Quarter 1, Week 2 Module 2 Solving Quadratic Equations

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MathematicsQuarter 1, Week 2 – Module 2Solving Quadratic Equations

Mathematics - Grade 9Alternative Delivery ModeQuarter 1, Week 2 - Module 2: Solving Quadratic EquationsFirst Edition, 2020Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice 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 ofroyalty.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 fromtheir respective copyright owners. The publisher and authors do not represent nor claimownership over them.Published by the Department of Education – Region 10Regional Director: Dr. Arturo B. Bayocot, CESO IIIAssistant Regional Director: Dr. Victor G. De Gracia Jr., CESO VDevelopment Team of the ModuleAuthor:Evaluators/Editor:Randulph J. CagulaBrenda A. YordanDr. Antonio N. LegaspiNatividad FinleyStephanie Mae R. LanzaderasDr. Renielda Dela ConcepcionPriscilla C. LuzonAnnabelle De GuzmanIllustrator/Layout Artist:Management TeamChairperson:Co-Chairpersons:Dr. Arturo B. Bayocot, CESO IIIRegional DirectorDr. Victor G. De Gracia Jr. CESO VAsst. Regional DirectorRoy Angelo E. Gazo, PhD, CESO VSchools Division SuperintendentNimfa R. Lago,PhD, CESEAssistant Schools Division SuperintendentMala Epra B. Magnaong, Chief ES, CLMDMembersNeil A. Improgo, EPS-LRMSBienvenido U. Tagolimot, Jr., EPS-ADMHenry B. Abueva OIC-CID ChiefExquil Bryan P. Aron, EPS-MathSherlita L. Daguisonan, LRMS ManagerMeriam S. Otarra, PDO IICharlotte D. Quidlat, Librarian IIPrinted in the Philippines byDepartment of Education – Region 10Office Address:Zone 1, DepEd Building, Masterson Avenue, Upper BalulangCagayan de Oro CityContact Number:(088) 880 7072E-mail Address:region10@deped.gov.ph

9MathematicsQuarter 1 Week 2 – Module 2Solving Quadratic EquationsThis instructional material is collaboratively developed and reviewed byeducators from public schools. We encourage teachers and other educationstakeholders to email their feedback, comments, and recommendations to theDepartment of Education-Region 10 at region10@deped.gov.ph.Your feedback and recommendations are highly valued.Department of Education Republic of the Philippines

Table of ContentsWhat This Module is About . iWhat I Need to Know . iHow to Learn from this Module . .iIcons of this Module . iiLesson 2:Solving Quadratic Equations . 1What I Need to Know. 1What I Know . 1Lesson 2a:Solving Quadratic Equations by Extracting Square Roots . 4What I Need to Know. 4What’s In. 4What’s New . 5What Is It . 8What’s More . 12What I Have Learned. 12What I Can Do . 13SummaryLesson 2b:Solving Quadratic Equations by Factoring . 15What I Need to Know. 15What’s In. 15What’s New . 16What Is It . 17What’s More . 21What I Have Learned. 22What I Can Do . 23SummaryLesson 2c:Solving Quadratic Equations by Completing the Square . 25What I Need to Know. 25What’s In. 25

What’s New . 27What Is It . 29What’s More . 34What I Have Learned. 34What I Can Do . 35SummaryLesson 2d:Solving Quadratic Equations by Quadratic Formula . 37What I Need to Know. 37What’s In. 37What’s New . 39What Is It . 40What’s More . 43What I Have Learned. 44What I Can Do . 44SummaryAssessment 47Key to Answers . 49References . 56

What This Module is AboutThis module consists of four lessons on Solving Quadratic Equations.As you go through each part of this module, you will be able to demonstrateunderstanding of the key concepts of solving quadratic equations by (a) extractingsquare roots, (b) factoring (c) completing the square and (d) using quadratic formula.Furthermore, you will be able to formulate and solve real-life problems by usingthese four methods in solving Quadratic Equations.What I Need to KnowIn this module, you are expected to solve quadratic equations by (a)extracting square roots; (b) factoring; (c) completing the square; and (d) using thequadratic formula (M9AL-Ia-b-1). Specifically, you will:1. state the steps in solving quadratic equations by:(a) extracting square roots;(b) factoring;(c) completing the square; and(d) using the quadratic formula2. solve quadratic equations by:(a) extracting square roots;(b) factoring;(c) completing the square; and(d) using the quadratic formula3. use available or recyclable resources to perform the tasks set for you.How to Learn from this ModuleTo achieve the objectives of this module, you are to do the following:1. Take your time reading the lessons carefully.2. Follow the directions and/or instructions in the activities and exercisesdiligently.3. Answer all the given tests and exercises.i

Icons of this ModuleWhat I Need toKnowThis part contains learning objectives thatare set for you to learn as you go along themodule.What I knowThis is an assessment as to your level ofknowledge to the subject matter at hand,meant specifically to gauge prior relatedknowledgeThis part connects previous lesson with thatof the current one.What’s InWhat’s NewAn introduction of the new lesson throughvarious activities, before it will be presentedto youWhat is ItThese are discussions of the activities as away to deepen your discovery and understanding of the concept.What’s MoreThese are follow-up activities that are intended for you to practice further in order tomaster the competencies.What I HaveActivities designed to process what youLearnedhave learned from the lessonWhat I can doThese are tasks that are designed to showcase your skills and knowledge gained, andapplied into real-life concerns and situations.ii

LessonSolving Quadratic Equations2What I Need to KnowYou have already learned the Illustrations of Quadratic Equations in theprevious module. Now, this module; Solving Quadratic Equations will enable you tofind the values of the variable in quadratic equations using the four different methodspresented in each lesson, namely:Lesson 2a. Solving Quadratic Equations by Extracting Square RootsLesson 2b. Solving Quadratic Equations by FactoringLesson 2c. Solving Quadratic Equations by Completing the SquareLesson 2d. Solving Quadratic Equations by Using the Quadratic FormulaWhat I KnowThis part will assess your prior knowledge of solving quadratic equationsusing the four different methods. Answer all items and take note of the items that youwere not able to answer correctly. Find the right answer as you go along this module.Pre-AssessmentDirections: Find out how much you already know about this module. Choose theletter of the correct answer.1. What method can we use to solve a quadratic equation that can be written inthe form x2 r?A. Quadratic FormulaC. Extracting Square RootsB. FactoringD. Completing the Square1

2. Which of the following states that if the product of two real numbers is zero,then either of the two is equal to zero or both numbers are equal to zero?A. Multiplication PropertyC. Identity PropertyB. Zero Product PropertyD. Transitive Property3. In the equation x2 5x – 14 0, the solutions are .A. 7 and -2B. -7 and 2C. 7 and 2D. -7 and -24. The roots of 4x2 12x – 16 0 are .A. 8 and -2B. -4 and 1C. 4 and -1D. -8 and 225. In the equation x 121 22x, the roots are .A. 9 and -9B. 12 and -12C. 11 and 11D. 8 and 136. Find the solutions of the equation x2 - 5x 14.A. 7 and -2B. -7 and 2C. 7 and 2D. -7 and -27. Find the solutions of the equation x2 - 3x – 40 0.A. -5 and -8B. 5 and -8C. -5 and 8D. 5 and 828. Solve for x in the equation x x 12.A. 6 and -2B. -3 and -4C. - 4 and 3D. -2 and 69. In the equation 2x2 -2x – 12 0, the values of x are .A. -6 and 2B. -3 and 4C. - 4 and 3D. -2 and 310. Solve for x in the equation x2 256 by extracting square roots.A. 14 and -14B. 23 and -23C. 16 and -16D. 18 and -18211. Solve by extracting square roots: 2x 162A. 7 and -7B. 9 and -9C. 9 and -11D. 11 and -1112. In the equation x2 18x 81 0, the roots are .A. 8 and -8B. - 9 and - 9C. 9 and 8D. 9 and 913. In the equation x2 – 5x – 14 0, the solutions are .A. 7 and -2B. -7 and 2C. 7 and 2D. -7 and -2214. The roots of 4x 12x – 16 0 are?A. 8 and -2B. - 4 and 1C. 4 and -1D. -8 and 215. In the equation x2 64 16x, the roots are .A. 9 and -9B. 12 and -12C.11 and 11D. 8 and 816. In the equation 4x2 - 16x 12 0, one of its roots is .A. 3B. 4C. -32D. 2

17. In the equation x2 - 8x 15 0, the solutions are .A. 3 and -3B. 5 and -3C. 5 and 3D. -5 and -318. In the equation x2 – 2x 7, the solutions are .A. 1 and 1-C. 1 and 1-B. 3 and 3 -D. 1 and 1-19. In the equation x2 14x 32, the solutions are?A. 9 and 2B. 2 and -16C. -9 and -220. In the equation x2 - 6x - 11 0, the solutions are?A. 3 B. 3 and 3 and 3 -C. 3 and 3-D. 3 and 3 -3D. -2 and 16

LessonSolving Quadratic Equationsby Extracting Square Roots2aWhat I Need to KnowSolving quadratic equations by extracting square roots is one of the fourmethods in solving quadratic equations. In this lesson, you are expected to learn thesteps on how to solve quadratic equations by extracting square roots, solvequadratic equations by extracting square roots and apply its properties.What’s InActivity 1: Extract Me Please!Directions: Determine the square roots of the following radicals. Answer carefullythe questions that follow.1.4.2. –5.3.6.Process Questions:a. How did you find each square root?b. How many square roots do these numbers have?c. Does a negative number have a square root? Why or why not?d. Which of these numbers are rational numbers?e. Which of these are irrational numbers?Activity 2: Notice My Roots!!!4

Directions: Give the square roots of each numbers in the box and answer thequestions below.,,,,,and.1. What kind of numbers do we have in this activity?2. How did you find the square roots of irrational numbers?In the next activity, you will be dealing with a situation. You will need torecall the knowledge you learn in writing mathematical sentences and othermathematics concepts to satisfy the conditions asked in the problem.What’s NewActivity 3: A Lot of Square!!!Directions: Read and analyze the situation given below. Answer the questions thatfollow.Mr. Mariano bought a square - shaped lot that measure 2,500 square metersfor his future dream house. Moreover, he wanted to put his dream house particularlyat the center of his property. The house has a dimension of 30m by 30m based onthe floor plan.a. Draw an actual diagram to show the given situation.b. Using the variable s as the length of one side of the lot, write an equation thatrepresents the area of the whole square-shaped lot.c. From your answer in b, how will you solve for the length of one side of thesquare-shaped lot? Provide a solution.d. What is the area of the lot used to build the house?5

e. What is the remaining area of the square-shaped lot that is not used to buildthe house? How will you obtain its area?f. Using the values you obtain and the variable s as the length of one side of thesquare-shaped lot, write an equation that represents the area of the remaininglot in terms of s?The activity you just have done shows how a real - life situation can berepresented by a mathematical sentence. Were you able to represent the givensituation by a quadratic equation? To further give you more ideas on solvingquadratic equations. Perform the next activity.Activity 4: I am Quadratic!!!Directions: Use the quadratic equations below to answer the questions that follow.x2 81b2 – 49 03c2 – 75 01. Compare the three equations and make a statement to describe them.2. Solve each quadratic equation using any method you can think of.3. How will you know whether the values you obtained from solving really satisfy theequation?Were you able to determine the values of the variable that make eachequation true? Let us increase your understanding of quadratic equations anddiscover more about their solutions by performing the next activity.6

Activity 5: Real or Not RealDirections: Find the solutions of each of the following quadratic equations andanswer the questions that follow.x2 16x2 10 10x2 20 161. How did you obtain the solutions of each equation?2. Which of the equations have two solutions? Are the solutions real or not real?3. Which of the equations have only one solution? Is the solution real or not real?4. Which of the equations have no real solution? Why do you say so?5. What conclusion can you make base on what you have observed with theobtained solutions?Were you able to determine the values of the variable that make eachequation true? Were you able to find other ways of solving each equation? Let usincrease your understanding of quadratic equations and discover more about theirsolutions by performing the next activity. Before doing these activities, read andunderstand first some important notes on solving quadratic equations by extractingsquare roots and the examples presented.7

What Is ItQuadratic Equations that can be written in the form x2 r, where r could beany real number, can be solved by the method called Extracting Square Roots.This method is used with the following properties as a guide:Property 1. If r 0, then x2 r has two real solutions or roots: x .Example 1: Find the solutions of the equation x2 – 36 0 by extracting squareroots.Solutions:Rewrite x2 – 36 0 in the form x2 rx2 – 36 0by adding both sides of the equation byx2 – 36 36 0 36x2 3636.Since r 36 which is greater than 0(r 0), we need to use Property 1 whichstates “ If r 0, then x2 r has two real” to find thesolutions or roots: x x2 36x2 36x x x 6 or x - 6values of x that will make the equationx2 – 36 0 true.To check if the values we obtained is correct, we just substitute the values ofx in the original equation.Checking:For x 6:For x - 6x2 – 36 0x2 – 36 0( 6 )2 – 36 0(- 6 )2 – 36 036 – 36 036 – 36 00 008 0

Both values of x satisfy the given equation.Thus x2 – 36 0 is true when x 6 and x -6.Answer: The equation x2 – 36 0 has two solutions: x 6 and x -6.Note: A quadratic equation can have two or only one real solution(s).In some cases, it can also have no real solutions.Property 2. If r 0, then x2 r has one real solution or root: x 0.Example 2: Solve the equation m2 0.Solutions:m2 02The equation m 0 is already inthe form x2 r.Since r 0, we need to use the secondProperty 2 which states “If r 0, then x2 r has one real solution or root: x 0.”That is, m 0.To check, we substitute the value of m in the original equation.Checking:For m 0:m2 0(0)2 00 0Answer: The equation m 2 0 has only one solution which is x 0.9

Property 3. If r 0, then x2 r has no real solutions or roots.Example 3: Find the roots of the equation x2 9 0.Solutions:Rewrite x2 9 0 in the formx2x2 9 0x2 9 – 9 0 – 9 r by adding both sides of thex2 - 9equation by -9.Since r - 9 which is less than 0(r 0), we need to use Property 3 whichAnswer: The equation x2 9 0 hasstates “If r 0, then x2 r has no realno real solutions or roots.solutions or roots”. Because there is noreal number that gives - 9 when squared.In the next example, other mathematical concepts you previously learned areused along with the property needed to solve the quadratic equation by extractingsquare roots. Study the steps to help you with the activities that follow.Example 4: Find the solutions of the equation (x – 3)2 – 81 0.Solutions:To solve (x – 3)2 – 81 0, add 81(x – 3)2 – 81 81 0 81to both sides of the equation andsimplify.(x – 3)2 81Extract the square roots of both sides ofthe equation.(x – 3) The result gives us two equations:x – 3 9 and x – 3 - 9x–3 910,x–3 -9

Solve each equation to find the solutions.For x – 3 9For x – 3 - 9,x–3 3 9 3x – 3 3 -9 3x -6x 12To check, substitute the values of x in the original equation.Checking:For x 12For x - 6(x – 3)2 – 81 0(x – 3)2 – 81 0(12 - 3)2 – 81 0(-6 - 3)2 – 81 0(9)2 – 81 0(-9)2 – 81 081 – 81 081 – 81 0 00 00Both values of x satisfy the given equation.The equation (x – 3)2 – 81 0 is true when x 12 and x -6.Therefore, the equation (x – 3)2 – 81 0 has two solutions: x 12 and x - 6.Your goal in this section is to apply key concepts of solving quadratic equationsby extracting square roots. Use the mathematical ideas and the examples presentedto answer the next activities.11

What’s MoreActivity 6: Label and Dig Me Out!Directions: Solve each of the following quadratic equations by extracting squareroots. Label every steps of your solution with the steps of solving byextracting square roots as presented previously in the examples.1. x2 – 100 04. x2 2. x2 1215. (x – 2)2 – 4 03. 2x2 50Were you able to extract the roots of each equation? I’m sure you did!Now, deepen your understanding of solving quadratic equations byextracting square roots further by doing the next activities.What I Have LearnedActivity 7: Strengthen Your Understanding!Directions: Read and analyze each item below. Provide a solution is necessary.Write your answer in your Mathematics notebook.1. Give examples of quadratic equations that can be solved by extracting theroot witha. two real solutionsb. one real solutionc. no real solution.2. Were the steps of solving quadratic equations by extracting square rootshelpful to you? Why?12

Now that you have deeper understanding of the topic, you are ready todo a practical task in which you will demonstrate your understanding ofsolving quadratic equations by extracting square roots.What I Can DoActivity 8: You Can Do It!Directions: Read and analyze each item carefully to answer. Provide solutions ifneeded and write your answer in your Mathematics notebook.1. Write a quadratic equation that represents the area of each square. Then findthe length of its side using the equation formulated. Answer the questions thatfollow.SSArea 225 cm2SS2. Gather square objects of different sizes. Using these square objects,formulate quadratic equations that can be solved by extracting square roots.Find the solutions or roots of these equations.13

SummaryThis lesson was about solving quadratic equations by extracting square roots.The lesson provided you with opportunities to describe quadratic equations andsolve these by extracting square roots. You were also able to find out how suchequations are illustrated in real life. Moreover, you were given the chance todemonstrate your understanding of the lesson by doing practical tasks. Yourunderstanding of this lesson and other previously learned mathematics concepts andprinciples will enable you to learn about the wide applications of quadratic equationsin real life.14

Solving Quadratic Equationsby FactoringLesson2bWhat I Need to KnowStart Lesson 2b of this module by assessing your knowledge of the differentmathematics concepts previously studied and your skills in performing mathematicaloperations. These knowledge and skills will help you understand solving quadratic equationsby factoring. If you find any difficulty in answering the exercises, seek the assistance of yourteacher or peers or refer to the modules and lessons you have gone over earlier. You maycheck your answers with your teacher.What’s InActivity 1: Deal with my Factor!Directions: Factor each of the following polynomial expressions and answer thequestions that follow.1. 2x2 – 6x4. 4t2 8t 42. -3x2 21x5. 4x2 - 93. x2 -10x 246. 2y2 – 3y – 14Process Questions:a. How did you factor each polynomial expression?b. What factoring technique did you use to come up with the factors of eachpolynomial expression? Justify your method or technique.c. How did you check if the factors you obtained are correct?d. Which of the polynomial expressions you find difficult to factor? Why?15

What do you think of the activity? Were you able to recall and apply thedifferent mathematics concepts or principles in factoring polynomials? I’m sureyou were good at it. The activity was a preparation for the next lesson.What’s NewActivity 2: My Zero Products!Directions: Use the equations inside the box to answer the questions that follow.x–2 0( x- 2) (x – 9) 0x – 9 01. How would you compare the three equations?2. What value(s) of x would make each equation true?3. How would you know if the value of x that you got satisfies each equation?4. Compare the solutions of the given equations and state your observation.5. Are the solutions of x – 2 0 and x – 9 0 the same as the solutions of(x – 2 ) (x – 9) 0? Why?6. How will you interpret the meaning of (x – 2 ) (x – 9) 0?How did you find the activity? Are you ready to learn about solvingquadratic equations by factoring? I know you are always prepared toexplore new challenges just like in real life. But how does finding solutionsof quadratic equations help in solving real life problems and in makingdecisions? You will find this out in the next activity. Before engaging theseactivities, read and understand first some important notes on solvingquadratic equations by factoring and the examples presented.16

What Is ItSome quadratic equations can be solved easily by factoring. These type ofquadratic equations is said to be factorable. To solve such quadratic equations, thefollowing steps can be followed:1. Transform the quadratic equation into standard form if necessary.2. Factor the quadratic expression.3. Apply the zero product property by setting each factor of the quadraticexpression equal to 0.Zero Property PropertyIf the product of two real numbers is zero, then either of the two is equal to zeroor both numbers are equal to zero4. Solve each resulting equation to get the value of the variable.5. Check the values of the variable obtained by substituting each in theoriginal equation.Example 1: Find the solutions of x2 7x - 6 by factoring.StepsSolutions1. Transform the equation intox2 7x - 6x2 7x 6 0x2 7x 6 0(x 6) (x 1) 02standard form ax bx c 0.2. Factor the quadratic expression.Recall: A quadratic trinomial is a productof two binomials. Thus, we can check ifthe factor (x 6) (x 1) is the right factor.If it is, we should get x2 7x 6 afterapplying FOIL method.3. Apply the zero product property by(x 6) (x 1) 0setting each factor of the quadraticx 6 0 , x 1 0expression equal to 0.17

4. Solve each resulting equation to getx 6 0the value of the variablex -6 0–6x 1 0x 1–1 0–1x -1x -65. Check to determine if the values areChecking:correct by substituting it from theFor x -6:original equation.x2 7xFor x -1: -6x2 7x -6(-6)2 7(-6) -6(-1)2 7(-1) -636 – 42 -6 - 6 -61 – 7 -6-6 -6Both values of x satisfy the given equation.Thus x2 7x - 6 is true when x -6 and x -1.Answer: The equation x2 7x - 6 has two solutions: x - 6 and x -1.Example 2: Factor 4x2 – 9 0 and solve for x.StepsSolutions1. Transform the equation into4x2 – 9 02standard form ax bx c 0.In this case, the quadratic equationis already in standard form.2. Factor the quadratic expression.4x2 – 9 0(2x – 3)(2x 3) 0Recall: The expression x2 – y2 is aDifference of Two Squares and its factoris the expression ( x – y ) ( x y ).In this case, we can rewrite 4x2 – 9 to(2x)2 – 32 which is an example of adifference of two squares. Hence, itsfactor is (2x – 3)(2x 3).18

3. Apply the zero product property by(2x – 3) (2x 3) 0setting each factor of the quadratic2x – 3 0 , 2x 3 0expression equal to 0.4. Solve each resulting equation to get2x – 3 02x 3 0the value of the variable2x – 3 3 0 32x 3 - 3 0 - 32x 32x - 3 x x 5. Check to determine if the values areChecking:correct by substituting it from theoriginal equation.For x :For x 4x2 – 9 04x2 – 9 044–9 0–9 0–9 044:–9 09–9 09–9 0 00 0 0Both values of x satisfy the given equation.Thus 4x2 – 9 0 is true when x and x .Answer: The equation 4x2 – 9 0 has two solutions: x 19and x .

Example 3: Solve 4y2 36 - 24y.Steps1. Transform the equation into2standard form ax bx c 0.2. Factor the quadratic expression.Solutions4y2 36 - 24y4y2 24y 36 04y2 24y 36 0(2y 6) (2y 6) 0(2y 6) 2 0In this case, the quadratic expression4y2 24y 36 is a Perfect SquareTrinomial, therefore its factors arerepeated.Recall: A Perfect Square Trinomialx2 2xy y2 has a factor in the form(x y) (x y) or (x y)2.Since 4y2 24y 36 is a Perfect SquareTrinomial, we can rewrite it to(2y)2 2(2y)(6) 62 and its factor is theexpression (2y 6) (2y 6) or (2y 6) 23. Apply the zero product property by(2y 6) (2y 6) 0setting each factor of the quadratic2y 6 0 , 2y 6 0expression equal to 0.Note: We can apply extracting squareroots method if we choose to use thefactor (2y – 6) 2.4. Solve each resulting equation to get2y 6 02y 6 0the value of the variable.2y 6 – 6 0 – 62y 6 – 6 0 – 62y - 62y - 6y -3y -320

In this case, we can say that thequadratic equation has only one realsolution since the two equations obtainedthe same value which is y - 3.5. Check to determine if the value isChecking:correct by substituting it from the4y2 36 - 24yoriginal equation.4( - 3 )2 36 - 24( - 3 )4(9) 36 7236 36

Lesson 2a. Solving Quadratic Equations by Extracting Square Roots Lesson 2b. Solving Quadratic Equations by Factoring Lesson 2c. Solving Quadratic Equations by Completing the Square Lesson 2d. Solving Quadratic Equations by Using the Quadratic Formula What I Know This part will assess your prior knowledge of solving quadratic equations

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