter IV – Fourth Grading Period Module 7 – Triangle Trigonometry Grade 9 Mathematics
  15. 15. Curriculum Guide Legend Sample: M9AL-Ic-d-1 M9 AL I c-d 1 Math 9 Algebra Quarter 1 Week 3-4 Competency 1
  16. 16. Curriculum Guide Legend Sample: M9AL-IIg-2 M9 AL II g 2 Math 9 Algebra Quarter 2 Week 7 Competency 2
  17. 17. Domain/Component Code Number Sense NS Geometry GE Patterns and Algebra AL Measurement ME Statistics and Probability SP
  18. 18. MODULE 1 QUADRATIC EQUATIONS AND INEQUALITIES LIVE C. ANGGA
  19. 19. Module 1 QUADRATIC EQUATIONS AND INEQUALITIES LM -pages 1 – 118 TG- pages 1 – 78 CG-pages - 11
  20. 20. Quadratic Equations, Quadratic Inequalities, and Rational Algebraic Equations Illustrations of Quadratic Equations Solving Quadratic Equations Nature of Roots of Quadratic Equations Sum and Product of Roots of Quadratic Equations Extracting Square Roots Extracting Square Roots Extracting Square Roots Extracting Square Roots M O D U L E M A P
  21. 21. Equations Transformable to Quadratic Equations Applications of Quadratic Equations and Rational Algebraic Equations Quadratic Inequalities Rational Algebraic Equations Illustrations of Quadratic Inequalities Solving Quadratic Inequalities Application of Quadratic Inequalities M O D U L E M A P
  22. 22. Group Number Module 1 Activity Group Lessons 1 Group Lesson 2a Group Lesson 2b Group Lesson 2c Group Lesson 2d Group Lesson 3 Group Lesson 4 Group Lesson 5 Group Lesson 6 Group Lesson 7 Group Assignments
  23. 23. Lessons Coverage and its Objective: Lesson I. Illustrations of Quadratic Equations Objective: * Illustrate Quadratic Equations
  24. 24. Lessons Coverage and Objective: Lesson 2- Solving Quadratic Equations Extracting Square Roots Factoring Completing the Square Quadratic Formula Objective: * Solve Quadratic Equations by: a. Extracting square roots b. factoring c. completing the squares d. using quadratic formula
  25. 25. Lesson 3. Nature of roots of Quadratic Equations Objective: * characterize the roots of a quadratic equation using the discriminant.
  26. 26. Lesson 4: Sum and Product or Roots of Quadratic Equations Objective: describe the relationship between the coefficient and the roots of a quadratic equation
  27. 27. Lesson 5 : Equations Transformable to Quadratic Equations ( Including Rational Algebraic Equations) Objective: * solve equations transformable to quadratic equations (Including rational algebraic equations)
  28. 28. Lesson 6 : Applications of Quadratic Equations and Rational Algebraic Equations Objective: * solve problems involving quadratic equations and rational algebraic equations.
  29. 29. Lesson7 : Quadratic Inequalities Objective: * Illustrate quadratic inequalities * solve quadratic inequalities and * solve problems involving quadratic inequalities
  30. 30. Pretest • Group Activity: 5 groups Group 1 – answer item 1-7 Group 2 – answer item 8-14 Group 3 – answer item 15-21 Group 4 – answer item nos. 22-28 Group5 – answer Part II items 1-7 Lesson No. Topic What to Know What to Process What to Reflect What to Transf er Tot al 1 Illustrations of Quadratic Equations ( 1,2,3) = 3 (4,5,6) = 3 ( 7) = 1 ( 8 ) = 1 8
  31. 31. Lesson No. Topic What to Know What to Process What to Reflect What to Transf er Tot al Lesson 2A Solving Quadratic Equations by Extracting the Square Roots (1,2,3,4,5) = 5 ( 6,7) = 2 ( 8, 9, ) = 2 ( 10) = 1 10
  32. 32. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 2B Solving Quadratic Equations by Factoring ( 1,2,3) = 3 (4,5) = 2 ( 6) = 1 (7) = 1 7
  33. 33. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 2C Solving Quadratic Equations by Completing the Square (1,2,3,4) = 4 ( 5,6) = 2 ( 7) = 1 ( 8 ) = 1 8
  34. 34. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 2D Solving Quadratic Equations by Using Quadratic Formula ( 1,2,3,4) = 4 ( 5,6 ) = 2 ( 7 ) = 1 ( 8 ) = 1 8
  35. 35. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 3 The Nature of the Roots of a Quadratic Equation (1,2,3,4, 5,6) = 6 ( 7,8) = 2 ( 9 ) = 1 ( 10 ) = 1 10
  36. 36. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 4 The Sum and the Product of Roots of Quadratic Equations (1,2,3,4) = 4 ( 5,6) = 2 ( 7 , 8) = 2 ( 9 ) = 1 9
  37. 37. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 5 Equations transformable to Quadratic Equations (1,2,3) = 3 ( 4,5,6,7) = 4 ( 8 ) = 1 ( 9 ) = 1 9
  38. 38. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 6 Solving Quadratic Equations by Using Quadratic Formula (1,2,3) = 4 (4) = 3 (5) = 1 ( 6,7) = 2 7
  39. 39. Lesson No. Topic What to Know What to Process What to Reflect What to Transfer Total 7 Quadratic Inequalities ( 1,2,3) = 3 ( 4,5,6, 7,8) = 5 ( 9) = 1 ( 10 ) = 1 10
  40. 40. Let’s do the Activity …
  41. 41. Norms to Follow During the Presentation of Outputs A – accurate (exact) B – Brief (short duration) C – Concise (using only few words clearly stated) D – direct (easy to understand or respond to)
  42. 42. Time Frame 20 minutes – simultaneous group preparation 10 minutes - group presentation 5 minutes - interaction
  43. 43. Lesson 1 –Illustrations of Quadratic Equations • What to know Activity I: Do you Remember these Products? Answer the Questions Refer: LM. pp.11 TG. pp. 14 Activity 2 :Another Kind of Equation Answer the Questions Refer: LM. pp12 TG. Pp.14 Activity 3: A Real Step to Quadratic Equations Refer: LM. pp12 TG. pp. 15
  44. 44. Continuation of Lesson 1 • What to Process Activity 4: Quadratic or Not Quadratic Refer: LM. pp.14 TG. pp. 16 Activity 5: Does it Illustrate Me? Refer: LM. pp.14 TG. pp. 16 Activity 6 : Set Me to Your Standard Refer: LM. pp.15 TG. pp. 17
  45. 45. Continuation of Lesson 1 • What to Reflect or Understand Activity 7: Dig Deeper Refer: LM. pp.16 TG. pp. 18 • What to Transfer Activity 8 Where in the Real World Refer: LM. pp.18 TG. pp. 18 * Summary/Synthesis/Generalization Refer: LM pp. 17 TG pp. 18
  46. 46. Abstraction Quadratic Equations in one variable is a mathematical sentence of degree 2 that can be written in the general form: ax2+bx+c =0 • A quadratic equation is an equation equivalent to one of the form • Where a, b, and c are real numbers and a  0 a is the quadratic coefficient b is a linear coefficient c is the constant term or free term Include in the standard form: ax2 = 0 ax2+bx = 0 ax2+c = 0 Note: If a = 0 can’t be a quadratic equation
  47. 47. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  48. 48. Lesson 2A –Solving Quadratic Equations by Extracting Square Roots • What to Know Activity 1. Find My Roots Refer: LM pp.18 TG pp. 19 Activity 2 . What Would Make A Statement True? Refer: LM pp.18 TG pp. 19 Activity 3. Air Out!!! Refer: LM pp.18 TG pp. 19 Activity 4. Learn to Solve Quadratic Equations!!! Refer: LM pp. 20 TG pp. 20 Activity 5 . Anything Real or Nothing Real? Refer: LM pp. 20 TG pp. 21
  49. 49. Continuation…. Lesson 2A • What to Process Activity 6: Extract Me!!! LM pp. 23 TG pp. 21 Activity 7 : What Does a Square Have? LM pp. 24 TG pp. 22 • What to Reflect and Further Understand Activity 8: Extract Then Describe Me! LM pp. 25 TG pp. 22 Activity 9: Intensify your Understanding LM pp. 25 TG pp. 22
  50. 50. Continuation…. Lesson 2A What to Transfer: With activity 10.in the TG: What More can I do? LM pp. 23 TG pp. 26 Summary/ Synthesis/Generalization LM pp. 26 TG pp.23
  51. 51. Abstraction: a. Extracting Square Roots • An alternate method of solving a quadratic equation is using the Principle of Taking the Square Root of Each Side of an Equation If x2 = a, then x = + a
  52. 52. Ex 1: Solve by taking square roots 5(x – 4)2 = 125 First, isolate the squared factor: 5(x – 4)2 = 125 (x – 4)2 = 25 Now take the square root of both sides: 25)4( 2 x 254 x x – 4 = + 5 x = 4 + 5  x = 4 + 5 = 9 and x = 4 – 5 = – 1
  53. 53. Lesson 2B Solving Quadratic Equations by Factoring • What to Know Activity 1: What Made Me? LM pp. 27 TG pp. 24 Activity 2: The Manhole LM pp. 28 TG pp. 24 Activity 3: Why is the Product Zero? LM pp. 28 TG pp. 25
  54. 54. Continuation…Lesson 2BSolving Quadratic Equations by Factoring • What to Process Activity 4 : Factor Then Solve! LM pp. 31 TG pp. 25 Activity 5: What Must be My Length and Width? LM pp. 32 TG pp. 26 • What to Reflect and Further Understand Activity 6. How well Did I Understand? LM pp. 33 TG pp. 26 • What to Transfer Activity 7. Meet My Demands!!! ( TG) LM pp. 34 TG pp 27 * Summary/ synthesis/ Generalization
  55. 55. b. Factoring • Ex 1: Solve x2 + 5x + 4 = 0 Quadratic equation  factor the left hand side (LHS) x2 + 5x + 4 = (x + )(x + )1  x2 + 5x + 4 = (x + 4)(x + 1) = 0 Now the equation as given is of the form ab = 0  set each factor equal to 0 and solve x + 4 = 0 x + 1 = 0 x = – 4 x = – 1 Solution: x = - 4 and –1 x = {-4, -1} 4
  56. 56. Ex 2: Solve x2 -10x = - 25 Quadratic equation but not of the form ax2 + bx + c = 0 x2 - 10x + 25 = (x - )(x - )5 5  x2 - 10x + 25 = (x - 5)(x - 5) = 0 Now the equation as given is of the form ab = 0  set each factor equal to 0 and solve x - 5 = 0 x = 5 x - 5 = 0 x = 5 Solution: x = 5  x = { 5}  repeated root Quadratic equation  factor the left hand side (LHS) Add 25  x2 – 10x + 25 = 0
  57. 57. Ex 3: Solve 5x2 = 4x Quadratic equation but not of the form ax2 + bx + c = 0 5x2 – 4x = x( )5x – 4  5x2 – 4x = x(5x – 4) = 0 Now the equation as given is of the form ab = 0  set each factor equal to 0 and solve x = 0 5x – 4 = 0 5x = 4 Solution: x = 0 and 4/5  x = {0, 4/5} Quadratic equation  factor the left hand side (LHS) Subtract 6x  5x2 – 4x = 0 x = 4/5
  58. 58. Explain: Zero property and Factoring procedure: Zero Property = If the product of two real numbers is zero, then either of the two is equal to zero or both numbers are equal to zero. Procedure: 1. Transform quadratic equation into standard form if necessary. 2. Factor the quadratic expression 3. Apply the zero property by setting ach factor of the quadratic expression equal to zero 4. Solve each resulting equation. 5. Check the values of the variable obtained by substituting each in the original equation.
  59. 59. Lesson 2C. Solving Quadratic Equations by Completing the Square • What to Know Activity 1: How Many Solutions Do I have? LM pp. 35 TG pp. 28 Activity 2: Perfect Square Trinomial to Square of a Binomial LM pp. 36 TG pp. 29 Activity 3: Make it Perfect LM pp. 37 TG pp. 29 Activity 4. Finish the Contract LM pp. 37 TG pp. 29
  60. 60. Continuation: Lesson 2C. • What to Process Activity 5. Complete Me! LM pp. 42 TG . 30 Activity 6. Represent then Solve! LM pp. 43 TG. Pp. 33 • What to Reflect and Further Understand Activity 7 . What Solving Quadratic Equations by Completing the Square Means to Me… LM pp. 44 TG pp. 31
  61. 61. Continuation: Lesson 2C. • What to Transfer: Activity 8. Design Packaging Boxes LM pp 45 TG pp. 32 • Summary/ Synthesis/Generalization LM pp. 46 TG pp. 32
  62. 62. Completing the Square • Recall from factoring that a Perfect-Square Trinomial is the square of a binomial: Perfect square Trinomial Binomial Square x2 + 8x + 16 (x + 4)2 x2 – 6x + 9 (x – 3)2 • The square of half of the coefficient of x equals the constant term: ( ½ * 8 )2 = 16 -----------------64/4 =16 [½ (-6)]2 = 9 ------------------36/4 = 9
  63. 63. • Write the equation in the form x2 + bx = c • Add to each side of the equation [½(b)]2 • Factor the perfect-square trinomial x2 + bx + [½(b)] 2 = c + [½(b)]2 • Take the square root of both sides of the equation • Solve for x
  64. 64. Further explanation: • Quadratic equation ax2+bx+c = 0 can be transformed into (x-h)2=k where k≥0. • K should not be negative.. Why? • Explain how to transform general form to standard or vertex form.
  65. 65. Steps in completing the square: LM page 38 1. Divide both sides of the equation by a then simplify. 2. Write the equation such that the terms with variables are on the left side of the equation and the constant term is on the right side. 3. Add the square of one-half of the coefficient of x on both sides of the resulting equation. The left side of the equation becomes a perfect square trinomial. 4. Express the perfect square trinomial on the left side of the equation as a square of the binomial. 5. Solve the resulting quadratic equation by extracting the square root. 6. Solve the resulting linear equation. 7. Check the solutions obtained against the original equation.
  66. 66. Lesson 2D. Solving Quadratic Equations by Using Quadratic Formula • What to Know Activity 1: It’s Good to be Simple! LM pp. 47 TG pp 33 Activity 2 Follow the Standard LM pp. 48 TG pp. 34 Activity 3. Why do the Gardens Have to be Adjacent? LM pp.48 TG pp. 35 Activity 4 Lead Me to the Formula LM pp. 49 TG pp. 35
  67. 67. Continuation…Lesson 2D. • What to Process Activity 5: Is the Formula Effective? LM pp. 52 TG pp. 36 Activity 6. Cut to Fit! LM pp. 52 TG pp. 36 • What to Reflect and Further Understand Activity 7 : Make the Most Out of It! LM pp. 53 TG pp. 37 • What to Transfer: Activity 8. Show Me the Best Floor Plan? LM pp. 55 TG pp. 38 • Summary/ Synthesis/Generalization LM pp. 55 TG pp38
  68. 68. Abstraction: d. The Quadratic Formula • Consider a quadratic equation of the form ax2 + bx + c = 0 for a nonzero • Completing the square 2 ax bx c   2 b c x x a a    2 2 2 2 2 b b c b x x a 4a a 4a     
  69. 69. The Quadratic Formula Solutions to ax2 + bx + c = 0 for a nonzero are 2 2 2 b b 4ac x 2a 4a        2 b b 4ac x 2a     2 2 2 2 2 2 b b 4ac b x x a 4a 4a 4a     
  70. 70. Ex: Use the Quadratic Formula to solve x2 + 7x + 6 = 0 Recall: For quadratic equation ax2 + bx + c = 0, the solutions to a quadratic equation are given by a2 ac4bb x 2   Identify a, b, and c in ax2 + bx + c = 0: a = b = c =1 7 6 Now evaluate the quadratic formula at the identified values of a, b, and c
  71. 71. )1(2 )6)(1(477 x 2   2 24497 x   2 257 x   2 57 x   x = ( - 7 + 5)/2 = - 1 and x = (-7 – 5)/2 = - 6 x = { - 1, - 6 }
  72. 72. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  73. 73. Lesson 3. The Nature of the Roots of a Quadratic Equation • What to Know Activity 1. Which are Real? Which are Not? LM pp. 56 TG pp.39 Activity 2: Math in A,B,C? LM pp . 57 TG pp. 40 Activity 3: Math My Value? LM pp. 57 TG 40 Activity 4: Find my Equation and Roots LM pp. 58 TG pp. 40 Activity 5: Place Me on the Table LM pp.58 TG pp. 41 Activity 6: Let’s Shoot that Ball! LM pp. 59 TG pp. 41
  74. 74. Continuation….Lesson 3 • What to Process Activity 7: What is My Nature? LM pp. 42 TG pp. 62 Activity 8: Lets Make a Table! LM pp. 63 TG pp. 43 • What to Reflect and Further Understand Activity 9: How Well Did I Understand the Lesson? LM pp. 63 TG pp. 43 • What to Transfer: Activity 10 . Will It or Will It Not? LM pp. 64 TG PP. 44 • Summary/ Synthesis/Generalization Lm PP. 65 TG PP. 44
  75. 75. Abstraction: Explain Discriminant and its nature of roots. It is the value of the expression b2-4ac of the quadratic equation ax2+bx+c = 0; It describes the nature of the roots of the quadratic equation ; It can be: * zero * positive and perfect square * positive but not perfect square * negative
  76. 76. Nature of Roots LM: Page 59-61 Value of D Nature of Roots Roots D=0 Real and equal Each root = to – b/2a D˃0 and a perfect square rational and are not equal {-b+√D/2a} D˃0 but not perfect square Irrational and are not equal {-b+√D/2a} D˂0 No real roots none
  77. 77. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  78. 78. Lesson 4. The Sum and the Product of Roots of Quadratic Equations • What to Know Activity 1: Let’s Do Addition and Multiplication! ML pp. 66 TG pp.45 Activity 2: Find My Roots! LM pp. 67 TG pp. 45 Activity 3: Relate Me to My Roots LM pp. 67 TG pp. 46 Activity 4 : What the Sum and Product Mean to Me.. LM pp. 68 TG pp. 46
  79. 79. Continuation…..Lesson 4. • What to Process Activity 5: This is My Sum and this is My Product. Who Am I? LM pp. 71 TG pp. 47 Activity 6. Here Are the Roots. Where is the Trunk? LM pp. 72 TG pp. 48 * What to Reflect and Further Understand Activity 7. Fence My Lot!! LM pp. 73 TG pp. 48 Activity 8. Think of These Further! LM pp. 74 TG pp 49
  80. 80. Continuation…..Lesson 4. • What to Transfer: • Activity 9: Lets Make a Scrap Book! LM pp. 75 TG pp. 49 • Summary/ Synthesis/Generalization LM pp. 76 TG pp. 49
  81. 81. Abstract • Solving quadratic equations by factoring,
  82. 82. Consider the general quadratic equation: where Multiply to create a leading coefficient of 1: Represent the roots of the equation as and :
  83. 83. Comparing the equations, it can be seen that: or and Our investigation reveals that there is a definite relationship between the roots of a quadratic equation and the coefficient of the second term and the constant term. The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term divided by the leading coefficient. The product of the roots of a quadratic equation is equal to the constant term divided by the leading coefficient.
  84. 84. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  85. 85. Lesson 5. Equations Transformable to Quadratic Equations • What to Know Activity 1: Let’s Recall LM pp. 77 TG pp. 50 Activity 2: Let’s Add and Subtract! LM pp. 77 TG pp. 50 Activity 3: How Long Does It Take To Finish Your Job? LM pp. 78 TG pp. 51
  86. 86. Lesson 5. Equations Transformable to Quadratic Equations ( Continuation) • What to Process Activity 4: View Me in Another Way! LM pp. 83 TG pp. 51 Activity 5: What Must be The Right Value? LM pp. 83 TG pp. 52 Activity 6: Let’s Be True! LM pp. 84 TG pp. 52 Activity 7: Let’s Paint the House! LM pp. 84 TG pp. 52
  87. 87. Lesson 5. Equations Transformable to Quadratic Equations ( Continuation) What to Reflect and Further Understand Activity 8: My Understanding of Equations Transformable to Quadratic LM pp. 85 TG pp. 53 • What to Transfer: Activity 9: A Reality of Rational Algebraic Equation LM pp. 86 TG pp. 53 • Summary/ Synthesis/Generalization LM pp. 87 TG pp. 53
  88. 88. Abstraction An equation is said to be in a quadratic form if its original variable is in the highest degree of 2. Example: ax2+bx+c = 0 is said to be a quadratic form because the variable x has a highest degree of 2.
  89. 89. Example: By factoring • Solve: x2-34x+ 225 = 0 Solution: (x-9) (x-25) = 0 (x-9) = 0 and (x-25) = 0 x = 9 x = 25 Solution set : {9, 25}
  90. 90. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  91. 91. Lesson 6. Solving Quadratic Equations by Using Quadratic Formula • What to Know Activity 1: Find My Solution! LM pp. 88 TG pp. 54 Activity 2: Translate into…. LM pp. 88 TG pp. 54 Activity 3: What are my Dimensions? LM pp. 89 TG pp. 55 • What to Process Activity 4 : Let Me Try LM pp. 92 TG56
  92. 92. Lesson 6. Solving Quadratic Equations by Using Quadratic Formula ( Continuation) • What to Reflect and Further Understand Activity 5: Find Those Missing! LM pp. 93 TG pp. 56 • What to Transfer: Activity 6: Let’s Draw! LM pp. 94 TG pp. 57 Activity 7: Play the Role of … LM pp. 94 TG pp. 57 • Summary/ Synthesis/Generalization LM PP. 95 tg PP. 57
  93. 93. Abstraction Quadratic Formula: For The solutions of some quadratic equations, ( ), are not rational, and cannot be obtained by factoring. Note: The quadratic formula can be used to solve ANY quadratic equation, even those that can be factored.
  94. 94. By factoring (this equation is factorable): By Quadratic Formula: a = 1, b = 2, c = -8 Hints: Be careful with the signs of the values a, b and c. Don't drop the sign when substituting into the formula. Also remember your rules for multiplying and adding signed numbers as you solve the formula. MSJC ~ San Jacinto Campus Math Center Workshop Series
  95. 95. Hints: Remember that a negative value under the radical is creating an imaginary number (a number with an i). Example 2: This equation cannot be solved by factoring. By Quadratic Formula: a = 1, b = 4, c = 5 MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur
  96. 96. Example 3. This equation cannot be solved by factoring. By Quadratic Formula: a = 3, b = -10, c = 5 Hints: Notice how the value for b was substituted into the formula using parentheses (-10). This helps you to remember to deal with the negative value of b. Also, notice how the (-10)2 is actually a positive value. When you square a value, the answer is always positive. If needed, these answers can be estimated as decimal values, such as (rounded to 3 decimal places): x = 2.721; x = 0.613 The radical answers are the "exact" answers. The decimal answers are "approximate" answers. MSJC ~ San Jacinto Campus Math Center Workshop Series Janice Levasseur
  97. 97. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  98. 98. Lesson 7. Quadratic Inequalities • What to Know Activity 1 : What Makes Me True? LM pp. 96 TG pp. 58 Activity 2 Which are Not Quadratic Equations? LM pp. 97 TG pp. 59 Activity 3: Let’s Do Gardening LM pp. 97 TG pp. 59
  99. 99. Lesson 7. Quadratic Inequalities ( Continuation) • What to Process Activity 4: Quadratic Inequalities or Not? LM pp. 106 TG pp. 60 Activity 5. Describe My Solutions! LM pp. 107 TG pp. 60 Activity 6: Am I a Solution or Not? LM pp. 107 TG pp. 61 Activity 7: What Represents Me? LM pp. 108 TG pp. 62 Activity 8: Make It Real! LM pp. 110 TG pp. 62
  100. 100. Lesson 7. Quadratic Inequalities ( Continuation) • What to Reflect and Further Understand Activity 9: How Well I Understood… LM pp. 111 TG pp. 63- 65 • What to Transfer: Activity 10: Investigate Me! LM pp. 112 TG pp. 66 Activity 11. How Much Would It Cost to Tile a Floor? LM pp. 112 TG pp. 66 • Summary/ Synthesis/Generalization LM pp. 114 TG pp. 66
  101. 101. Abstraction Quadratic inequalities can be solved graphically or algebraically. The graph of an inequality is the collection of all solutions of the inequality. The trick to solving a quadratic inequality is to replace the inequality symbol with an equal sign and solve the resulting equation. The solutions to the equation will allow you to establish intervals that will let you solve the inequality. Plot the solutions on a number line creating the intervals for investigation. Pick a number from each interval and test it in the original inequality. If the result is true, that interval is a solution to the inequality.
  102. 102. Example 1 (one variable inequality): Answer: x < -3 or x > 4
  103. 103. Example 2 (two variable inequality): • Begin by graphing the corresponding equation . • (Use a dashed line for < or > and a solid line for < or >.) • Test a point above the parabola and a point below the parabola into the original inequality. Shade the entire region where the test point yields a true result. • The parabola graph was drawn using a solid line since the inequality was "greater than or equal to". • The point (0,0) was tested into the inequality and found to be true. • The point (0,-2) was tested into the inequality and found to be false. • The graph was shaded in the region where the true test point was located. ANSWER: The shaded area (including the solid line of the parabola) contains all of the points that make this inequality
  104. 104. When you solved quadratic equations, you created factors whose product was zero, implying either one or both of the factors must be equal to zero. When solving a quadratic inequality, you need to take more options into consideration. Consider these two different problems
  105. 105. Solving a quadratic inequality From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive) or below the x-axis (negative). So we can see from the graph the interval or intervals where the inequality is positive. But how can we find this out without graphing the quadratic? We can simply test the intervals around the zeros in the quadratic inequality and determine which make the inequality true.
  106. 106. Solving a quadratic inequality For the quadratic inequality, we found zeros 3 and –2 by solving the equation . Put these values on a number line and we can see three intervals that we will test in the inequality. We will test one value from each interval. 062 xx 062  xx -2 3
  107. 107. Solving a quadratic inequality Interval Test Point Evaluate in the inequality True/False  2,  3,2  ,3     06639633 2      066416644 2  3x 0x 4x True True False 062  xx 062  xx 062  xx (0)2- (0)-6= 0-0-6=-6˃0
  108. 108. Example 2: Solve First find the zeros by solving the equation, 0132 2  xx 0132 2  xx 0132 2  xx    0112  xx 01or012  xx 1or 2 1  xx
  109. 109. Example 2: Now consider the intervals around the zeros and test a value from each interval in the inequality. The intervals can be seen by putting the zeros on a number line. 1/2 1
  110. 110. Forms of Quadratic Inequalities y<ax2+bx+c y>ax2+bx+c y≤ax2+bx+c y≥ax2+bx+c • Graphs will look like a parabola with a solid or dotted line and a shaded section. • The graph could be shaded inside the parabola or outside.
  111. 111. Steps for graphing 1. Sketch the parabola y=ax2+bx+c (dotted line for < or >, solid line for ≤ or ≥) ** remember to use 5 points for the graph! 2. Choose a test point and see whether it is a solution of the inequality. 3. Shade the appropriate region. (if the point is a solution, shade where the point is, if it’s not a solution, shade the other region)
  112. 112. Example: Graph y ≤ x2+6x- 4 3 )1(2 6 2      a b x * Vertex: (-3,-13) * Opens up, solid line 134189 4)3(6)3( 2  y 9-5- 12-4- 13-3- 12-2- 9-1- yx •Test Point: (0,0) 0≤02+6(0)-4 0≤-4 So, shade where the point is NOT! Test point
  113. 113. Graph: y>-x2+4x-3 * Opens down, dotted line. * Vertex: (2,1) 2 )1(2 4 2       a b x 1384 3)2(4)2(1 2   y y * Test point (0,0) 0>-02+4(0)-3 0>-3 x y 0 -3 1 0 2 1 3 0 4 -3 Test Point
  114. 114. Last Example! Sketch the intersection of the given inequalities. 1 y≥x2 and 2 y≤-x2+2x+4 • Graph both on the same coordinate plane. The place where the shadings overlap is the solution. • Vertex of #1: (0,0) Other points: (-2,4), (-1,1), (1,1), (2,4) • Vertex of #2: (1,5) Other points: (-1,1), (0,4), (2,4), (3,1) * Test point (1,0): doesn’t work in #1, works in #2. SOLUTION!
  115. 115. Application • Journal Writing/ Self-Reflection: • I realize that I need to do the following in order to improve the delivery of the lessons in ____________. • ___________________________________________ • ___________________________________________ • ___________________________________________
  116. 116. Thoughts to Remember • Speak 6 lines to yourself everyday: 1. I am blessed 2. I can do it 3. I am a winner 4. Today is my day 5. God is always with me and 6. I am a child of God Be a blessing with others committed in sharing knowledge, skills and abilities ,nurturing learners, promoting better education. God bless us all

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