Math-9-q1-m2 (1)

Mathematics

9

Quarter 1 Self-Learning Module 2 Solving Quadratic Equation by Factoring

Mathematics – Grade 9 Alternative Delivery Mode Quarter 1 – Self-Learning Module 2: Solving Quadratic Equation by Factoring 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 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 - Schools Division of Pasig City Development Team of the Self-Learning Module Writer: Maricel A. Sediarin Editors: Maria Pilita M. Evangelista, Cecilia M. Marcelo Reviewers: Ma. Cynthia P. Badana; Ma. Victoria L. Peñalosa Illustrator: Layout Artist: Management Team: Ma. Evalou Concepcion A. Agustin OIC-Schools Division Superintendent Aurelio G. Alfonso EdD OIC-Assistant Schools Division Superintendent Victor M. Javeña EdD Chief, School Governance and Operations Division and OIC-Chief, Curriculum Implementation Division Education Program Supervisors Librada L. Agon EdD (EPP/TLE/TVL/TVE) Liza A. Alvarez (Science/STEM/SSP) Bernard R. Balitao (AP/HUMSS) Joselito E. Calios (English/SPFL/GAS) Norlyn D. Conde EdD (MAPEH/SPA/SPS/HOPE/A&D/Sports) Wilma Q. Del Rosario (LRMS/ADM) Ma. Teresita E. Herrera EdD (Filipino/GAS/Piling Larang) Perlita M. Ignacio PhD (EsP) Dulce O. Santos PhD (Kindergarten/MTB-MLE) Printed in the Philippines by Department of Education – Schools Division of Teresita P. Tagulao EdD (Mathematics/ABM) Pasig City

Mathematics

9

Quarter 1 Self-Learning Module 2 Solving Quadratic Equation by Factoring

Introductory Message!

For the facilitator: Welcome to Mathematics 9 Self Learning Module on Solving Quadratic Equation by Factoring! This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling. This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances. In addition to the material in the main text, you will also see this box in the body of the module:

Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

For the learner: Welcome to Mathematics Self Learning Module on Solving Quadratic Equation by Factoring! The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands! This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner. This module has the following parts and corresponding icons:

Expectations – These are what you will be able to know after completing the lessons in the module Pretest – This will measure your prior knowledge and the concepts to be mastered throughout the lesson. Recap – This section will measure what learnings and skills that you understand from the previous lesson. Lesson – This section will discuss the topic for this module.

Activities – This is a set of activities you will perform.

Wrap Up – This section summarizes the concepts and applications of the lessons. Valuing – This part will check the integration of values in the learning competency.

Posttest – This will measure how much you have learned from the entire module.

EXPECTATIONS 1. Solve quadratic equations by factoring. 2. Apply different patterns on factoring.

PRETEST Directions: Read each item carefully. Choose the letter that corresponds to the correct answer. 1. Which of the following can be solved by factoring? A. 3𝑥² − 5𝑥 − 1 = 0 C. 𝑥² − 𝑥 − 56 = 0 B. 2𝑥² + 32 = 0 D. 𝑥² + 6𝑥 + 3 = 0 2. What are the factors of 𝑥² − 8𝑥 + 7 = 0? A. (x + 7)(x + 1)=0 B. (x + 7)(x - 1)=0

C. (x - 7)(x - 1)=0 D. (x - 7)(x + 1)=0

3. What is the factored form of 5𝑥² − 15𝑥 = 0? A. 𝑥(5𝑥 − 15) = 0 B. 5𝑥(𝑥 − 3) = 0

C. 𝑥 (𝑥 + 3) = 0 D. 5𝑥(𝑥 + 3) = 0

4. Which of the following are the roots of 11𝑥² − 33𝑥 = 0? A. 0 and -3 C. 0 and -1 B. 0 and 3 D. 0 and 1 5. What are the roots of 5𝑥2 – 9𝑥 = 2 A.

1 5

B.

1 − 5

and 2 and -2

C.

1 5

and-2 1 5

D. − and 2

RECAP

How can you tell when a quadratic expression is factorable or not?

Patterns on Factoring: Examples: Greatest Common Monomial Factor: 2𝑥² − 8𝑥 = 2𝑥 (𝑥 – 4) 3𝑥 2 + 9𝑥 = 3𝑥 (𝑥 + 3) 5𝑥² − 5 = 5𝑥² (𝑥 – 1) Factoring Difference of Two squares: 𝑥² − 36 = (𝑥 + 6)(𝑥 − 6) 𝑥 2 − 100 = (𝑥 + 10)(𝑥 − 10) 4𝑥 2 − 144 = (2𝑥 + 12)(2𝑥 − 12) Factoring Perfect Square Trinomial: 𝑥 2 − 10𝑥 + 25 = (𝑥 − 5)2 𝑥 2 + 10𝑥 + 25 = (𝑥 + 5)² 9𝑥 2 + 24𝑥 + 16 = (3𝑥 + 4)² Quadratic Expression 𝑥² + 11𝑥 𝑥² + 5𝑥 + 2 𝑥² + 25 𝑥² + 12𝑥 + 36 𝑥² − 12𝑥 − 28 4𝑥 2 − 36 𝑥2 + 3𝑥 − 4 𝑥2 − 6𝑥 + 9

Can it be factored?

Write the factors.

LESSON Solutions or Roots of a Quadratic equation are values of the variable/s that make a quadratic equation true.

Zero Product Property If (x + r1)(x + r2) = 0 then x + r1 must be 0 or x + r2 must be 0. Examples:

1. 2x(5x − 1) = 0

Equate each factors to 0

2x = 0

5x − 1 = 0 0 2 x= 2 2

MPE

x = 0 The roots are 0 and

5x − 1 + 𝟏 = 0 + 𝟏

APE

5 1 x = 5 5

MPE

x =

1 5

1 5

2. (x + 1)(x - 5) = 0 x+1=0 𝑥 + 1 − 1 = 0 − 1 APE

Equate each factors to 0 x-5=0 𝑥 − 5 + 5 = 0 + 5 APE

𝑥 = −1

𝑥 = 5

The roots of the equation are -1 and 5 3. (2x + 1)(3x + 4) = 0 2x + 1 = 0 2x + 1-1 = 0 -1 2x = -1 2 2

x =

𝑥 = −

−1 2

1.Equate each factor to 0 3x + 4 = 0 APE

3x + 4 − 4 = 0 − 4 APE 3𝑥 = −4 3

x= 3

MPE

1 2

The roots of the equation are

−4

𝑥= − 1 2

and

−4 3

3

4 3

MPE

FINDING THE ROOTS OF A QUADRATIC EQUATION BY FACTORING To factor a quadratic equation of the form 𝑎𝑥² + 𝑏𝑥 + 𝑐 = 0 write it as (x + r1) (x + r2) = 0 where c = r1⦁ r2 and b = r1 + r2 Examples: 1. Solve for r. 𝑟² + 13𝑟 + 12 = 0

The c term is 12, so you need to find a pair of factors with a product of 12. The b term is 13 so you need to find a pair of factors with a sum of 13. Since the product is positive (12) and the sum is positive (13), you need both factors to be positive.

Solutions: Step 1: Make a list of the possible factor pairs with a product of 12, and then find the one with a sum of 13. Factor pairs of c = 12 1*12 = 12 2*6 = 12 3*4 = 12

Sum of factor pairs 1 + 12 = 13 2+6=8 3+4=7

Step 2: Use 1 and 12 to factor the quadratic inside the parenthesis. (r + 1 ) (r +12) =0 Step 3: Use the zero product property to solve each linear equations. 𝑟 + 12 = 0 𝑟 + 1 = 0 𝑟 + 12 − 12 = 0 − 12 𝑟 + −1 = 0– 1 𝑟 = −12 𝑟 = −1 The roots of the equation are -12 and -1 2. Given: 2m² − 20m = 0, solve for m. Factor out the Greatest Common Monomial Factor 2m(m − 10) = 0 Equate each factors to 0 and solve each linear equation. 2m = 0 m – 10 = 0

1 1 (2𝑚 = 0) 2 2 𝑚 = 0 The roots of the equation are 0 and 10

m – 10 + 10 = 0 + 10 m = 10

3. Solve for n. 3n² + 14n − 5 = 0 Solutions: Because b is positive and c is negative, the factors of c have different signs.

Factors of 3

Factors of -5

Possible factors

Middle Term when Multiplied

1,3 1,3 1,3 1,3

1, -5 -1, 5 5, -1 -5, 1

(n + 1) (3n-5) (n -1) (3n + 5) (n +5) (3n-1) (n - 5) (3n+1)

-5n + 3n = -2n 5n - 3n =2n -n + 15n = 14n n - 15n = -14n

Write the factors (n + 5) (3n − 1) = 0 Apply zero product property then solve each linear equation. n +5=0

3n − 1 = 0

n + 5 −5= 0−5

3n − 1 + 1 = 0 + 1

n = −5

3 1 x= 3 3

The roots of the equation are -5 and

1 3

To check substitute the roots in the original equation. Checking: 3n² + 14n − 5 = 0

if n = 1/3

if n = −5 3(−5)2 + 14(−5) − 5 = 0

1 2 1 3 ( ) + 14 ( ) − 5 = 0 3 3

3(25) − 70 − 5 = 0

 1 14   + −5 = 0 3 3 

75 − 75 = 0

15 −5 = 0 3

0=0

5−5= 0 0=0

no no yes no

ACTIVITIES ACTIVITY 1: PRACTICE Direction: Solve each equation of the form 𝑎𝑥² + 𝑏𝑥 = 0 and check. 1. 𝑥² = −9𝑥

Roots:____________________

✓ Checking:

2. 12𝑥² + 4𝑥 = 0

Roots:____________________

✓ Checking:

Direction: Use factoring to solve each equation and check. 1.a² − 8a + 16 = 0 Solution:

2. 2b2 − 7b − 4 = 0 Solution:

3. 9c 2 − 1 = 0 Solution:

ACTIVITY 2: KEEP PRACTICING Directions: Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas. Covid – 19 death rates show that this virus is contagious. The number of death continually shifts as the outbreak progresses. What term is used to describe the ratio of people with a disease and died of the disease? Match the quadratic equation in column A with its roots in column B. Write the letter of the answer in the answer box. A

B

1. 2𝑥² − 𝑥 = 0

T

𝑥 = 0, 𝑥 = 1

2. 𝑥² − 9𝑥 + 20 = 0

S

𝑥 = 0, 𝑥 = 11

3. 𝑥² − 11𝑥 = 0

E

𝑥 = 0, 𝑥 = −13

4. 𝑥² = −13𝑥

A

𝑥 = 5, 𝑥 = 4

5. 2𝑥² = 5𝑥

F

𝑥 = 0, 𝑥 =

6. (𝑥 − 7)(𝑥 − 1) = 0

L

𝑥 = −1, 𝑥 = 3

7. (𝑥 − 3)(𝑥 + 3) = 0

A

𝑥 = −5, 𝑥 = −1

8. 𝑥² + 6𝑥 + 5 = 0

T

𝑥 = 3, 𝑥 = −3

9. (𝑥 + 1)(𝑥 − 3) = 0

C

𝑥 = 0, 𝑥 =

10. (2𝑥 − 3)(𝑥 + 1) = 0

E

𝑥 = 1

11. 𝑥² + 3𝑥 = 0

R

𝑥 = −7, 𝑥 = 3

12. 3𝑥² + 1 = −4𝑥

I

𝑥 =

13. 𝑥² + 4𝑥 = 21

A

𝑥 = 7, 𝑥 = 1

14. 2𝑥 2 − 5𝑥 = −2

Y

𝑥 = −1, 𝑥 = − 3

15. 𝑥² − 𝑥 = 0

T

𝑥 = 0, 𝑥 = −3

16. 𝑥 2 + 1 =2x

E

𝑥 = −3, 𝑥 = 2

A

𝑥 = 2, 𝑥 =

3 , 2

5 2

1 2

𝑥 = −1

1

1 2

Answer Box

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

ACTIVITY 3: TEST YOURSELF Direction: Read each question carefully. Choose the letter that corresponds to the correct answer. 1. Which of the following quadratic equations may be solved more appropriately by factoring? A. 2𝑥² = 7² C. 𝑥² + 12 𝑥 + 36 = 0 B. 𝑥² + 64 = 0 D. 2𝑥² − 𝑥 + 2 = 0 2. What are the roots 𝑜𝑓 𝑛² − 3𝑛 = 0? A. 𝑛 = 1, 𝑛 = 3 C. 𝑛 = 1, 𝑛 = −3 B. 𝑛 = 0, 𝑛 = 3 D. 𝑛 = 1, 𝑛 = −3 3. What is the factored form of 𝑥² − 10𝑥 − 24 = 0? A. (𝑥 − 8)(𝑥 + 3) = 0

C. (𝑥 − 6)(𝑥 + 4) = 0

B. (𝑥 − 12)(𝑥 + 2) = 0

D. (𝑥 − 24)(𝑥 + 1) = 0

4. Which of the following equations cannot be solved by factoring? A. 𝑥² + 𝑥 − 3 = 0

C. 𝑥² + 10𝑥 + 25 = 0

B. 3𝑥² − 5𝑥 = 0

D. 𝑥 2 − 36 = 0

5.Which of the following are the solutions for 3m2 − 5m − 2 = 0? A.𝑚 = 1/3, 𝑚 = 2

C. 𝑚 = −1/3, 𝑚 = 2

B. 𝑚 = 3, 𝑚 = 2

D. 𝑚 = −3, 𝑚 = −2

WRAP UP Steps in using Factoring Method: 1. Write the equation in standard form. 2. Factor the left-hand side of the equation. 3. Set each factor equal to zero using the Principle of Zero Product. 4. Solve each resulting linear equation. 5. Check the results in the original equation. Since it is a second degree equation, it has two solutions. The solutions of a quadratic equation are called the roots of the quadratic equation.

VALUING In life, listing down the factors to our problems makes it easier for us to see and clearly understand how we can go about in solving it. One must have perseverance, optimism and calm spirit to evaluate our problems more effectively. With COVID 19 afflicting our country, make a short skit where you can show how can you be a factor in lessening the number of positive cases in our barangay?

POSTTEST Read the questions carefully. Encircle the letter of the best answer. 1. Which of the following can be solved by factoring? A. 𝑥² + 3𝑥 − 18 = 0

C. 𝑥² + 6𝑥 − 14 = 0

B. 𝑥² + 100 = 0

D. 𝑥² + 2𝑥 − 7 = 0

2. What are the factors of 𝑥 2 – 12𝑥 + 36 = 0 ? A.(𝑥 + 6)(𝑥 + 6) = 0

C. (𝑥 − 6)(𝑥 − 6) = 0

B. (𝑥 + 4)(𝑥 − 3) = 0

D. (𝑥 − 4)(𝑥 + 3) = 0

3. What is the factored form of 3𝑥² + 12𝑥 = 0? A. 𝑥 (3𝑥 − 12) = 0

C. 𝑥 (𝑥 + 4) = 0

B. 3𝑥 (𝑥 − 4) = 0

D. 3𝑥 (𝑥 + 4) = 0

4. Which of the following are the roots of 𝑥² + 10𝑥 = 0? A. 0 and -10

B. 0 and 10

C. 0 and -1

D. 0 and 1

5. What are the roots of 𝑥² + 9𝑥 – 10 = 0? A.-10 and 1

B.10 and 1

C. -10 and -1

D. 10 and -1

1. C

2. B

3. B

4. A

1.A

2. D

3.D

4.A

5.A

POST TEST

1. 4 and 4(equal roots)

5. C

2. -1/2, 4 ACTIVITY 3:

3. 1/3 and – 1/3

CASE FATALITY RATE

1. 0,-9 2. 0, -1/3 ACTITVITY 2 KEEP PRACTICING:

1. C

ACTIVITY 1

2. C

3.B

4.B

5.B

PRE TEST ANSWERS:

KEY TO CORRECTION

REFERENCES 1. www.montereyinstitute.org/courses/DevelopmentalMath 2. www.purplemath.com/modules/solvquad.htm 3. www.mathsisfun.com/algebra/factoring-quadratics.html 4. www.ixl.com/math/skill-plans#test-preparation-skill-plans 5. www.dadsworksheets.com/puzzles/3x3-magic-square.html 6. Larson, R., Bowell, L., Kanold, T., Stiff, L. (2007). “Algebra 1 Texas Edition”. McDouglas Littell,Houghton Miffin Company.www.mcdougalittell.com

7. Mathematics 9 Learner’s Module (1st Edition 2015). Department of Education, Philippines.

8. Nivera, G. C., & Lapinid, M. R. C. "Grade 9 Mathematics Patterns and Practicalities." Panizales, V., Zuniga, E., Mcabales, E., Natividad, M., & Villas, N. (2013). Salessiana Books, Don Bosco Press, Inc., Chino Roces Ave., Makati.

9. Oronce, O., & Mendoza, M., (2015). E-Math Worktext in Mathematics, Rex Bookstore, CM Recto Ave., Revised Edition.

10. Bitmoji App, Google Play Store