This is the Grade 9 Mathematics self-learning module for Quarter 1, Module 1, titled “Illustrating Quadratic Equation.” It is meant for Week 1 and covers learning competency M9AL-Ia-1. It builds on the linear equations learners studied in Grade 8. By the end, learners can recognize a quadratic equation, write it in standard form, name the values of a, b, and c, and turn real-life situations into quadratic equations.
The module was developed under DepEd’s Alternative Delivery Mode (ADM), First Edition 2020. The writers are Analynn M. Argel, Rowena F. Reyes, and Queenie Pearl E. Domasig.
On this page 6 sections
What learners will learn
The module has one learning competency:
- Illustrate quadratic equation (M9AL-Ia-1)
In practice, this means learners will be able to:
- tell whether an equation is quadratic or not
- rewrite a quadratic equation in standard form, ax² + bx + c = 0
- identify the quadratic term, linear term, and constant term, and the values of a, b, and c
- write a quadratic equation that represents a word problem about area, volume, or numbers
What is a quadratic equation?
A quadratic equation in one variable is a second-degree equation. This means the highest exponent of the variable is 2. It can be written in standard form:
ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
| Part | Name | What it can be |
|---|---|---|
| ax² | Quadratic term | a can be positive or negative, but never zero |
| bx | Linear term | b can be positive, negative, or zero |
| c | Constant term | c can be positive, negative, or zero |
The vegetable garden example
The module opens with Aling Tuding, who wants to turn her rectangular backyard lot into a vegetable garden. The lot has an area of 15 m², and its length is 3 meters longer than its width.
- Let x be the width, so the length is x + 3.
- Using A = lw: 15 = x(x + 3)
- By the Distributive Property: 15 = x² + 3x
- By the Addition Property of Equality: x² + 3x − 15 = 0
This equation has degree 2, so it is quadratic. In standard form, a = 1, b = 3, and c = −15.
Checking if an equation is quadratic
To check an equation, multiply out any products, move all terms to one side, and combine similar terms. Then look at the highest exponent. The module gives three examples:
- 3x² + 2x = 9 is quadratic but not yet in standard form. In standard form it is 3x² + 2x − 9 = 0, so a = 3, b = 2, and c = −9.
- (x − 5)(2x + 3) = 7 becomes 2x² − 7x − 8 = 0. It is quadratic, with a = 2, b = −7, and c = −8.
- 2x² − 15 = 2(x² + 7x) simplifies to −14x − 15 = 0. The x² terms cancel, so the degree is 1. This is a linear equation, not a quadratic one.
Writing quadratic equations from situations
The module shows how to represent the unknowns with a variable and then form an equation:
- Rectangular prism: The height is 3 m, the length is three times the width, and the volume is 18 m³. Using V = lwh, 18 = (3w)(w)(3), which gives 9w² − 18 = 0.
- Model house window: The area is 315 cm², and the length is 6 cm more than the width. This gives x(x + 6) = 315, or x² + 6x − 315 = 0.
- Two numbers: The product is 48 and the sum is 16. This gives x(16 − x) = 48, or −x² + 16x − 48 = 0.
- Consecutive even integers: The product is 288. This gives x(x + 2) = 288, or x² + 2x − 288 = 0.
Parts of the module
- What I Need to Know: the competency and an overview
- What I Know: a 10-item multiple-choice pretest
- What’s In: a review of multiplying polynomials, with six practice items
- What’s New: the Vegetable Garden situation
- What Is It: the definition, the standard form, and worked examples
- What’s More: Test Yourself exercises on identifying quadratic equations and writing them from situations
- What I Can Do: 10 equations to classify, with a, b, and c to identify
- What I Have Learned: a one-sentence summary of the key idea
- Assessment: a 10-item multiple-choice post-test
- Additional Activities: write your own real-life word problem that leads to a quadratic equation
- Problem-Based Worksheet: Mang Toti’s vegetable garden (area 12 m², length 4 m longer than the width), with solutions
- Answer Key: for What’s In, What I Know, What’s More, What I Can Do, and the Assessment
How to use this module
For learners: Start with the pretest in What I Know and write your answers on a separate answer sheet. After checking, note the items you missed and look for the right answers as you go through the module. Then work through the parts in order. The polynomial review in What’s In matters, because most items ask you to multiply binomials before deciding if an equation is quadratic. Finish with the Assessment and the Additional Activities.
For parents: Help your child keep a separate answer sheet. Encourage them to try each activity before looking at the answer key at the end of the module.
For teachers: The module includes an answer key and a problem-based worksheet with full solutions. The Additional Activities task, where learners write and illustrate their own word problem, can serve as a performance output. When checking, keep in mind:
- In Example 2a, one line is misprinted as 19w² − 18 = 0. The correct equation is 9w² − 18 = 0.
- In What’s More, the key gives x² + 8x = 100. In standard form, this is x² + 8x − 100 = 0.
- For Assessment item 8, the key lists D. However, choice B (“the length of each window is 6 cm more than the width”) is the statement that forms a quadratic equation, matching the module’s own window example.
Frequently asked questions
What grade and quarter is this module for?
It is for Grade 9 Mathematics, Quarter 1, Module 1, and is intended for Week 1. It covers learning competency M9AL-Ia-1, illustrating quadratic equations.
What is the standard form of a quadratic equation?
The standard form is ax² + bx + c = 0, where a, b, and c are real numbers and a is not zero. Here, ax² is the quadratic term, bx is the linear term, and c is the constant term.
How do I know if an equation is quadratic?
Simplify it by multiplying out products, moving all terms to one side, and combining similar terms. If the highest exponent of the variable is 2, it is quadratic. If the x² terms cancel out, as in 2x² − 15 = 2(x² + 7x), the equation is linear.
Does the module have an answer key?
Yes. The answer key covers What’s In, What I Know, What’s More, What I Can Do, and the Assessment. The problem-based worksheet also comes with worked solutions.
What real-life situations does the module use?
It uses a backyard vegetable garden, a rectangular prism’s volume, a model house window, the product and sum of two numbers, and consecutive even integers.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 1-Module 1 Illustrating Quadratic Equation
Week 1 Learning Code -M9AL-Ia-1
[illustration: people learning with laptops, books, and speech bubbles]
GRADE 9 Learning Module for Junior High School Mathematics
Development Team of the Module
Writers: Analynn M. Argel- MTII; Rowena F. Reyes- T1; Queenie Pearl E. Domasig – TII
Editor: Sally C. Caleja– Head Teacher VI; Abigail T. Laureano – Head Teacher VI; Mary Joy T. Villas – Teacher II
Validators: Remylinda T. Soriano, EPS, Math; Angelita Z. Modesto, PSDS; George B. Borromeo, PSDS
Illustrator: Writers
Layout Artist: Writers
Management Team: Malcolm S. Garma, Regional Director; Genia V. Santos, CLMD Chief; Dennis M. Mendoza, Regional EPS in Charge of LRMS and Regional ADM Coordinator; Maria Magdalena M. Lim, CESO V, Schools Division Superintendent; Aida H. Rondilla, Chief-CID; Lucky S. Carpio, Division EPS in Charge of LRMS and Division ADM Coordinator
Quarter 1 – Module 1 – New Normal Math for G9 First Edition 2020 Copyright © 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties.
Borrowed materials (i.e. songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this module are owned by their respective copyright holders. Every effort has been exerted to locate and seek permission to use these materials from their respective copyright owners. The publisher and authors do not represent nor claim ownership over them.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
GRADE 9 Learning Module for Junior High School Mathematics
Mathematics Quarter 1-Module 1 Illustrating Quadratic Equation
What I Need to Know
MODULE 1 ILLUSTRATING QUADRATIC EQUATION
In the previous year level, you have learned about linear equations and how to find their solutions. You were also given some applications to solve some real-life problems using this concept. Now, since you are in a new grade level, you will acquire knowledge and skills about quadratic equations. In this module, you will learn different ways to illustrate quadratic equation and it will be helpful to understand the succeeding topics.
LEARNING COMPETENCY The learners will be able to:
- illustrate quadratic equation. M9AL-Ia-1
What I Know
Find out how much you already know about quadratic equation as presented in this module. Write the letter that you think is the best answer to each question on your answer sheet. Answer all items. After taking and checking this short test, take note of the items that you were not able to answer correctly and look for the right answer as you go through this module.
What do you call a second-degree polynomial equation 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
- B. Linear Inequality
- C. Quadratic Equation
- D. Quadratic Inequality
Which of the following is not quadratic equation?
- A. 8𝑘 – 3 = 12
- B. 9𝑟² – 25 = 0
- C. 𝑥² – 5𝑥 + 3 = 0
- D. (2𝑥 + 5) (𝑥 – 1) = −6
Find the linear term in the quadratic equation 3𝑥(𝑥 − 2) = −7𝑥 + 1.
- A. 13x
- B. -13x
- C. x
- D. –7x
What is the constant term of (𝑥 − 5) (2𝑥 + 3) = 7?
- A. 7
- B. -8
- C. -15
- D. -22
Which of the following quadratic equations is equivalent to 𝑥² + 9𝑥 = −8?
- A. (x + 1) (x + 8) = 0
- B. (x – 1) (x + 8) = 0
- C. (x -1) (x – 8) = 0
- D. (x+ 1) (x – 8) = 0
GRADE 9 Learning Module for Junior High School Mathematics
Which situations illustrates quadratic equation?
- A. The length of a rectangular board is 3m longer than its width and its perimeter is 25m.
- B. Joey paid at least ₱2,000 for the shirt and pants. The cost of pants is ₱700 more than the shirt.
- C. A garden’s length is 7m longer that its width and the area is 18 square meters.
- D. A lot cost ₱4,000 per square meter and the area is 120 square meters.
If (𝑥 + 2)² = 3(𝑥 + 2) is written in standard form, the value of b is _____.
- A. 7
- B. 5
- C. 3
- D. 1
When the quadratic equation (2𝑥 + 5) (𝑥 − 1) = −6 is written in standard form, what are the values of a, b, and c?
- A. 𝑎 = −2, 𝑏 = 3, 𝑐 = 1
- B. 𝑎 = 2, 𝑏 = 3, 𝑐 = 1
- C. 𝑎 = 2, 𝑏 = −3, 𝑐 = 1
- D. 𝑎 = −2, 𝑏 = −3, 𝑐 = 1
The dimensions of a rectangle with an area of 56 meters² are (3x – 1) meters by (x + 4) meters. Which of the following quadratic equation in standard form represents the situation?
- A. 3x² + 11x – 60 = 0
- B. 3x² + 11x + 60 = 0
- C. 3x² + 11x – 52 = 0
- D. 3x² + 11x + 52 = 0
When 7 – 5𝑥 = 3𝑥² was written in standard form, Ana and Elsa got the following answers: Ana: 3𝑥² + 5𝑥 – 7 = 0 Elsa: −3𝑥² – 5𝑥 + 7 = 0. Who do you think got the correct answer?
- A. Ana
- B. Elsa
- C. Both
- D. None
What’s In
Let us recall how to multiply polynomials. Remember that to multiply two polynomials, multiply each term of one polynomial by each term of the other polynomial then simplify by combining similar terms if needed.
Study the examples below:
a. 𝑥 ( 𝑥 + 5 ) = 𝑥² + 5𝑥 b. (𝑥 + 3) ( 𝑥 – 2) = 𝑥² + 𝑥 – 6
Try this!
Multiply and simplify, if needed.
- 𝑥 ( 2𝑥 – 7)
- (𝑥 + 5)( 𝑥 + 9)
- (2𝑥 – 1)( 𝑥 + 3)
- (𝑥 + 6)²
- (5𝑥 + 4) (5𝑥 – 4)
- (3𝑥 – 8) (𝑥 + 2)
Let us analyze!
a. How will you describe the products obtained? b. What is the degree of each product?
GRADE 9 Learning Module for Junior High School Mathematics
What’s New
Communication, character building and collaboration
There are situations that represent other function that is equally important as linear function. Perform the activity below to find out.
Vegetable Garden
In the middle of a crisis where establishments are closed and prime commodities are hard to find, we think of having our own vegetable garden. This way, we can get our supplies of vegetable right in our own backyard.
Aling Tuding is a resourceful person that is why she is planning to convert her rectangular vacant lot at the back of her house into a vegetable garden. She remembered that the area of her vacant lot is 15m². She also recalled that the length is 3 meters longer than the width. Without even starting to cultivate the soil, Aling Tuding is already excited to harvest her favorite vegetables soon.
- What equation would represent the attributes of the rectangular garden? Do you think you can use this equation to find the dimensions of the rectangular garden? Why or why not?
Despite of the crisis, we should be like Aling Tuding. We must make this crisis an opportunity to develop and strengthen ourselves.
What Is It
Communication, Critical Thinking, and Collaboration
Let’s investigate the measurement of Aling Tuding’s vegetable garden.
First, let us identify and represent the unknowns in the problem. If we let x be the width of the rectangular garden, then the length will be x + 3 since it is 3 more than the width. That is,
x = width of the garden x + 3 = length of the garden 15 (square meters) = area of the rectangular garden
Then we can represent the dimensions as shown in the figure:
[chart: diagram; rectangle figure with labels “x + 3” (top), “Area = 15m²” (inside), and “x” (right side)]
GRADE 9 Learning Module for Junior High School Mathematics
The area (A) of any rectangle can be solve by the formula: 𝐴 = 𝑙𝑤, where 𝑙 is the length and the w is the width. Hence, in the given figure:
𝐴 = 𝑙𝑤 15 = 𝑥(𝑥 + 3) Substituting to the area formula 15 = 𝑥² + 3𝑥 by Distributive Property 𝑥² + 3𝑥 – 15 = 0 by Addition Property of Equality
What is the degree of the equation obtained?
What do you call this kind of equation?
The equation in one variable obtained from the situation, 𝑥² + 3𝑥 – 15 = 0, has a degree of 2. It means the highest exponent of the variable is 2. This kind of second-degree equation is also called quadratic equation.
A quadratic equation in one variable is of the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 where 𝑎, 𝑏, and 𝑐 are real numbers and 𝑎 ≠ 0. This form of quadratic equation is written in standard form. In this equation, 𝑎𝑥² is the quadratic term, 𝑏𝑥 is the linear term, and 𝑐 is the arithmetic or constant term. In addition,
- 𝑎 in the quadratic term is the coefficient of x² which may be positive or negative but not equal to zero.
- 𝑏 in the linear term is the coefficient of x which may be positive, negative or zero.
- 𝑐 is the constant term which may be positive, negative or zero.
Going back to our equation 𝑥² + 3𝑥 – 15 = 0, since it is written in standard form, we can easily identify the values of a, b and c. Look at the table below to understand it better.
| Quadratic Equation in Standard Form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 | Quadratic Term 𝑎𝑥² | Linear Term 𝑏𝑥 | Constant Term 𝑐 |
|---|---|---|---|
| 𝑥² + 3𝑥 – 15 = 0 | x² | 3x | -15 |
| Therefore, | a = 1 since 1 is the coefficient of x² | b = 3 since 3 is the coefficient of x | c = -15 since -15 is the constant term |
GRADE 9 Learning Module for Junior High School Mathematics
Example 1:
Determine if the given equation is quadratic. If yes, identify the values of 𝑎, 𝑏 and 𝑐.
a. 3𝑥² + 2𝑥 = 9 b. (𝑥 − 5) (2𝑥 + 3) = 7 c. 2𝑥² – 15 = 2(𝑥² + 7𝑥)
Solutions:
a. The equation 3𝑥² + 2𝑥 = 9 is quadratic but not in standard form. The standard form is 3𝑥² + 2𝑥 – 9 = 0 with 𝑎 = 3, 𝑏 = 2 and 𝑐 = −9
b. To check if the equation is quadratic, multiply the left side of the equation and simplify by combining similar terms.
(𝑥 − 5) (2𝑥 + 3) = 7 2𝑥² – 7𝑥 – 15 = 7 2𝑥² – 7𝑥 – 8 = 0
Since the degree of the equation is 2, it is a quadratic equation. The value of 𝑎 = 2, 𝑏 = −7, and 𝑐 = −8.
c. To check if the equation is quadratic, simplify the left side of the equation then combine similar terms.
2𝑥² – 15 = 2(𝑥² + 7𝑥) 2𝑥² – 15 = 2𝑥² + 14𝑥 2𝑥² – 2𝑥² – 14𝑥 – 15 = 0 − 14𝑥 – 15 = 0
Since the resulting equation has a degree of 1, then it is not quadratic. Thus, − 14𝑥 – 15 = 0 is a linear equation.
Check your understanding by completing the table below.
| Quadratic Equations | Standard Form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 | a | b | c |
|---|---|---|---|---|
| Example: 𝑥 ( 2𝑥 – 7) = 0 | 2x² – 7x = 0 | 2 | -7 | 0 |
| 1. (𝑥 + 5)( 𝑥 + 9) = 20 | ||||
| 2. (2𝑥 – 1)( 𝑥 + 3) = -2x | ||||
| 3. (𝑥 + 6)² = 12x | ||||
| 4. (5𝑥 + 4) (5𝑥 – 4) = 0 | ||||
| 5. (3𝑥 – 8)(𝑥 + 2) = 2𝑥² |
GRADE 9 Learning Module for Junior High School Mathematics
Example 2: Represent each situation using quadratic equation.
a. A rectangular prism has a height of 3 m. Its length is three times its width. The volume of the prism is 18 m³.
Solution:
Given: Rectangular Prism Height = 3 m Volume = 18 m³
Representation of unknown: Let w – be the width Let 3w – be the length (since, the length is 3 times the width)
Solution: Volume of the Rectangular Prism = Length x Width x Height or 𝑉 = 𝑙𝑤ℎ 18 = (3w)(w)(3) 18 = 9w² 19w² – 18 = 0
[transcriber’s note: the line above is printed as “19w2 – 18 = 0” in the original, and the next line gives “9w2 -18 = 0”]
Thus, the quadratic equation that described the situation is 9w² -18 = 0.
b. Jody is constructing a model house. He wants each window to have an area of 315 cm², and he wants the length of each window to be 6 cm more than the width.
Solution:
Given: Area of the rectangular window = 315 cm²
Representation of unknown: Let x – be the width Let x + 6 – be the length
Solution: A = length x width or A = L x W 315 = x(x+6) x² + 6x = 315 x² + 6x – 315 = 0
Thus, the quadratic equation that described the situation is x² + 6x – 315 = 0.
c. The product of two numbers is 48 and their sum is 16.
Solution:
Given: Product of two numbers is 48 Sum of two numbers is 16
Representation of unknown: Let x be the 1st unknown number Let 16 − x be the 2nd unknown number
Solution: 1st number · 2nd number = 48 x(16 − x) = 48 16x − x² = 48 −x² + 16x − 48 = 0 Therefore, the quadratic equation for the described numbers is −x² + 16x − 48 = 0
d. The product of two consecutive even integers is 288. Solution: Given: Numbers are two consecutive even integers Product of two numbers is 288
Representation of unknown: Let x be the 1st even number Let x + 2 be the 2nd even number
Solution: 1st number · 2nd number = 288 x(x + 2) = 288 x² + 2x = 288 x² + 2x − 288 = 0 Therefore, the quadratic equation for the described numbers is x² + 2x − 288 = 0
WHAT’S MORE
Critical Thinking
TEST YOURSELF!
A. Which of these equations describe a quadratic equation?
- A = πr²
- x + 4x² = 0
- (x − 2)² − 5 = 0
- (x + 3) + 8 = 0
- x² = 0
B. Represent the following situations using quadratic equation in standard form.
- If the square of a number is added to 8 times the number, the result is 100.
- Mr. Apoloan wants to lay out a rectangular playground with an area of 30 square feet. The desired length will be 7 times the width.
Which of the following is a quadratic equation? For those that are, identify the value of a, b and c.
- x² – 6x + 2 = 0
- 3x + 5 = 0
- 2x² + 7x = 15
- (x + 1)² = 2(x – 3)
- x² + 2x + 1 = 5x + 6
- (x − 1)(x + 2) = x(x + 5)
- (x + 2)(x – 3) = 5
- x(x² + 3x – 10) = 0
- (x – 1)² + 3 = 2x + 1
- (x + 2)³ = x(x² – 10x + 25)
WHAT I HAVE LEARNED
A quadratic equation in one variable is a second-degree equation that can be written in the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
WHAT I CAN DO
Critical Thinking
ASSESSMENT
Write the letter of the correct answer on your answer sheet.
Given a, b, and c are real numbers and a ≠ 0, which of the following is the standard form of quadratic equation?
A. ax² + bx + c = 0 C. a = bx + c
B. ax + b + c = 0 D. ax + b² + c = 0Which of the following is a quadratic equation?
A. 3x + 5 = 0 C. (x + 2)(x – 3) = 5
B. (x + 1)(x – 2) = x² D. 2x³ + x² – 5 = 0Given (x – 1)² + 3 = 2x + 1, find the value of the constant term.
A. 1 B. 3 C. -1 D. -3Which of the following is equivalent to x² + 4x – 7 = 0?
A. (x + 1)(x – 7) = 0 C. (x + 5)(x – 3) = 6
B. (x + 4)(x – 3) = 5 D. (x + 1)(x + 3) = 10When the quadratic equation (x + 5)(2x – 3) = 2(x + 1) is written in standard form, what are the values of a, b, and c.
A. a = 4, b = 19, c = −17 C. a = 1, b = −5, c = 17
B. a = 2, b = 5, c = −17 D. a = 2, b = 10, c = 17Which of the following situations illustrates quadratic equation?
A. A garden’s length is 7m longer that its width and the area is 18 square meter.
B. A lot cost P4,000 per square meter and the area is 120 square meters.
C. Joey paid at least P2,000 for the shirt and pants. The cost of pants is P700 more than the shirt.
D. The length of a rectangular board is 3m longer than the width and the perimeter is 25m.Express the statement in quadratic equation: “A rectangular field with an area of 120m² has its length 12m. longer than the width.”
A. x² − 12x − 120 = 0 C. x² + 12x – 120 = 0
B. x² + 12x + 120 = 0 D. x² – 12x + 120 = 0Ryan is constructing a play house. He wants each window to have an area of 315 cm². Which of the statements below forms a quadratic equation?
A. The length of each window is 4 less than the square of the width.
B. The length of each window is 6 cm more than the width.
C. The area of the window is twice its width.
D. The sum of the length and width is twice its area.The area of a square with side s is 144 cm². Which quadratic equation represents the area of the square?
A. 4s = 144 B. s² = 144 C. s(s + 1) = 144 D. (s + 1)² = 144A rectangular lot has an area of 132m² and a perimeter of 46m. Which of the following quadratic equations illustrates the given situation?
A. x² − 23x + 132 = 0 C. x² + 46x – 132 = 0
B. x² + 23x + 132 = 0 D. x² – 46x + 132 = 0
ADDITIONAL ACTIVITIES
Communication, Critical Thinking, Creativity and Character Building
After dealing with examples of quadratic equations and possible situation that can represent quadratic equation, think and reflect. How will quadratic equations help you in solving real-life problems and making decisions?
Write a real-life word problem that will lead you to forming quadratic equation. Provide an illustration of the problem and guide questions such that it will be expressed as quadratic equation in standard form.
E-Search
You may also check the following link for your reference and further learnings on illustrating quadratic equations.
- https://examples.yourdictionary.com/examples-of-quadratic-equation.html
- https://www.mathsisfun.com/algebra/quadratic-equation.html
- https://mathbitsnotebook.com/Algebra1/Quadratics/QDquadequations.html
- https://www.math-only-math.com/introduction-to-quadratic-equation.html
https://www.mathsisfun.com/algebra/quadratic-equation.html - https://bit.ly/2W94SG7
REFERENCES
Dilao, S. J., & Bernabe, J. G. (2009). Intermediate Algebra. pp 52-54. Quezon CIty: SD Publicaton.
MacKeague, Charles, Intermediate Algebra, Concepts and Graphs. Saunders College Publishing, USA
Mathematics 9 Learner’s Material, Department of Education
Orines, Fernando B., et. al. Next Century Mathematics (Intermediate Algebra). Quezon City, Philippines: Phoenix Publishing House
Oronce, Orlando A. & Mendoza, Marilyn O. Exploring Mathematics II (Intermediate Algebra). Manila, Philippines: Rex Book Store Inc.
Ogena, Ester, et. al. Our Math Grade 9. Mc Graw Hill, Vibal Group. Inc.
PROBLEM – BASED WORKSHEET
The Vegetable Garden
Mang Toti joined an organization that encourages people to go into backyard farming. He owns a rectangular piece of vacant lot adjacent to his house. He is planning to convert his lot into vegetable garden for additional income.
[illustration: a small house beside a tree and a sun, with a shaded lot in front]
LET’S ANALYZE
If the area of the vacant lot is 12m² and the length is 4m longer than the width, how will you represent the length if the width is represented by w? __________________
What expression represents the area of the of the vacant lot? _____________________
What quadratic equation in general form represents the situation? _______________
Supposing Mang Toti bought 16m of fencing materials and plan to place it all around his rectangular land, what equation describes the situation? ________________
transcriber’s note: this page is printed upside down; the text below is transcribed in reading order after rotating it.
PROBLEM – BASED WORKSHEET
Solutions:
Illustrate the problem:
Let w = width (W)
w + 4 = length (L)w(w + 4)
Since area of rectangle is LW, then
w(w + 4) = 12
w² + 4w = 12
w² + 4w − 12 = 0
Therefore the quadratic equation that describes the situation is w² + 4w − 12 = 0.
If there will 16m of fencing material to be placed around the lot, then the perimeter is 16m. Thus,
2L + 2W = 16,
finding the semi-perimeter, we have
L + W = 8,
solve the equation in terms of L,
L = 8 − WThe equation that describes the area we will be,
substitute L = 8 − w, we have
LW = 12
(8 − w) w = 12
−w² + 8w = 12
by addition property of equality,
simplify,
−w² + 8w − 12 = 0
w² − 8w + 12 = 0
transcriber’s note: this page is printed upside down, and its unrotated fragments overlap the rotated answer key; the answer key below is transcribed from the rotated text.
MODULE 1
WHAT’S IN
transcriber’s note: the upright fragments of the “WHAT’S IN” items are jumbled and cannot be reliably ordered; the answer key items are given below.
ANSWER KEY
WHAT’S IN
- 2x² − 7x
- x² + 14x + 45
- 2x² + 5x − 3
- x² + 12x + 36
- 25x² − 16
- 3x² − 2x − 16
a. The products are all polynomials. b. The degree of the product is 2.
WHAT I CAN DO
- Quadratic; a = 1, b = −6, c = 2
- Not quadratic
- Quadratic; a = 2, b = 7, c = −15
- Quadratic; a = 1, b = 0, c = 7
- Quadratic; a = 1, b = −3, c = −5
- Not quadratic
- Quadratic; a = 1, b = −1, c = −11
- Not quadratic
- Quadratic; a = 1, b = −4, c = 3
- Quadratic; a = 16, b = −13, c = 8
WHAT’S MORE
I. 1, 2, 3, 5 are quadratic equations II. 1. x² + 8x = 100 2. 7x² − 30 = 0
WHAT I KNOW
- C
- A
- C
- D
- A
- C
- D
- B
- A
- A
ASSESSMENT
- A
- C
- B
- D
- B
- A
- C
- D
- B
- A
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