Math 9 Quarter 1 Module 1 Illustrating Quadratic Equation 2020 PDF

Mathematics Quarter 1-Module 1 Illustrating Quadratic Equation Week 1 Learning Code -M9AL-Ia-1  GRADE 9 Learning Module for Junior High School Mathematics Quarte...

Mathematics Quarter 1-Module 1 Illustrating Quadratic Equation Week 1 Learning Code -M9AL-Ia-1  GRADE 9 Learning Module for Junior High School Mathematics 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 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  GRADE 9 Learning Module for Junior High School Mathematics 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. WHAT I NEED TO KNOW PPREPREVIER! LEARNING COMPETENCY The learners will be able to: illustrate quadratic equation.M9AL-Ia-1 WHAT I KNOW PPREPREVIER ! 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. 1. What do you call a second-degree polynomial equation that can be written in the form 𝑎𝑥 2 + 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 not quadratic equation? A. 8𝑘 – 3 = 12 C. 𝑥 2 – 5𝑥 + 3 = 0 B. 9𝑟 2 – 25 = 0 D. (2𝑥 + 5) (𝑥 – 1) = −6 3. Find the linear term in the quadratic equation 3𝑥(𝑥 − 2) = −7𝑥 + 1. A. 13x B. -13x C. x D. –7x 4. What is the constant term of (𝑥 − 5) (2𝑥 + 3) = 7? A. 7 B. -8 C. -15 D. -22 5. Which of the following quadratic equations is equivalent to 𝑥 2 + 9𝑥 = −8? A. (x + 1) (x + 8) = 0 C. (x -1) (x – 8) = 0 B. (x – 1) (x + 8) = 0 D. (x+ 1) (x - 8) = 0 1  GRADE 9 Learning Module for Junior High School Mathematics 6. 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. 7. 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 8. 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 C. 𝑎 = 2, 𝑏 = −3, 𝑐 = 1 B. 𝑎 = 2, 𝑏 = 3, 𝑐 = 1 D. 𝑎 = −2, 𝑏 = −3, 𝑐 = 1 9. The dimensions of a rectangle with an area of 56 meters 2 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 C. 3x2 + 11x – 52 = 0 B. 3x + 11x + 60 = 0 2 D. 3x2 + 11x + 52 = 0 10. When 7 – 5𝑥 = 3𝑥 2 was written in standard form, Ana and Elsa got the following answers: Ana: 3𝑥 2 + 5𝑥 – 7 = 0 Elsa: −3𝑥 2 – 5𝑥 + 7 = 0. Who do you think got the correct answer? A. Ana B. Elsa C. Both D. None WHAT’S IN PPREPREV IER! 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 ) = 𝑥 2 + 5𝑥 b. (𝑥 + 3) ( 𝑥 – 2) = 𝑥 2 + 𝑥 – 6 Try this! Multiply and simplify, if needed. 1. 𝑥 ( 2𝑥 – 7) 2. (𝑥 + 5)( 𝑥 + 9) Let us analyze! 3. (2𝑥 – 1)( 𝑥 + 3) a. How will you describe the products 4. (𝑥 + 6)2 obtained? 5. (5𝑥 + 4) (5𝑥 – 4) b. What is the degree of each product? 6. (3𝑥 – 8) (𝑥 + 2) 2  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 15m 2. 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. Communication, Critical WHAT IS IT 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: 3  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 = 𝑥 2 + 3𝑥 by Distributive Property 𝑥 2 + 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, 𝑥 2 + 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 𝑎𝑥 2 + 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, 𝑎𝑥 2 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 𝑥 2 + 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 Quadratic Term Linear Term Constant Term Standard Form ax2 bx c 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 𝑥 2 + 3𝑥 – 15 = 0 x2 3x -15 Therefore, a=1 b=3 c = -15 since 1 is the since 3 is the since -15 is the coefficient of x2 coefficient of x constant term 4  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. 3𝑥 2 + 2𝑥 = 9 b. (𝑥 − 5) (2𝑥 + 3) = 7 c. 2𝑥 2 – 15 = 2(𝑥 2 + 7𝑥) Solutions: a. The equation 3𝑥 2 + 2𝑥 = 9 is quadratic but not in standard form. The standard form is 3𝑥 2 + 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 2𝑥 2 – 7𝑥 – 15 = 7 2𝑥 2 – 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. 2𝑥 2 – 15 = 2(𝑥 2 + 7𝑥) 2𝑥 2 – 15 = 2𝑥 2 + 14𝑥 2 2 2𝑥 – 2𝑥 – 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 a b c 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 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) = 2𝑥 2 5  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 19w – 18 = 0 2 Thus, the quadratic equation that described the situation is 9w 2 -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 cm 2 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 x + 6x – 315 = 0 2 Thus, the quadratic equation that described the situation is x 2 + 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 6  GRADE 9 Learning Module for Junior High School Mathematics 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 − 𝑥 2 = 48 2 −𝑥 + 16𝑥 − 48 = 0 Therefore, the quadratic equation for the described numbers is −𝑥 2 + 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 𝑥 2 + 2𝑥 = 288 2 𝑥 + 2𝑥 − 288 = 0 Therefore, the quadratic equation for the described numbers is 𝑥 2 + 2𝑥 − 288 = 0 WHAT’S MORE Critical Thinking TEST YOURSELF! A. Which of these equations describe a quadratic equation? 1. 𝐴 = 𝜋𝑟 2 2. 𝑥 + 4𝑥 2 = 0 3. (𝑥 − 2)2 − 5 = 0 4. (𝑥 + 3) + 8 = 0 5. 𝑥 2 = 0 B. Represent the following situations using quadratic equation in standard form. 1. If the square of a number is added to 8 times the number, the result is 100. 2. 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. 7  GRADE 9 Learning Module for Junior High School Mathematics WHAT I HAVE LEARNED A quadratic equation in one variable is a second-degree equation that can be written in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎, 𝑏 , and 𝑐 are real numbers and 𝑎 ≠ 0. Critical Thinking WHAT I CAN DO Which of the following is a quadratic equation? For those that are, identify the value of a, b and c. 1. 𝑥 2 – 6𝑥 + 2 = 0 6. (𝑥 − 1) (𝑥 + 2) = 𝑥(𝑥 + 5) 2. 3𝑥 + 5 = 0 7. (𝑥 + 2) (𝑥 – 3) = 5 3. 2𝑥 2 + 7𝑥 = 15 8. 𝑥(𝑥 2 + 3𝑥 – 10) = 0 4. (𝑥 + 1) = 2(𝑥 – 3) 2 9. (𝑥 – 1)2 + 3 = 2𝑥 + 1 5. 𝑥 2 + 2𝑥 + 1 = 5𝑥 + 6 10. (𝑥 + 2)3 = 𝑥(𝑥 2 – 10𝑥 + 25) ASSESSMENT Write the letter of the correct answer on your answer sheet. 1. Given a, b, and c are real numbers and 𝑎 ≠ 0, which of the following is the standard form of quadratic equation? A. 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 C. 𝑎 = 𝑏𝑥 + 𝑐 B. 𝑎𝑥 + 𝑏 + 𝑐 = 0 D. 𝑎𝑥 + 𝑏 2 + 𝑐 = 0 2. Which of the following is a quadratic equation? A. 3𝑥 + 5 = 0 C. (𝑥 + 2) (𝑥 – 3) = 5 B. (𝑥 + 1) (𝑥 – 2) = 𝑥 2 D. 2𝑥 3 + 𝑥 2 – 5 = 0 3. Given (𝑥 – 1)2 + 3 = 2𝑥 + 1, find the value of the constant term. A. 1 B. 3 C. -1 D. -3 4. Which of the following is equivalent to 𝑥 2 + 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) = 10 5. When 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 = 17 6. Which 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. 8  GRADE 9 Learning Module for Junior High School Mathematics 7. Express the statement in quadratic equation: “A rectangular field with an area of 120m2 has its length 12m. longer than the width.” A. 𝑥 2 − 12𝑥 − 120 = 0 C. 𝑥 2 + 12𝑥 – 120 = 0 B. 𝑥 + 12𝑥 + 120 = 0 2 D. 𝑥 2 – 12𝑥 + 120 = 0 8. Ryan 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. 9. 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 = 144 10. A rectangular lot has an area of 132𝑚 and a perimeter of 46𝑚. Which of the 2 following quadratic equations illustrates the given situation? A. 𝑥 2 − 23𝑥 + 132 = 0 C. 𝑥 2 + 46𝑥 – 132 = 0 B. 𝑥 + 23𝑥 + 132 = 0 2 D. 𝑥 2 – 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 9  GRADE 9 Learning Module for Junior High School Mathematics 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. https://www.freepik.com/free-vector/woman-with-long-hair-teaching- online_7707557.htm https://www.freepik.com/free-vector/kids-having-online-lessons_7560046.htm https://www.freepik.com/free-vector/illustration-with-kids-taking-lessons-online- design_7574030.htm 10  GRADE 9 Learning Module for Junior High School Mathematics 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. LET’S ANALYZE 1) If the area of the vacant lot is 12m 2 and the length is 4m longer than the width, how will you represent the length if the width is represented by w? __________________ 2) What expression represents the area of the of the vacant lot? _____________________ 3) What quadratic equation in general form represents the situation? _______________ 4) Supposing Mang Toti bought 16m of fencing materials and plan to place it all around his rectangular land, what equation describes the situation? ________________ 11  GRADE 9 Learning Module for Junior High School Mathematics 12  13 ANSWER MODULEKEY 1 WHAT’S IN 1. 2x2 – 7x a. The products are all polynomials. 2. x2 + 14x + 45 b. The degree of the product is 2. 3. 2x2+ 5x – 3 4. x2 + 12x + 36 5. 25x2 – 16 6. 3x2 – 2x – 16 WHAT I CAN DO 1. Quadratic; a = 1, b = -6, c = 2 2. Not Quadratic 3. Quadratic; a = 2, b = 7, c = -15 4. Quadratic; a = 1, b = 0, c = 7 5. Quadratic; a = 1, b = -3, c = -5 6. Not Quadratic 7. Quadratic; a = 1, b = -1, c = -11 8. Not Quadratic 9. Quadratic; a = 1 , b = -4, c = 3 10. Quadratic; a = 16, b = -13, c = 8 WHAT’S MORE I. 1,2,3,5 are quadratic equations II. 1. 𝑥 2 + 8𝑥 = 100 2. 7𝑥 2 − 30 = 0 WHAT I KNOW ASSESSMENT 1. C 1. A 2. A 2. C 3. C 3. B 4. D 4. D 5. A 5. B 6. C 6. A 7. D 7. C 8. B 8. D 9. A 9. B 10. A 10. A Learning Module for Junior High School Mathematics GRADE 9