Algebra / Geometry II: Unit 2- Quadratic Equations

SUCCESS CRITERIA:

1) Write a Quadratic Equation in Standard Form and in Vertex 2) Graph a Quadratic Equation in Standard Form and Identify Key Features3) Solve a Quadratic Equation with real or imaginary roots

4) Determine the appropriate solution in a Real World example.

INSTRUCTOR: Craig Sherman Hidden Lake High School

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Westminster Public Schools

EMPOWER Recorded TARGET SCALE THEME

MA.10.EE.06.04 Create Equations that Describe Numbers/Relationships

MA.10.F.03.04 Graphing Functions

MA.10.EE.04.04 Equivalent Expressions

MA.10.F.01.04 Properties of Functions

PROFICIENCY SCALE:

SCORE REQUIREMENTS

4.0 In addition to exhibiting Score 3.0 performance, in-depth inferences and applications that go BEYOND what was taught in class.

Score 4.0 does not equate to more work but rather a higher level of performance.o Compare and contrast the graphing in Standard Form verses Vertex Form, ORo Compare and contrast the three methods for solving a Quadratic Equation

3.5 In addition to Score 3.0 performance, in-depth inferences and applications with partial success.o Determine the appropriate solution in a Real World example.

3.0 The learner exhibits no major errors or omissions regarding any of the information and processes (simple or complex) that were explicitly taught.

o Write a Quadratic Equation in Standard Form and in Vertex Form, ANDo Graph a Quadratic Equation in Standard Form and in Vertex Form identifiying axis of symmetry, vertex, y-intercept, and zeros., ANDo Graph a Quadratic Equation in Standard Form and in Vertex Form, ANDo Solve a Quadratic Equation with real or imaginary roots

2.0 Can do one or more of the following skills / concepts: There are no major errors or omissions regarding the simpler details and processes as the

learner…o Put an equation in Standard Form, ORo Put an equation in Vertex Form, ORo Graph an Equation in Standard Form identifying axis of symmetry, vertex, y-intercept, and zeros. , ORo Graph an Equation in Vertex Form identifying axis of symmetry, vertex, y-intercept, and zeros. , ORo Solve a Quadratic Equation using the factor method, ORo Solve a Quadratic Equation using the square root and complete the square method, ORo Solve a Quadratic Equation using the Quadratic Formula, ORo Determine if the solution to a Quadratic Equation has real or imaginary roots

1.0 Know and use the vocabulary

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Identify the Basic ElementsWith help, a partial understanding of some of the simpler details and process

GRAPHING & FINDING ZEROES (STANDARD FORM)

WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

variable

coefficient

constant

Standard Form

axis of symmetry

y-intercept

x-intercept

zeroes

INSTRUCTION 1: VIRTUAL NERD: HOW TO GRAPH PARABOLA in STANDARD FORM

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GIVEN: -5x - 4 = 2x2

Class Worka) Put equation in Standard Form

b) Identify a, b, and c

c) Does the graph open up or down? d) Is the graph wider or narrower than the parent

equation of y=x2?

e) What is the y-intercept?

f) Find the axis of symmetry.

g) Find the vertex.

h) Graph the equation

i) Name the zeroes

1. y= x2 +3x - 4

2. -x 2 + 5x = 6

3. y= x2 -4x -2

4. -2x2 - 3 =6x

5. y= 3x2 -4x -2

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6. y= x2 +2x -8

7. -x2 - 3x = 2

8. y= x2 -5x -1

9. -5x - 4 = 2x2

10. y= 3x2 -2x

Homeworka. Put equation in Standard Form

b. Identify a, b, and c

c. Does the graph open up or down?

d. Is the graph wider or narrower than the

parent equation of y=x2?

e. What is the y-intercept?

f. Find the axis of symmetry

g. Find the vertex.

h. Graph the equation

i. Name the zeroes

11. y = 2x2 +3x -412. y = -.7x2 -4x +313. y = -1.2x2 +614. y = 3x2 +3x15. y = -4x2

16. y =-.6x2 +3x -617. y = 1.7x2 -4x +518. y = -1.02x2 +819. y = 1.3x2 +4x20. y = 5

SOLVING: FactoringWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

Zero Product Property

INSTRUCTION 1: ZERO PRODUCT PROPERTY INSTRUCTION 2: KHAN ACADEMY TUTORIAL(factoring)

EXEMPLAR 1: EXEMPLAR 2:GIVEN: (x – 4) (x + 3) = 0 GIVEN: x2 –x – 12 = 0

STEP 1: Set each factor = o x – 4 = 0 and x + 3 = 0 STEP 1: Factor (x + 3)(x – 4) = 0

STEP 2: Solve x = 4 and x = - 3 STEP 2: Set each factor = o x + 3 = 0 and x – 4 = 0

STEP 3: Solve x = - 3 and x

= 4

Class Work

Solve for the variable:

21. ( x+3 ) ( x−2 )=022. ( x−4 ) (x−4 )=023. ( x+5 ) ( x−5 )=024. ( x−6 ) ( x+10 )=0

25. ( x ) ( x−6 )=026 (2 x−4 ) (2x+5 )=027 (3 x−7 ) (2 x+7 )=028 (4 x−8 ) (4 x+10 )=0

- - - - -

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29.a2 +4a +3= 0

31.b2 -4b -5= 0

33.-c2 -6c = -7

35. d2 +8d = -12

37.-e2 +9 = 0

30.f2 +4f +4 = 0

32.–g2 +5g = 6

34.2h2 +7h +6= 0

36.3j2 -4j = -1

38. A garden has a length of (x + 2) feet and a width of (2x - 1) feet. The garden’s total area is 88 square feet. Find the

length.

Home WorkSolve for the variable:

39. ( z+7 ) ( z−9 )=041. ( y−10 ) ( y−10 )=043. (w+8 ) (w−8 )=045. ( v−9 ) (v+1 )=0

40. (u ) (u−8 )=042. (2 t−6 ) (2t +9 )=044. (3 s−10 ) (2 s+11)=0

46. (6 r−15 ) (6 r+10 )=0- - - - -

47. a2 +6a +5= 0

48. b2 -b -6= 0

49. c2 -6c = -8

50. d2 +7d = -10

51. -e2 +16 = 0

52. f2 +6f +9 = 0

53. –g2 +7g = 6

54. 2h2 +8h +6= 0

55. 3j2 -7j = -4

56. A garden has a length of (x - 4)feet and a width of (2x +3)feet. The garden’s total area is 76 square feet. Find the

length.

SOLVING: Square Roots Method

INSTRUCTION 1: KHAN ACADEMY TUTORIAL INSTRUCTION 2: SOPHIA TUTORIAL

EXEMPLARS:GIVEN: w 2 = 9 GIVEN: 3(w +4)2 -4 = 44

STEP 1: Square Root w = ± 3 or w = 3 and – 3 STEP 1: Add 4 3(w +4)2 = 48

STEP 2: Divide by 3 (w +4)2 = 16

STEP 3: Square Root w +4 = 4 and - 4

STEP 4: Subtract 4 w = 0 and - 8

Class Work Solve by using the square roots method

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57. m2 = 16

58. n2 = 25

59. 3p2 = 12

60. 5q2 = 80

61. r2 -3 =6

62. s2 +8 =17

63. 2t2 -6 = -4

64. 3u2 +5 = 17

65. (v -7)2 -5 = 11

66. 2(w -3)2 +6 = 56

67. The square of six less than a number is twenty-

five. Write an equation that models this

situation. Solve the equation.

Homework

Solve by using the square roots method

68. m2 = 36

69. n2 = 64

70. 3p2 = 27

71. 5q2 = 20

72. r2 -3 = 13

73. s2 +8 = 24

74. 2t2 -6 = 12

75. 3u2 +5 = 8

76. (v -2)2 +4 = 13

77. 3(w +4)2 -4 = 44

78. Two times the square of five more than a

number is seventy-two. Write an equation that

models this situation. Solve the equation.

SOLVING: Completing the Square

INSTRUCTION 1: KHAN ACADEMY TUTORIAL INSTRUCTION 2: SOPHIA TUTORIAL

EXEMPLAR: GIVEN: 6n + 90 = - n2

STEP 1: Standard Form n2 + 6n + 90 = 0

STEP 2: Constant on other side n2 + 6n = - 90

STEP 3: Find the square (n+ 3)2 = - 90 + 9

(n+ 3)2 = - 81

STEP 4: Square Root n + 3 = ± 9i

STEP 5: Solve for Variable n = - 3 ± 9i

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Class WorkSolve by completing the square.

79. h2 + 6h =16

80. j2 - 8j = -7

81. k2 + 9 = -10k

82. m2 - 13 = 12m

83. 14n + 20 = -n2

84. 8p + p2 = 0

85. 2q2 - 8q = 40

86. 3r2 + 36r = 12

87. A toy rocket launched into the air has a height (h

feet) at any given time (t seconds) as h = -16t2 +

96t until it hits the ground. At what time(s) is it at

a height of 7 feet above the ground?

HomeworkSolve by completing the square.

88. h2 + 4h =12

89. j2 - 10j = -9

90. k2 + 13 = -14k

91. m2 - 21 = 20m

92. 2n + 80= -n2

93. 6p + p2 = 0

94. 2q2 - 12q = -22

95. 3r2 + 15r = 18

96. A toy rocket launched into the air has a height (h

feet) at any given time (t seconds) as

h = -16t2 + 160t until it hits the ground. At what

time(s) is it at a height of 9 feet above the

ground?

SOLVING: Quadratic Formula & the DiscriminantWORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

Quadratic Formula

INSTRUCTION 1: KHAN ACADEMY TUTORIAL INSTRUCTION 2: YOUTUBE TUTORIAL

EXAMPLARS:

GIVEN: g2 -5g +3 =0 GIVEN: 3z – 6z2 = -8

STEP 1: Standard Form g2 -5g +3 =0 STEP 1: Standard Form – 6z2 +3z + 8 =0

STEP 2: Substitute −(−5)±√(−5)2−4 (1)(3)

2(1)STEP 2: Substitute

−(3)±√(3)2−4 (−6)(8)2(−6)

STEP 3: Calculate 5¿±√13¿¿ ¿2 STEP 3: Calculate −3 ±√153

−12

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5± 3.6

2−3 ±12.4

−125+336

2 and

5−3362

−3+12.4

−12 and

5−12.4−12

8.62

and 1.42

9.4

−122 and

1.42

STEP 4: Solutions 4.3 and 0.7 -0.8 and 0.6

WORD or CONCEPT DEFINITION or NOTES EXAMPLE or GRAPHIC REPRESENTATION

Discriminant

EXAMPLAR:

GIVEN: 7x – 6 = 2x2

STEP 1: Standard Form 0 = 2x2 – 7x + 6

STEP 2: Substitute (- 7)2 – 4(2)(6)

D > 0 (positive) D = 0 D < 0 (negative) STEP 3: Calculate 1

2 Real Roots 1 Real Root No Real Roots STEP 4: Roots 2 Real Roots

INSTRUCTION 1: DISCRIMINANT TUTORIAL

Class WorkSolve the quadratic formula. Round answers to hundredth place.

109.

97. x2 +8x -6 =0

98. g2 -4g +2 =0

99. 3d2 + 4d -3 =0

100. -2m2 + 3m = 1

101. 4w2 -8 = 5w

102. 7z – 9z2 = -4

103. An employee makes (2x + 3) dollars an hour for x hours. If the employer wants to pay no more than $120 a day,

what is the maximum number of hours the employee can work? (Round to the nearest quarter hour)

Homework

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Solve the quadratic formula. Round answers to hundredth place.

104. x2 +7x -5 =0

105. g2 -5g +3 =0

106. 2d2 + 5d -3 =0

107. -3m2 + 4m = -5

108. 5w2 -2 = 5w

109. 3z – 6z2 = -8

110. An employee makes (3x - 5) dollars an hour for x hours. If the employer wants to pay no more than $200 a day,

what is the maximum number of hours the employee can work? (Round to the nearest quarter hou

Mixed Application ProblemsClass WorkSolve the following problems using any method.

111. The product of two consecutive positive integers is 272, find the integers.

112. The product of two consecutive positive even integers is 528, find the integers.

113. The product of two consecutive odd integers is 255, find the integers.

114. Two planes leave airport at the same time (from different runways). If three hours later they are 500 miles apart

and the plane flying south has traveled 200 miles farther, how far did the one flying west travel?

115. Two cars leave a gas station at the same time, one traveling north and one traveling east. One hour later they are

80 miles apart and the one traveling east went 10 miles farther, how far is it from the gas station?

116. A square has its length increased by 4 feet and its width by 5 feet. If the resulting rectangle has an area of 132

square feet what was the perimeter of the original square?

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117. A rectangular parking lot has a width 30 feet more than its length. The owners are able to increase the width by

20 feet and the length by 40. The new lot has an area of 27,200 square feet, what is the area of the original lot?

HomeworkSolve the following problems using any method.

118. The product of two consecutive positive integers is 272, find the integers.

119. The product of two consecutive positive even integers is 342, find the integers.

120. The product of two consecutive odd integers is 483, find the integers.

121. Two planes leave airport at the same time (from different runways). If three hours later they are 600 miles apart

and the plane flying south has traveled 100 miles farther, how far did the one flying west travel?

122. Two cars leave a gas station at the same time, one traveling north and one traveling east. One hour later they are

90 miles apart and the one traveling east went 15 miles farther, how far is it from the gas station?

123. A square has its length increased by 6 feet and its width by 8 feet. If the resulting rectangle has an area of 239.25

square feet what was the perimeter of the original square?

124. A rectangular parking lot has a width 20 feet more than its length. The owners are able to increase the width by

20 feet and the length by 40. The new lot has an area of 7225 square feet, what is the area of the original lot?

SOLVING Quadratics summary

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FACTOR

set = 0

factor trinomial

set each factor = 0

solve for X

SQUARE ROOT

set = 0

square root each side

answer is both + and -

solve for X

COMPLETE the SQUARE

move constant to other side

add to both sides

square root each side

answer is both + and -

solve for X

QUADRATIC FORMULA

set = 0

plug-in a,b,c

calculate

Quadratic Equations Unit Review

Multiple Choice– Choose the correct answer for each question. No partial credit will be given.

1. Comparing the graph of y = 5x2 + 4x - 2 to its parent function, it:A) opens down and is wider than the parent

function graph.B) opens down and is narrower than the parent

function graph.

C) opens up and is wider than the parent function graph.

D) opens up and is narrower than the parent function graph.

2. What is the equation of the axis of symmetry of y = -3x2 - 12x – 5?A) x = -2B) x = -4

C) x = 4D) x = 2

3.  What a r e t h e vertex a n d a x i s of s y m m e t r y o f t h e parabola?

y = x2 + 4x + 3?

A) vertex: (2,−1); axis of symmetry: x = 2 B) vertex: (2,1); axis of symmetry: x = 2C) vertex: (−2,−1); axis of symmetry: x = −2 D) vertex: (−2,1); axis of symmetry: x = −2

4.What is the y- intercept of y = -2x2 + 2x – 3?

A) (0 , 5)B) B (3 , 0)

C) C (0 , -3)D) D (-3, 0)

5.Which graph(s) has more than one zero?

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6.Which of the following is a step in solving y = 2x2 - x - 3 by the Factoring Method?A) 2x + 1 = 0 or x + 3 = 0B) 2x - 3 = 0 or x + 1 = 0

C) 2x + 3 = 0 or x - 1 = 0D) 2x - 1 = 0 or x - 3 = 0

7.The solution to (x + 2)2 = 16 isA) 14 and -18B) -14 and 18

C) -6 and 2D) -2 and 6

8.What value goes in the blank to complete the square: x2 - 6x + ___?A) -9B) -36

C) 36D) 9

9.What is the discriminant of 2x2 + 6x + 2 = 0?A) 2B) 6

C) 20D) 28

10. How many real zeros does an equation have if the discriminant is -4?A) 0B) 1

C) 2D) Not enough information

11. Solve 3x2 + 5x + 1 =0

X

A)

B)

C)

12. Given the height of a rocket as h = -16t2 + 160t + 896, where t is in seconds. At what time t, does the rocket hit the ground?

A) - 14 and 4 secondsB) 14 seconds

C) - 4 secondsD) the rocket will not hit the ground

13. Solve 3x2 + 7x + 4 = 0

A) - 1 and - B) 1 and C) -7 and - 6

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D) 7 and 6

Short Constructed Response – Write the correct answer for each question.

14. What value goes in the blank to complete the perfect square trinomial: x2 + __x + 36?

15. Solve 6x2 + 13x + 6 = 0

16. Solve (3x- 7)(x+3)=0

Extended Constructed Response – Solve the problem, showing all work.

17. A rectangle has a length 6 more than its width. If the width is decreased by 2 and the length decreased by 4, the resulting rectangle has an area of 21 square units. What is the length of the original rectangle? What is ratio of the original rectangle’s area to the new rectangle’s area? What is the perimeter of the new rectangle?

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