9 NOT Mathematics 11 Quarter 1 - Module 4: Solving Equations Transformable into Quadratic Equations Department of Education ● Republic of the Philippines Math- Grade 9 Alternative Delivery Mode Quarter 1 - Module 4: Solving Equations Transformable into Quadratic Equations First Edition, 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 royalty. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book 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 – Division of Iligan City Schools Division Superintendent: Roy Angelo L. Gazo, PhD.,CESO V Development Team of the Module Author/s: Daryl G. Bastatas Evaluators/Editor: Priscilla G. Luzon, Natividad B. Finley Illustrator/Layout Artist: Management Team Chairperson: Roy Angelo E. Gazo, PhD, CESO V Schools Division Superintendent Co-Chairpersons: Nimfa R. Lago, PhD, CESE Assistant Schools Division Superintendent Members Henry B. Abueva, OIC - CID Chief Exquil Bryan P. Aron, EPS - Math Sherlita L. Daguisonan, EPS - LRMS Meriam S. Otarra, PDO II Charlotte D. Quidlat, Librarian II Printed in the Philippines by Department of Education – Division of Iligan City Office Address: General Aguinaldo, St., Iligan City Telefax: (063)221-6069 E-mail Address: [email protected] 9 Mathematics Quarter 1 - Module 4 Solving Equations Transformable into Quadratic Equation 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 at action@ deped.gov.ph. We value your feedback and recommendations. Department of Education ● Republic of the Philippines 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: Equations Transformable to Quadratic Functions .................................................................... 1 What I Need to Know..................................................................................................... 1 What I Know ................................................................................................................... 1 What’s In ............................................................................................................................ 2 What’s New ..................................................................................................................... 3 What Is It ........................................................................................................................... 3 What’s More…. ................................................................................................................ 6 What I Have Learned..................................................................................................... 6 What I Can Do ................................................................................................................. 6 Summary .................................................................................................................................................. 7 Assessment: (Post-Test) ................................................................................................................... 7 Key to Answers ...................................................................................................................................... 8 References ............................................................................................................................................... 9 What This Module is About In this module, the lesson starts in assessing your understanding of the various mathematics principles and concepts studied previously, and enhance skills in performing mathematical operations. All these skills and knowledge may help you in solving Equations transformable into Quadratic Equations. As you embrace through this lesson, be aware of this significant question: “How are these Algebraic Equations be transformed into Quadratic Equations?”. To discover the answers to this vital question, the set of activities must be performed. If difficulty ascends, you may ask your teacher, peer, or friends to help you in revisiting the modules completed over earlier. Your teacher can help you measure your answers. What I Need to Know In this lesson you will learn to: Solve Algebraic Equations which can be transformed to Quadratic Equations; Check for extraneous solutions. i 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. ii Lesson Equations Transformable into Quadratic Equations 1 What I Need to Know In this Lesson, you will have a chance to build up your skills in solving equations which can be transformed to quadratic equations. What I Know Pre-Assessment Test Directions: Find out how much you already know about this module. Solve the following problems. 1. Express in simplest form. A B. C. D. 2. Which of the following is the standard form of quadratic equations? A. C. B. D. y = mx +b 3. Express (w+7)(w-2) =0 in standard form A. C. B. D. 4. Find the roots of the equation . A. -5 , -2 B. 3 , 4 c. 5 , 2 D. 2 , -3 5. Which of the following rational algebraic equations is transformable to a quadratic equation? A. = C. = B. = 6m D. = 7 6. Which of the following equations have extraneous roots or solutions? A. - x x x C. - x x x B. - D. x (x +3) = 28 x 1 What’s In Activity 1: Who Said It? Direction: Answer the puzzle below by simplifying the following expressions. Then shade the box containing the corresponding answer. The unshaded boxes will show the answer to this puzzle. Which great mathematician and scientist said: “Do not worry about difficulties in Mathematics. I can assure you that mine are still greater.” 1. 7. 2. 8. 3. 9. 4. 10. 5. 11. 6. 12. C E I A N H S H A T R E W L I E T N P O 2 What’s New In learning this module, you will need your skills in adding and subtracting rational algebraic expressions. Because of that, let’s recall these first. Example: Find the sum of and . x Step 1: Write the expression x x Step 2: Find the Least Common Denominator (LCD) of the rational algebraic expressions, and . x 2x = 2 · x 6x = 2 · 3 · x 2 · 3 · x = 6x Therefore, the LCD is 6x. Step 2: Rewrite the expression using LCD, 6x. or x x x x x x Now, you’re ready! Let’s try having the activity below. What Is It Activity 2: Let’s Add and Subtract! Direction: Perform the indicated operation then express your answer in simplest form. 1. 4. 2. 3. Were you able to add or subtract the rational expressions and simplify the results? Suppose you were given a rational algebraic equation, how would you find its solution/s? You will learn this in the succeeding activities. 3 Solving Quadratic Equations That Are Not Written in Standard Form 2 Standard Form of Quadratic Equation: ax + bx + c = 0, where x is the variable and a, b and c are constants (a ≠ 0) Example 1: Solve This quadratic equation is not written in standard form. To write the quadratic equation in standard form: write the given equation = 36 simplify the left side of the equation x2 - 5x = 36 transform to standard form of quadratic equation Use any of the four methods in finding the solutions of the quadratic equation Try factoring in finding the roots