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Mathematics

Quarter 1, Wk. 1 - Module 1

Illustrations of Quadratic Equations

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Department of Education ● Republic of the Philippines

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Math- Grade 9

Alternative Delivery Mode Quarter 1,Wk. 1 - Module 1: Illustrations of Quadratic Equations

First Edition, 2020

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Published by the Department of Education – Division of Iligan City

Schools Division Superintendent: Roy Angelo L. Gazo, PhD.,CESO V

Development Team of the Module

Author/s: Angela P. Canoy Evaluators/Editor: Priscilla G. Luzon, Natividad B. Finley

Illustrator/Layout Artist: Joe Marie P. Perez, Beverly D. Sarno

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Mathematics

Quarter 1, Wk.1 - Module 1

Illustrations of Quadratic Equations

This instructional material was collaboratively developed and reviewed

by educators from public and private schools, colleges, and or/universities.

We encourage teachers and other education stakeholders to email their

feedback, comments, and recommendations to the Department of Education

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We value your feedback and recommendations.

Department of Education ● Republic of the Philippines

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Table of Contents

What This Module is About ....................................................................................................................... i

What I Need to Know .................................................................................................................................. i

How to Learn from this Module............................................................................................................... ii

Icons of this Module ................................................................................................................................... ii

Lesson 1:

Illustrations of Quadratic Equations .............................................................................. 1

What I Need to Know..................................................................................................... 1

What I Know ................................................................................................................. 1

What’s In ........................................................................................................................ 2

What’s New ................................................................................................................... 3

What Is It ........................................................................................................................... 4

What’s More .................................................................................................................... 5

What I Have Learned..................................................................................................... 8

What I Can Do ................................................................................................................. 9

Summary

Key to Answers ...................................................................................................................................... 11

References ............................................................................................................................................... 16

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What This Module is About

Real-world problems can be solved with the help of mathematical equations

such as where a rocket ship will land, predicting the maximum height of the fireworks, how much to charge for a product and a wide variety of applications

This module is about quadratic equations and how they are illustrated in real-life. The lesson provided you with opportunities to discuss quadratic equations using practical situations and their mathematical representations. Moreover, you were

given the chance to formulate quadratic equations as illustrated in some real-life situations. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson,

solving quadratic equations.

What I Need to Know

This module primarily deals with defining and illustrating quadratic equations; differentiating quadratic equation from the linear equation; writing a quadratic

equation in standard form and identifying the values of a, b, and c; formulating a quadratic equation to represent the given real-life situations; and appreciating the uses of quadratic equation in real-life situations. These concepts are important for

solving real-life problems.

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How to Learn from this Module

To achieve the objectives cited above, you are to do the following:

• Take your time reading the lessons carefully.

• Follow the directions and/or instructions in the activities and exercises diligently.

• Answer all the given tests and exercises.

Icons of this Module

What I Need to This part contains learning objectives that

Know are set for you to learn as you go along the

module.

What I know This is an assessment as to your level of

knowledge to the subject matter at hand,

meant specifically to gauge prior related

knowledge

What’s In This part connects previous lesson with that

of the current one.

What’s New An introduction of the new lesson through

various activities, before it will be presented

to you

What is It These are discussions of the activities as a

way to deepen your discovery and under-

standing of the concept.

What’s More These are follow-up activities that are in-

tended for you to practice further in order to

master the competencies.

What I Have Activities designed to process what you

Learned have learned from the lesson

What I can do These are tasks that are designed to show-

case your skills and knowledge gained, and

applied into real-life concerns and situations.

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1

Lesson

Illustrations of Quadratic Equations

1

What I Need to Know

Let’s start this module by assessing your knowledge and skills of the different mathematics concepts that you previously learned. This knowledge and skills will help you in learning quadratic equations by defining and illustrating quadratic equations; differentiating quadratic equation from the linear equation; writing a quadratic equation in standard form and identifying the values of a, b, and c; formulating a quadratic equation to represent the given real-life situations; and appreciating the uses of quadratic equation in real-life situation.

What I Know

Pre - Assessment Directions: Find out how much you already know about this module. Choose the letter that you think best answers the question. Please answer all items. Take note of

the items that you were not able to answer correctly and find the right answer as you go through this module.

1. It is a polynomial equation of degree two that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. a. Linear Equation c. Quadratic Equation

b. Linear Inequality d. Quadratic Inequality

2. Which of the following is a quadratic equation?

a. 3s2 + s – 4 c. 2x – 1 = 5

b. m2 – 8m – 1 = 0 d. 5y2 + 4y 7

3. In the quadratic equation 2x2 – 9x – 5 = 0, which is the quadratic term?

a. 2x2 b. x2 c. – 9x d. – 5

4. In the quadratic equation 2x2 – 9x – 5 = 0, which is the linear term?

a. 2x2 b. x2 c. – 9x d. – 5

5. In the quadratic equation 2x2 – 9x – 5 = 0, which is the constant term?

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a. 2x2 b. x2 c. – 9x d. – 5 6. In the quadratic equation x2 + 8x – 2 = 0, what are the values of a, b, and c?

a. a = 0, b = 3, c = -1 c. a = - 3, b = 0, c = -1 b. a = 1, b = 8, c = -2 d. a = 3, b = 0, c = 1

7. In the quadratic equation 3x2 – 1 = 0, what are the values of a, b, and c? a. a = 0, b = 3, c = -1 c. a = - 3, b = 0, c = -1 b. a = 3, b = 0, c = -1 d. a = 3, b = 0, c = 1

8. In the quadratic equation (y + 5)(y – 5) = 4, what are the values of a, b, and c?

a. a = 1, b = 5, c = -5 c. a = 1, b = 0, c = -29

b. a = 1, b = - 5, c = 5 d. a = 1, b = 0, c = 25

9. What is the standard form of the quadratic equation 3x(x – 3) = 7?

a. 3x2 – 9x = 7 c. 3x2 – 9x + 7 = 0 b. 3x2 – 3x – 7 = 0 d. 3x2 – 9x – 7 = 0

10. What is the standard form of the quadratic equation 2x + (x – 4)(x + 1) = 9? a. x2 – x – 13 = 0 c. x2 – 5x + 5 = 0 b. x2 + x + 13 = 0 d. x2 – 5x – 13 = 0

What’s In

Activity 1: Find My Partner

Directions: Solve the indicated product of the following and find your answer shown on the right side to find its partner.

Process Questions:

a. Were you able to find the indicated product? b. What mathematical concepts or principles did you use to find your answer? c. What common characteristics can you see in the products?

3(x + 7)

2x(x – 4)

7(x + 1) – 2x

(x - 3)(x + 1)

(x + 4)2

2x2 – 8x

x2 + 8x + 16

x2 – 2x - 3

3x + 21

5x + 7

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Were you able to recall and apply different mathematical concepts or principles in finding the product? Why do you think there is a need to perform such mathematical tasks? You will find this out as you go through this lesson.

What’s New

Activity 2: Post Me In the Right Path!

Directions: Post the following equations to its corresponding path.

Linear Equation Not Linear Equation

Process Questions: a. Which of the given equations are linear? b. How do you describe linear equation?

c. Which of the equations are not linear? Why? d. How are these equations different from those which are linear? e. What common characteristics do these equations have?

From the activities that you have done, you were able to describe equations other than linear equations, and these are quadratic equations. But how are quadratic equations used in real-life problems and in making decisions? You will find

these out in the activities in the next section. Before doing these activities, read and understand first some important notes on quadratic equations and the examples presented.

1. 2m2 – 8m = 0

2. r2 + 8r + 16 = 0

3. x2 – 2x – 3 = 0

4. 3s + 21 = 0

5. 5t + 7 = 0

6. x2 – 5x + 3 = 0

7. 8k – 3 = 0

8. 9 – 4x = 0

9. r2 - 16 = 0

10. 4x2 + 4x + 1 = 0

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What Is It

A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the following standard form.

ax2 + bx + c = 0, where a, b, and c are real numbers and a 0.

In the equation, ax2 is the quadratic term, bx is the linear term, and c is the constant term.

Why do you think a must not be equal to zero? What happens to the equation when a is equal to zero?

The value of a must not be equal to zero because it is the numerical coefficient of the quadratic term.

Example 1: 3x2 + 7x – 1 = 0 is a quadratic equation in standard form with

a = 3, b = 7, c = -1.

Example 2: 2x(x - 4) = 12 is a quadratic equation. However, it is not written in

standard form. To transform it to its standard form, use distributive

property and make one side of the equation zero as shown below.

2x(x - 4) = 12 2x2 – 8x = 12 , by distributive

property

2x2 – 8x – 12 = 12 – 12 , by subtraction

property

2x2 – 8x – 12 = 0

The standard form of 2x(x - 4) = 12 is 2x2 – 8x – 12 = 0 where a = 2, b

= -8, and c = -12.

Example 3: (x + 1)(x - 8)= -9 is also a quadratic equation but is not written in

standard form. Just like in Example 2, transform it to its standard form

using distributive property and make one side of the equation zero as

shown below.

(x + 1)(x - 8)= -9 x2 – 7x - 8 = -9 , by distributive

property

x2 – 7x – 8 + 9 = -9 + 9 , by addition property

x2 – 7x + 1 = 0

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The standard form of (x + 1)(x - 8)= -9 is x2 – 7x + 1 = 0 where a = 1, b

= -7, and c = 1.

When b = 0 in the equation ax2 + bx + c = 0, it results to a quadratic equation

of the form ax2 + c = 0.

Examples: Equations such as 3x2 – 1 = 0, -5x2 + 3 = 0, and x2 - 1 = 0 are quadratic

equations of the form ax2 + c = 0. In each equation, the value of b = 0.

While, equations such as 4x2 – x = 0, -x2 + 5x = 0, and -2x2 – 5x = 0

are quadratic equation of the form ax2 + bx = 0. In each equation, the

value of c = 0.

Your goal in this section is to apply key concepts of quadratic equations use the

mathematical ideas and the examples presented in the preceding section to answer

activities provided.

What’s More

Activity 3: Am I Quadratic or Not? Directions: Determine whether each equation is Quadratic or Not quadratic. Write Q

if it is quadratic and N if it is not quadratic.

1. C = d 6. (s + 1) = 0

2. 2(x + 3) = 0 7. (t + 4)(t+7) = 0 3. x + 3x2 = 0 8. (x – 5)2 – 3 = 0

4. 5x – 4 = 0 9. x2 – 7 = 0 5. -2m2 + m = 1 10. 5(m – 8) + 1 = 0

Process Questions:

a. Were you able to identify which equations are quadratic and not quadratic? b. What makes the equation not quadratic? Differentiate a quadratic equation

from an equation that is not quadratic.

In the next activity, you will write quadratic equations in standard form. Activity 4: Set Me to My Standard!

Directions: Write each quadratic equation in standard form, ax2 + bx + c = 0 then identify the values of a, b, and c.

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1. x2 – 6x – 7 = 0 6. 2x(x + 1) = 0 2. x2 – 64 = 0 7. (x + 4)2 = 5

3. -2x +8 = -9x2 8. (x + 3)(x – 3) = x(4x + 5) 4. x2 = 3x + 10 9. (x + 5)(x - 9) = 21 5. 4x2 – 2 = 0 10. x(2x – 4) = (x – 2)(x – 2)

Process Questions:

a. What mathematics concepts or principles did you apply to write each

quadratic equation in standard form? Discuss how you applied these mathematics concepts or principles.

b. Write the steps in transforming a quadratic equation to its standard form.

c. Which quadratic equations did you find difficult to write in standard form? Why?

You have already familiarized the basic concepts and definitions of quadratic equations. This time, let’s try to see if quadratic equations can be used in solving real-life problems.

Activity 5: Discovery Method!

Directions: Consider the situation below and fill in the table.

Staying at home is the best way to keep yourself safe during this COVID-19 pandemic. To keep you and your siblings cool at home during this summer break, your mother decided to construct a swimming pool in your backyard. She asked you

to make a layout of a rectangular pool whose area is 28m2. She specified that the length of the pool must be 3 m more than its width.

Guide Questions Your Answer

1. If x represents the width of the rectangular pool, how would you represent the length?

Width = x Length = _______

2. You know that the formula for the area of the rectangle is A = lw. How would you represent the

area of the rectangular pool?

A = 28, w = x, l = ___

A = lw ____________

Width = x

Length = 3m more than its width

Area = 28m2

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3. What are the dimensions of the

pool?

Process Questions: a. How did you find the activity? b. What if the length of the pool is 5 meters more than its width, what equation

would represent the given situation? c. How would you describe the equation formulated? d. Do you think you can use the equation formulated to find the length and the

width of the pool? Justify your answer.

From the activities that you have done, you were able to find out how a particular quadratic equation is illustrated and used in solving real life problems and making decisions. To understand more about quadratic equation and its application to real-life, let’s have our next activity.

Activity 6: Let’s Get Online! Why do we need to study quadratic equation? Please watch this video online

(https://www.youtube.com/watch?v=BjbyqgUEbAE) to know where and when we can use quadratic equation.

Process Questions:

a. Based on the video, what are some examples of real life situations where we

can use quadratic equation? b. Where and when can we use quadratic equations? c. Is it important for us to learn about quadratic equations? Why?

For our next activity, you will identify situations that illustrate quadratic equations and represent these by mathematical statements.

https://www.youtube.com/watch?v=BjbyqgUEbAE

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Activity 7: Does It Illustrate Me? Directions: Tell whether or not each of the following situations illustrates quadratic

equations. Justify your answer by representing each situation by a

mathematical sentence.

1. The square of a number is added to two times the number and the sum is 24.

2. The width of a rectangle card is 2cm less than the length and the area is

35cm2.

3. A flowerbed is to be 3m longer than its width. The flowerbed will have an

area of 70m2.

4. Angela is 4 years younger than Genesis. Four years later, Genesis will be

twice as old as Angela.

5. A rectangular bahay-kubo with the dimension of 11m more than its width built

in a rectangular backyard. The area of the bahay-kubo is 85m2.

Process Questions: a. Did you find the activity challenging?

b. Were you able to represent each situation by a mathematical statement?

Now that you know the important ideas about this topic, let’s go deeper by moving on to the next activity.

What I Have Learned Activity 8: Sum It Up

Directions: Let’s summarize all of the concepts or principles that you have learned

about quadratic equations using the diagram below.

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Activity 9: Dig Deeper!

Directions: Answer the following questions.

1. How are quadratic equations different from linear equations?

2. How do you write quadratic equations in standard form? Give at least 2

examples and identify the values of a, b, and c.

3. The following are the values of a, b, and c that Angela and Genesis got when

they expressed 4 – 7x = x2 in standard form.

Angela: a = -1; b = -7; c = 4 Genesis: a = 1; b = 7; c = -4

Who got the correct values of a, b, and c? Justify your answer.

4. Can the equation -1 + 9x = 4x2 be written in standard form in two different

ways? What are the two possible answers?

Now that you have deeper understanding of the topic, you are ready to do our mini task which will demonstrate your understanding of quadratic equations.

What I Can Do

Activity 10: Mini-task: Where in the real world? Directions: Completely fill in your Home Quarantine Pass before proceeding to the next lesson.

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Summary

This lesson is about quadratic equations and how they are illustrated in real-

life. The lesson provided you with opportunities to discuss quadratic equations using practical situations and their mathematical representations. Moreover, you were given the chance to formulate quadratic equations as illustrated in some real life

situations. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the next lesson, solving quadratic equations.

11

Key to Answers Pre - Assessment

Directions: Find out how much you already know about this module. Choose the letter that you think best answers the question. Please answer all items. Take note of the items that you were not able to answer correctly and find the right answer as you

go through this module.

1. It is a polynomial equation of degree two that can be written in the form

ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. a. Linear Equation c. Quadratic Equation b. Linear Inequality d. Quadratic Inequality

2. Which of the following is a quadratic equation?

a. 3s2 + s – 4 c. 2x – 1 = 5

b. m2 – 8m – 1 = 0 d. 5y2 + 4y 7

3. In the quadratic equation 2x2 – 9x – 5 = 0, which is the quadratic term? a. 2x2 b. x2 c. – 9x d. – 5

4. In the quadratic equation 2x2 – 9x – 5 = 0, which is the linear term? a. 2x2 b. x2 c. – 9x d. – 5

5. In the quadratic equation 2x2 – 9x – 5 = 0, which is the constant term? a. 2x2 b. x2 c. – 9x d. – 5

6. In the quadratic equation x2 + 8x – 2 = 0, what are the values of a, b, and c? a. a = 0, b = 3, c = -1 c. a = - 3, b = 0, c = -1 b. a = 1, b = 8, c = -2 d. a = 3, b = 0, c = 1

7. In the quadratic equation 3x2 – 1 = 0, what are the values of a, b, and c?

a. a = 0, b = 3, c = -1 c. a = - 3, b = 0, c = -1

b. a = 3, b = 0, c = -1 d. a = 3, b = 0, c = 1

8. In the quadratic equation (y + 5)(y – 5) = 4, what are the values of a, b, and c?

a. a = 1, b = 5, c = -5 c. a = 1, b = 0, c = -29 b. a = 1, b = - 5, c = 5 d. a = 1, b = 0, c = 25

9. What is the standard form of the quadratic equation 3x(x – 3) = 7? a. 3x2 – 9x = 7 c. 3x2 – 9x + 7 = 0 b. 3x2 – 3x – 7 = 0 d. 3x2 – 9x – 7 = 0

10. What is the standard form of the quadratic equation 2x + (x – 4)(x + 1) = 9?

a. x2 – x – 13 = 0 c. x2 – 5x + 5 = 0

b. x2 + x + 13 = 0 d. x2 – 5x – 13 = 0

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Activity 1: Find My Partner Directions: Solve the indicated product of the following and find your answer shown

at the right side to find its partner. =

=

= =

=

Activity 2: Post Me In the Right Path! Directions: Post the following equations to its corresponding path.

Linear Equation Not Linear Equation

Activity 3: Am I Quadratic or Not? Directions: Determine whether each equation is Quadratic or Not quadratic. Write Q

if it is quadratic and N if it is not quadratic.

1. C = d N 6. (s + 1) = 0 N

2. 2(x + 3) = 0 N 7. (t + 4)(t+7) = 0 Q

3. x + 3x2 = 0 Q 8. (x – 5)2 – 3 = 0 Q 4. 5x – 4 = 0 N 9. x2 – 7 = 0 Q 5. -2m2 + m = 1 Q 10. 5(m – 8) + 1 = 0 N

3s + 21 = 0

5t + 7 = 0

8k – 3 = 0

9 – 4x = 0

2m2 – 8m = 0

r2 + 8r + 16 = 0

x2 – 2x – 3 = 0

x2 – 5x + 3 = 0

r2 - 16 = 0

4x2 + 4x + 1 = 0

3(x + 7)

2x(x – 4)

7(x + 1) – 2x

(x - 3)(x + 1)

(x + 4)2

2x2 – 8x

x2 + 8x + 16

x2 – 2x - 3

3x + 21

5x + 7

13

Activity 4: Set Me to My Standard!

Directions: Write each quadratic equation in standard form, ax2 + bx + c = 0 then identify the values of a, b, and c.

1. x2 – 6x – 7 = 0 x2 – 6x – 7 = 0 ; a = 1; b = -6; c = 7

2. x2 – 64 = 0 x2 – 64 = 0 ; a = 1; b = 0; c = -641 3. -2x +8 = -9x2 9x2 - 2x + 8 = 0 ; a = 9; b = -2; c = 8

or -9x2 + 2x - 8 = 0 ; a = -9; b = 2; c = -8

4. x2 = 3x + 10 x2 - 3x – 10 = 0 ; a = 1; b = -3; c = -10 or -x2 + 3x + 10 = 0 ; a = -1; b = 3; c = 10

5. 4x2 – 2 = 0 4x2 – 2 = 0 ; a = 4; b = 0; c = -2

6. 2x(x + 1) = 0 2x2 + 2x = 0 ; a = 2; b = 2; c = 0 7. (x + 4)2 = 5 x2 + 8x + 11 = 0 ; a = 1; b = 8; c = 11

or -x2 - 8x - 11 = 0 ; a = -1; b = -8; c = -11

8. (x + 3)(x – 3) = x(4x + 5) -3x2 – 5x – 9 = 0 ; a = -3; b = -5; c = -9 or 3x2 + 5x + 9 = 0 ; a = 3; b = 5; c = 9

9. (x + 5)(x - 9) = 21 x2 – 4x – 66 = 0 ; a = 1; b = -4; c = -66

or -x2 + 4x + 66 = 0 ; a = -1; b = 4; c = 66 10. x(2x – 4) = (x – 2)(x – 2) x2 – 4 = 0 ; a = 1; b = 0; c = -4

or -x2 + 4 = 0 ; a = -1; b = 0; c = 4

Activity 5: Discovery Method! Directions: Consider the situation below and fill in the table.

Staying at home is the best way to keep yourself safe during this COVID-19 pandemic. To keep you and your siblings cool at home during this summer break,

your mother decided to construct a swimming pool in your backyard. She asked you to make a layout of a rectangular pool whose area is 28m2. She specified that the length of the pool must be 3 m more than its width.

Guide Questions Your Answer

1. If x represents the width of the

rectangular pool, how would you represent the length?

Width = x

Length = x + 3

2. You know that the formula for the

area of the rectangle is A = lw. How would you represent the area of the rectangular pool?

A = 28, w = x, l = x + 3 A = lw

28 = (x+3)(x)

Width = x

Area = 28m2

Length = 3m more than its width

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3. What are the dimensions of the

pool?

Solutions: (x+3)(x) = 28

x2 + 3x = 28 x2 + 3x – 28 = 28 - 28 x2 + 3x – 28 = 0

(x + 7)(x - 4) = 0 x + 7 = 0 x – 4 = 0 x + 7 – 7 = 0 – 7 x – 4 + 4 = 0 + 4

x = -7 x = 4 We will just take the positive value of x

since we are looking for a measurement.

So, x = 4. Thus, Width = x = 4 Length = x + 3 = 4 + 3 = 7.

Therefore, the dimensions of the swimming pool are 7m by 4m.

Activity 7: Does It Illustrate Me?

Directions: Tell whether or not each of the following situations illustrates quadratic

equations. Justify your answer by representing each situation by a

mathematical sentence.

1. The square of a number is added to two times the number and the sum is 24.

Answer: Quadratic; x2 + 2x = 24 x2 + 2x – 24 = 0

2. The width of a rectangle card is 2cm less than the length and the area is

35cm2.

Answer: Quadratic; (x)(x – 2) = 35 x2 - 2x – 35 = 0

3. A flowerbed is to be 3m longer than its width. The flowerbed will have an

area of 70m2.

Answer: Quadratic ; (x + 3)(x) = 70 x2 + 3x – 70 =

0

4. Angela is 4 years younger than Genesis. Four years later, Genesis will be

twice as old as Angela.

Answer: Not Quadratic; x + 4 = 2(x) x – 4 = 0

5. A rectangular bahay-kubo with the dimension of 11m more than its width built

in a rectangular backyard. The area of the bahay-kubo is 85m2.

Answer: Quadratic; (x + 11)(x) = 85 x2 + 11x – 85 = 0

15

Activity 8: Sum It Up Directions: Let’s summarize all of the concepts or principles that you have learned

about quadratic equations using the diagram below. Activity 9: Dig Deeper! Directions: Answer the following questions.

1. How are quadratic equations different from linear equations?

Answer: Quadratic equations are mathematical sentences of degree 2

while linear equations are mathematical sentences of degree 1.

2. How do you write quadratic equations in standard form? Give at least 2

examples and identify the values of a, b, and c. (Check students’ explanations and examples they will give. Their answers might be different and all are correct.)

3. The following are the values of a, b, and c that Angela and Genesis got when

they expressed 4 – 7x = x2 in standard form.

Angela: a = -1; b = -7; c = 4 Genesis: a = 1; b = 7; c = -4

Who got the correct values of a, b, and c? Justify your answer.

Answer: Angela and Genesis are both correct. The equation 4 – 7x = x2 can be written in standard form in two ways, -x2 – 7x + 4 = 0 or x2 + 7x - 4 = 0.

4. Can the equation -1 + 9x = 4x2 be written in standard form in two different

ways? What are the two possible answers?

Answer: Yes. -4x2 + 9x – 1 = 0 or 4x2 - 9x + 1 = 0.

16

References

Websites: Tumanova, Elena. “Straight Empty Road Through The Countryside. Summer

Landscape Stock Vector - Illustration of Illustration, Blue: 157847811.” Dreamstime, September 6, 2019. https://www.dreamstime.com/straight-empty-road-countryside-summer-landscape-green-hills-blue-sky-meadow-mountains-vector-illustration-

image157847811.

GraphicsRF, Shawn. “Curve Road in the Park.” Vecteezy, April 12, 2019.

https://www.vecteezy.com/vector-art/433447-curve-road-in-the-park.

https://www.dreamstime.com/straight-empty-road-countryside-summer-landscape-green-hills-blue-sky-meadow-mountains-vector-illustration-image157847811https://www.dreamstime.com/straight-empty-road-countryside-summer-landscape-green-hills-blue-sky-meadow-mountains-vector-illustration-image157847811https://www.dreamstime.com/straight-empty-road-countryside-summer-landscape-green-hills-blue-sky-meadow-mountains-vector-illustration-image157847811https://www.vecteezy.com/vector-art/433447-curve-road-in-the-park

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For inquiries and feedback, please write or call:

Department of Education – Division of Iligan City

Office Address: General Aguinaldo, St., Iligan City

Telefax: (063)221-6069

E-mail Address: iligan.city@deped.gov.ph

mailto:iligan.city@deped.gov.ph

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