Integrated Math 2Unit Topic: Quadratic Equations and Functions

General Instructions for Parents: This packet contains review topics to help fortify your student’s learning for this course. Students may feel free to use their textbook, prior notes, and other resources used during the year. This packet of materials is meant to support student learning that has already taken place. It will not be counted--toward or against a student’s grades. Do not have your student spend more than one hour on any one day’s assignment.

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Day Lesson Topic Student Activities

Day 1 Intro to Factoring Pages 2-3

Day 2 Factoring Quadratic Equations Pages 4-7

Day 3 Factoring Out GCFs; Difference of Squares Pages 8-10

Day 4 Graphing Quadratic, Part 1 Pages 11-13

Day 5 Vertices and Axes of Symmetry Pages 14-17

Day 6 Finding Solutions Pages 18-21

Day 7 Using the Discriminant and the Quadratic Formula Pages 22-26

Day 8 Graphing Quadratic Equations, Part 2 Pages 27-33

Day 9 Picking Methods and Solving Word Problems Pages 34-40

Day 10 Wrap It Up: Quadratics Pages 41-42

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Intro to Factoring

Set 1: Look at the numbers in these “diamonds.” a. b. c. d.

1. How are the numbers on the sides related to the numbers at the top and bottom? Explain the relationship you see.

Set 2: Using the diamond relationship you have discovered, fill in the missing numbers in the diamonds below.

a. b. c. d.

e. f. g. h.

Set 3: Using the diamond relationship you have discovered, fill in the missing numbers in the diamonds below.

a. b. c. d.

e. f. g. h.

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7

4

12

3

6

5

5

1

5

8

–24

–3

0

–7

–49

7

2 5 4 2 –4 2 4 –2

5 7 –5 7–5 –7 3 –3

14

12

24

–7

6

–6

–7

10

–5

3

–40

8

14

9

45

0

6

–36

8

5

15

–10

–6

24

Set 4: Fill in the missing numbers.

a. b. c. d.

e. f. g. h.

Set 5: Fill in the missing numbers.

a. b. c. d.

e. f. g. h.

i. j. k. l.

m. n. o. p.

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6

–2 5

–10

9

27

–1

8

–2

208

16

4

36 6

3

9

20

1

–20

–12

27

0

–81

12

20

–8

16

13

42

18

81

9

14

1

–30

–12

–28

0

–9

12

–28

–10

21

–13

42

18

80

(a)(c)

a

Quadratic Equations by Graphing and Factoring

The standard form of a quadratic equation is f(x) = ax2 + bx + c.The graph of a quadratic equation is a parabola.

ax2 + bx + c.The polynomial can be factored if there are two factors of (a)(c) whose sum is b.

To factor:

____ ____ a b

Example 1: Find 2 factors that wecan multiply to be 6, but add to be 7.

Example 2: Find 2 factors that wecan multiply to be -15, but add to be -2.

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To determine the x-intercepts of the parabola, we can solve the quadratic equation by factoring or quadratic formula.

The 2 factors that we can

multiply to be 6 but add to be 7

are _______ and ______ .

The 2 factors that we can multiply

to be -15 but add to be -2 are

_______ and ______ .

Example 3: Find 2 factors that wecan multiply to be -56, but add to be -1.

Example 4: Find 2 factors that wecan multiply to be 9, but add to be -10.

Example 5: Find 2 factors that wecan multiply to be -60, but add to be 7.

Example 6: Find 2 factors that wecan multiply to be 10, but add to be -11.

Example 7: Factor

f(x) = 2x2 + 7x + 3

f(x) = 2x2 + 7x + 3 in factored form is f(x) =( )( ).

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Example 8: Writef(x) = 5x2 - 7x + 2in factored form and find the solutions.

Example 9: Writey = 12x2 +7x +1 in factored form and find the solutions.

Example 10: Writef(x) = x2 - 12x + 27 in factored form and find the solutions.

Example 11: Write y = x2 – 9 in factored form and find the solutions.

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Factoring GCF and a = 15 x4+10 x3−240 x2

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

3 x5−21x 4+30x3

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

4 x7−64 x5

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

−3 x5+27x3

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

4 x2−32x+28

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

7 x2+21x−28

Factor out GCF:

Identify your new : a = b = c =

Make the diamond and find the factors:

Factored form: ________( )( )

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Factoring a > 1

2x² + 7x + 3 7x² + 8x + 1

3x² + 10x+ 7 3x² + x -10

3x² + 2x-8 4x² - 5x -6

8x² -14x +3 2 + 7x + 6x²

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Factoring GCF and Difference of Squares

What is the complete factorization of 54−6 z2?

A −6 (3+ z )(3−z )

B 6(3+ z )(3−z )

C −6 (3+ z )2

D 6(3−z )2

What is the complete factorization of 27−3 z2?

A −3(3+z )(3−z )

B 3(3+z )(3−z )

C −3(3+z )2

D 3(3−z )2

What is the complete factorization of5−5 z2?

A −5(1+z )(1−z )

B 5(1+z )(1− z )

C −5(1+z )2

D 5(1−z )2

What is the complete factorization of 4−4 z2?

A −4(1+z )(1−z )

B 4 (1+z )(1−z )

C −4(1+z )2

D 4 (1−z )2

What is the complete factorization of32−2 z2?

A −2( 4+z )( 4−z )

B 2(4+z )( 4−z )

C −2( 4+z )2

D 2(4−z )2

What is the complete factorization of16−z2?

A −(4+z )(4−z )

B (4+z )( 4−z )

C −(4+z )2

D (4−z )2

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Pick 2 points from your table, find the rate of change.

Pick another 2 points from your table, find the rate of change.

Are they the same or different? Why or why not?

Pick 2 points from your table, find the rate of change.

Pick another 2 points from your table, find the rate of change.

Are they the same or different? Why or why not?

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Graphing a Parabola Using a TableGraph each parabola using the table of x values provided. Don’t forget to follow

PEMDAS!!!

1¿ f (x )=x2−4 x+1f( )¿()2−4()+1

2¿ y=−3 x2+6 x+4y=−3()2+6 ( )+¿4

Pick 2 points from your table, find the rate of change.

Pick another 2 points from your table, find the rate of change.

Are they the same or different? Why or why not?

Pick 2 points from your table, find the rate of change.

Pick another 2 points from your table, find the rate of change.

Are they the same or different? Why or why not?

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3¿ f (x)=x2+6 x+5f( )¿()2+6()+5

4 ¿ y=2 x2+8 x−3y=2()2+8()−3

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Practice Sheet Vertex of a parabola

(− (b )2(a )

, y)1. What is the vertex of the graph of y = -x2- 6x + 5?

a = b = c=

x=−( ____ )2( ___ )

y=−( ___ )2−6( ___ )+5

Vertex: (_____ , ____)

2.What are the coordinates of the vertex of y = x2 + 2x + 2?

a = b = c=

x=−( ____ )2( ___ )

y=( ___ )2+2( ___ )+2

Vertex: (_____ , ____)

3. What are the coordinates of the vertex of y = - x2 - 4x?

a = b = c=

x=−( ____ )2( ___ )

y = - (____)2 - 4(___)

Vertex: (_____ , ____)

4. What is the vertex of the graph of y = 2x2 - 8x + 8?

a = b = c=x=−

( ____ )2( ___ )

y=2(___ )2−8( ___ )+8

Vertex: (_____ , ____)

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Finding the Vertex Writing Assignment

1. Explain in words how to find the vertex of a quadratic equation. How do you know if the vertex is a maximum or minimum?

2. Find the vertex: y=−x2−4 x+5

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Graph, Vertex Concept ConnectionA rocket team is using simulation software to create and study water bottle rockets.The team begins simulating the launch of a rocket without a parachute.The table gives the data for one rocket design.

A) Graph the data and connect the points. B) Does the function have a maximum or minimum?

What does that value represent (in words).

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Time (s) Height (m)

0 0 1 34.32 58.83 73.54 78.45 73.56 58.87 34.38 0

Identifying Solutions Given an Equation Practice Sheet1. Which quadratic function, when graphed,

has x intercepts of 3 and -7?

A) y = (x – 3)(x + 7) B) y = (x + 3)(x + 7) C) y = (x – 3)(x - 7) D) y = (x + 3)(x – 7)

2. Which quadratic function, when graphed, has solutions 1 and – 5?

A) (x + 1)(x + 5) B) (x – 1)(x – 5) C) (2x+10)(x – 1) D) (x + 1)(x - 5)

3. Which quadratic function, when graphed, has roots -8 and – 2?

A) (x + 8)(x – 2) B) (x – 8)(x – 2) C) (4x + 32)(2x + 4) D) (4x – 32)(x + 2)

4. Which quadratic function, when graphed, has zeroes -1 and 9?

A) (x – 1)(x + 9) B) (x + 1)(x + 9) C) (x – 1)(x – 9) D) (x + 1)(x – 9)

5. Which quadratic function, when graphed, has x intercepts of 2 and 3?

A) (x + 2)( x + 3) B) (2x – 4)(3x + 9) C) (2x – 4)(3x – 9) D) (2x + 4)(3x + 9)

6. Which quadratic function, when graphed, has solutions -2/3 and ¼?

A) (3x – 2)(4x- 1) B) (3x + 2)(4x + 1) C) (3x – 2)(4x + 1) D) (3x + 2)(4x – 1)

7. What are the solutions to (x – 1)(2x + 5)?

A) x = -1 and x = 5/2 B) x= 1 and x = -5/2 C) x= -1 and x = 2/5 D) x = 1 and x = -2/5

8. What are the solutions for the quadratic equation x²-4x=5?

A) -5, 1 B) -5, -1 C) 5, 1D) 5, -1

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Solutions to Graphs Practice Sheet1.

Solutions:

Vertex:

Equation:

2.

x – intercepts:

Vertex:

Equation:

3.

Value of x when y = 0:

Axis of Symmetry:

4.

Zeroes:

Axis of Symmetry:

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5.

x-intercepts:

Axis of Symmetry:

6.

Value of x when y = 0:

Axis of Symmetry:

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Investigating the Discriminant NOTES/DISCOVERY Review: The general form of a quadratic equation is .

The Quadratic Formula can be used to solve this equation. The quadratic formula is:

The discriminant is a part of the quadratic formula. The discriminant is the expression inside the radical.

1.

a. Based on the graph, how many real zeros are there? ______

b. What is the value of the discriminant? ______

2.

a. Based on the graph, how many real solutions are there? ______

b. What is the value of the discriminant? ______

3.

a. Based on the graph, how many realx-intercepts are there? ______

b. What is the value of the discriminant? ______

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4.

a. Based on the graph, how many real roots are there? ______

b. What is the value of the discriminant? ______

5.

a. Based on the graph, how many real zeros are there? ______

b. What is the value of the discriminant? ______

6.

a. Based on the graph, how many times does the parabola cut the x axis? ______

b. What is the value of the discriminant? ______

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Discriminant Practice Use the discriminant to determine the nature of the roots to the quadratic equation.

1. x2 – 5x + 7 = 0 2. x2 – 5x + 6 = 0 3. 2x2 – 5x + 5 = 0

4. x2 + 7x + 2 = 0 5. 2x2 + 7x + 6 = 0 6. 2x2 + 7x + 7 = 0

7. 2x2 – 7x + 6 = 0 8. 2x2 + 7x – 6 = 0 9. x2 + 6x + 9 = 0

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Quadratic Formula: ‘Step by Step’ Practice SheetProblem Simplify Solutions

1. y=2x2+3 x−5Write the equation in standard form:

A = 2 b = 3 c = -5

x=−(3 )±√(3 )2−4 (2)(−5)

2(2 )

2. y=x2−2 x−4

Write the equation in standard form:

a = b = c =

x=−(−2)±√()2−4 ()()

2()

3. −14+ x2=5 xWrite the equation in standard form:

a = b = c =

x=−()±√()2−4 ( )( )

2( )

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4. −12 x=−9 x2−4Write the equation in standard form:

a = b = c =

x=−()±√()2−4 ( )( )

2()

5. 2 x=3+2 x2

Write the equation in standard form:

a = b = c =

x=−(−2)±√( )2−4 ()()

2()

Quadratic Formula Check for Understanding1. What is the quadratic formula?

2. Using a complete sentence, what is the quadratic formula used to find/determine?

3. Using the quadratic formula, find the solutions of

y=2x2−7 x+3.

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Factoring vs. Quadratic Formula Practice Topic: Finding x-intercepts for quadratics using factoring and quadratic formula.

If the given function can be factored, then factor and provide the x-intercepts. If you cannot factor the function, then use the quadratic formula to find the x-intercepts.

1. A(x) = x2 + 4x – 21 2. B(x) = 5x2 + 16x + 3 3. C(x) = x2 – 4x + 1

4. D(x) = x2 – 16x + 4 5. E(x) = x2 + 3x – 40 6. F(x) = 2x2 – 3x – 9

7. G(x) = x2 – 3x 8. H(x) = x2 + 6x + 8 9. K(x) = 3x2 - 11

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Graphing Quadratics in Standard Form Notes1. Graph

a= b= c=

List the intervals where the function is increasing and decreasing

Domain: Range:

2. Opens UP or DOWN? Why?

3. Y-intercept?

4. Roots?

5. Coordinates of Vertex? Axis of Symmetry? Is it a maximum or minimum?

1. Graph

a= b= c=

List the intervals where the function is increasing and decreasing

Domain: Range:1. Opens UP or DOWN? Why?

2. Y-intercept?

3. Zeroes?

4. Coordinates of Vertex? Axis of Symmetry ? Is it a maximum or minimum?

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1. Graph

a= b= c=

List the intervals where the function is increasing and decreasing

Domain: Range:

2. Opens UP or DOWN? Why?

3. Y-intercept?

4. Solutions?

5. Coordinates of Vertex? Axis of Symmetry? Is it a maximum or minimum?

1. Graph

a= b= c=

List the intervals where the function is increasing and decreasing

Domain: Range:

2. Opens UP or DOWN? Why?

3. Y-intercept?

4. X intercepts?

5. Coordinates of Vertex? Axis of Symmetry? Is it a maximum or minimum?

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Graphing Quadratics Practice Sheet1. Find the Vertex y=x2−6x+8

Is it a maximum or minimum?

4. List the intervals where the function is increasing and decreasing.

2. Find the zeroes

3.Identify the y intercept

5.Graph the parabola

Domain: Range:1. Find the Vertex y=x2+2 x−15

Is it a maximum or minimum?

4. List the intervals where the function is increasing and decreasing.

2.Find the zeroes

3.Identify the y intercept

5.Graph the parabola

Domain: Range:

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1. Find the Vertex y=x2−8 x−20

Is it a maximum or minimum?

4. List the intervals where the function is increasing and decreasing.

2. Find the zeroes

3. Identify the y intercept

5.Graph the parabola

Domain: Range:

Parabolas in Standard Form and Factored Form PracticeGraphing Quadratic Equations

Equation (put in standard form)

Opening & y-intercept

Vertex & axis of symmetry

(AOS)

Roots (solutions, zeroes, etc)

Graph

1) −2 x2+ y=4 x−6 Opening:

y-intercept:

Vertex:

Is it a maximum or minimum?

AOS:Domain:

Range:

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Equation (put in standard form)

Opening & y-intercept

Vertex & axis of symmetry

(AOS)

Roots (solutions, zeroes, etc)

Graph

2)4 x+6+ y=2 x2

Opening:

y-intercept:

Vertex:

Is it a maximum or minimum?

AOS:

Domain:

Range:

3)2 x2−6+ y=4 x Opening:

y-intercept:

Vertex:

Is it a maximum or minimum?

AOS:

Domain:

Range:

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Equation (put in standard form)

Opening & y-intercept

Vertex & axis of symmetry

(AOS)

Roots (solutions, zeroes, etc)

Graph

4)

f(x) = (x – 3) (x + 2) Opening:

y-intercept:

Vertex:

Is it a maximum or minimum?

AOS: Domain:

Range:

1. The following problem was solved as follows:

Problem: x2 + 6x – 7 = 0Step 1: x2 + 6x = 7Step 2: x2+6 x+9=7+9 Step 3: (x + 3)2 = 16

a) What was done mathematically to get from step 1 to step 2

b) Describe what was done mathematically to get from step 2 to step 3.

c) Finish the problem completely.

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Identifying Features of a Graph Given a Table

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All 3 Forms

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Evaluating 4 Methods to Solving QuadraticsJuan, Lisa, Sol and Logan were asked to solve the quadratic equation: . Their work is shown below.

Juan’s Work: Lisa’s Work: Sol’s Work:

Logan’s Work:

Using your description skills identify various methods each used to solve the quadratic.

Juan:_______________________________ Lisa:____________________________

Sol: ________________________________ Logan: __________________________

Using your analysis skills, compare and contrast the four different methods of solving the quadratic equation above.

What is the same about the four methods?

What is different about the four methods?

Now, let’s evaluate these methods:Review each student’s work. Who solved the quadratic equation accurately?

Now, prove it by evaluating the validity of the solutions by checking if the solutions are correct by substituting the values into the original equation. Remember, the solutions should make the equation true.

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1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

123456789

10

–1–2–3–4–5–6–7–8–9

–10

y

Now, providing your reasoning, in complete sentences explain what each student did incorrectly:

Looking back at each method, which is your preferred method to solving quadratics for this particular problem? Explain why.

Okay, now let’s see you do it. Solve the quadratic equation by any method of your choice.

Which Method??For each of the given quadratic equations, find the solutions using any method. State the method you are using as well

as the solutions. You must use at least three different methods.1. x2 + 17x + 60 = 0 2. x2 + 16x + 39 = 0 3. x2 + 7x – 5 = 0

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4. 3x2 + 14x – 5 = 0 5. x2 - 12x = -8 6. x2 + 6x = 8

Graphing Quadratic Word Problems

The path of a baseball being thrown from a pitcher to a batter can be modeled by where s is the height of the ball after t seconds.

At what time does the ball reach its maximum height?

How long is the baseball in the air?

What is the domain and range of the function?

Graph the Quadratic function.

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Graphing Quadratic Word ProblemsThe path of a rocket being launched from the ground can be modeled by h (t )=100 t−25 t2 where h is the height of the rocket after t seconds.

At what time does the ball reach its maximum height?

How long is the rocket in the air?

What is the domain and range of the function?

Graph the Quadratic function.

Graphing Quadratics, Vertex Concept ConnectionA water bottle rocket is shot upward with an initial velocity, vi=45 ft/s from the roof of a school, which is hi, 50 ft above the ground. The

equation h=−12a t2+v i t+hi models the rocket’s height as a function

of time. The acceleration due to gravity, a, is 32 ft/s².

A) Write the equation for height as a function of time for this situation.

B) Find the vertex of this parabola.

C) Sketch the graph of this parabola. Label the vertex.

D) What do the coordinates of the vertex represent in terms of time and height?

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Graphing Real Life Application TaskYour ballet class is having a recital during December. They will be performing The Nutcracker. Your class chooses two primary dancers for the lead roles, one male and one female. One of the male dancer’s leaps can be modeled by:

h=32t−16 t 2 where h is the height in feet and t is the time in seconds. One of the female dancer’s leaps can be

modeled by: h=20 t−10t 2 .

1. Estimate the maximum height reached by the male dancer.

2. How many seconds does it take the male dancer to reach his maximum height?

3. Estimate the maximum height reached by the female dancer.

4. How many seconds does it take the female dancer to reach her maximum height?

5. Sketch the graphs of the equations for the male and female dancer.

Conceptual Problem Solving TaskThe height h in feet of a ball t seconds after being tossed upward is given by the formula: h=80 t−16 t2

A. Complete the table t 0 1 2 3 4 5h

B. Sketch a graph of the model

C. Use the graph to find a positive root if the equation 0=80 t−16 t2

D. After how many seconds will the ball hit the ground?

E. What is the domain and range of the function?

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Unit 5 Review – Graphing Quadratics1. Find the solutions to the equation:

4x2 + 12x + 5 = 0

2. An apple falling from a tree is h feet above the ground t seconds after it begins to fall, where . After how many seconds will the apple hit the ground?

3. 4.

5. What are the roots of the quadratic equation x2 – 8x = 9?

6. What are the coordinates of the vertex for the

parabola given by ?

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What are the real roots of the function? For what value or values of x if y = 0?

7. Which best

represents the graph of y = -x2 + 3?

8.

Which is the graph of y =

3x2 + 6x?

9. Find the roots and vertex of the quadratic equation below.

x2+5 x=2(x−1)

10. Circle the equations that are quadratic functions

Equation 1: 4 x+1= y

Equation 2: 3 x2−7 x= y

Equation 3: y−1=x−2 x2

Equation 4: 5 x2−x3+3= y

Equation 5: 4x2 +3 x−9= y

11. Which of the following equations have 12. Write a quadratic equation in standard form

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equivalent zeroes?

Equation A :

y=( 12x+7)(2x+2)

Equation B :y=x2+14

EquationC :y= (x+7 )(x+2)

EquationD :y=x2+9 x+14

whose roots are x = 2 and x = ½.

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