mathematics - fnhs.edu.ph

Mathematics Quarter 1 – Module 1:

Illustrations of Quadratic

Equations and Solving

Quadratic Equations

9

1

Mathematics – Grade 9 Self-Learning Module (SLM)

Quarter 1 – Module 1: Illustration of Quadratic Equation and Solving 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 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.

Printed in the Philippines by Department of Education – SOCCSKSARGEN Region Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal

Telefax: (083) 2288825/ (083) 2281893

E- mail Address: region12@deped.gov.ph

Development Team of the Module

Writers: Gienyfer T. Buhay, Joan A. Cagadas

Editors: Noel B. Wamar

Reviewers: Ronela S. Molina

Illustrator: Gienyfer T. Buhay, Joan A. Cagadas

Layout Artist: Gienyfer T. Buhay, Joan A. Cagadas

Cover Art Designer: Reggie D. Galindez

Management Team: Dr. Carlito Rocafort, CESO IV – Regional Director

Fiel Y. Almendra, CESO V – Assistant Regional Director

Dr. Natividad G. Ocon, CESO VI-OIC-Schools Division Superintendent

Dr. Meilrose B. Peralta-OIC-Assistant Schools Division Superintendent

Gilbert B. Barrera – Chief, CLMD

Arturo D. Tingson Jr. – REPS, LRMS

Peter Van C. Ang-ug – REPS, ADM

Jay- ar Lipura– REPS-Mathematics

Dr. Reynaldo M. Pascua, CESE– OIC - CID Chief

Dr. Hazel G. Aparece – EPS, LRMS

Antonio R. Pasigado – Division ADM Coordinator

Ronela S. Molina – EPS-Mathematics

2

Introductory Message This Self-Learning Module (SLM) is prepared so that you, our dear learners, can

continue your studies and learn while at home. Activities, questions, directions,

exercises, and discussions are carefully stated for you to understand each

lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step

as you discover and understand the lesson prepared for you.

Pre-test are provided to measure your prior knowledge on lessons in each SLM.

This will tell you if you need to proceed on completing this module, or if you need

to ask your facilitator or your teacher’s assistance for better understanding of

the lesson. At the end of each module, you need to answer the post-test to self-

check your learning. Answer keys are provided for each activity and test. We

trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teachers are also

provided to the facilitators and parents for strategies and reminders on how they

can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of

this SLM. Use a separate sheet of paper in answering the exercises and tests.

Read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the

tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

3

What I Need to Know

This is an introductory lesson on quadratic equation. A clear understanding

on the concept of the lesson will help you identify, describe quadratic equation

and illustrate it using proper and right representations. You will also

formulate quadratic equations as illustrated in real life situations.

This module is divided into two lessons, namely: Lesson 1: Illustration of Quadratic Equations Lesson 2: Solving Quadratic Equations

Specifically, you are expected to: 1. Illustrate quadratic equations

a. Identify and describe quadratic equation b. Write quadratic equations into standard form

c. Determine the values of a, b, and c in the quadratic

equation. 2. Solve quadratic equations by:

a. Extracting square roots; b. Factoring;

c. Completing the square; and d. Using the quadratic formula.

What I Know

PRE-ASSESSMENT

Directions: Let us find out how much you already know about this module.

Answer the following questions as much as you can by writing the letter of

your answer on a separate sheet of paper. Take note of the items that you

were not able to answer correctly and then let us find out the correct answer

as we go through this module.

1. It is a polynomial equation of degree two 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 C. Quadratic Equation B. Linear Inequality D. Quadratic Inequality

2. Which of the following is a quadratic equation?

A. 3x2 + 5r – 1 C. x2 + 4x -6 = 0 B. 5t – 8 = 9 D. 4x2 – 6x ≤ 2

4

3. In the quadratic equation 2x2 + 6x – 3 = 0, which is the quadratic term? A. x2 B. 6x C. 2x2 D. -3

4. What is a in the quadratic equation 2x2 + 8x + 10 = 0? A. 8x B. 2 C. 2x2 D. 2x

5. The length of the store lot is 2 m more than twice its width and its area is 48m2.

Which of the following equations represents the given situation?

A. x2 + x = 24 C. x2 + x = 48

B. x2 +2x =24 D. x2 + 2x = 48

6. What is 2x ( x - 6) = 10 in standard form of quadratic equation?

A. 2x2 + 2x +10=0 C. 2x2 – 12x -10 = 0 B. 2x2 +12x+10=0 D. 2x2 +12x +10 =0

7. What is the value of b in the equation 6 + 2x2 = 6x?

A. 6 B. -6 C. 2 D. -2

8. What is the most convenient method in solving the quadratic

equation(𝑥 − 2)2 = 1?

A. extracting square roots C. completing the square B. factoring D. quadratic formula

9. Which of the following methods can be used in solving quadratic equation

of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0?

I. Extracting Square Roots III. Completing the Square II. Factoring IV. Quadratic Formula

A. I, II and III C. II, III and IV

B. I, III and IV D. I, II, III and IV

10. What quadratic equation has roots of 3 𝑎𝑛𝑑 4? A. (𝑥 + 3)(𝑥 + 4) = 7 C. 𝑥2 + 4𝑥 = 12 − 3𝑥 B. (𝑥 − 3)(𝑥 + 4) = 7 D. 𝑥2 − 7𝑥 = −12

11. What is the quadratic equations formed by roots 3 𝑎𝑛𝑑 − 1

? 3

A. 3𝑥2 + 8𝑥 − 3 = 0 C. 3𝑥2 − 33𝑥 + 3 = 0 B. 3𝑥2 − 8𝑥 − 3 = 0 D. 3𝑥2 − 33𝑥 − 3 = 0

12. What number must be added to 4𝑥2 − 12𝑥 to make it a perfect square trinomial?

A. 4 B. 9 C. 16 D. 25 13. The product of two numbers is 32 and their quotient is 8. What are the

numbers?

A. 8 and 4 B. 16 and 2 C. 24 and 3 D. 32 and 4

14. Angela’s father is twice her age. If the product of their ages is 450, how

old is Angela now? A. 12 B. 15 C. 18 D. 20

15. The sum of two numbers is 8. The square of the larger number minus twice the square of the smaller number equals 7. What are the two numbers?

A. 4 and 4 B. 6 and 2 C. 5 and 3 D. 7 and 1

5

Lesson

1 Illustrations of Quadratic

Equations

What’s In

In this lesson, you are given the chance to use your previous understanding

and skills in learning quadratic equation. You are also given different activities

to process the knowledge and ability learned and transfers your

understanding of the different lessons.

Activity 1: Let’s Recall!

Find the indicated product.

1. 4(x2 + 5)

2. 3m (m-4)

3. (p+5) (p+3)

• How did you find the product?

• What mathematical concept did you apply to find each product?

• Are the products polynomial? If YES, describe.

What’s New

Activity 2: Lead Me to Quadratic Equations

Use the situation below to answer the questions that follow.

Peter was asked by his Filipino teacher to lay out a tarpaulin to be used as

back draft decoration during the Culmination Program of Buwan ng Wika. He

told Peter that the tarpaulin’s area must be 18 square feet.

6

Area = 18 𝑓𝑡2

1. Draw a diagram to illustrate the tarpaulin.

2. What are the possible dimensions of the tarpaulin? And how did you

determine such dimensions?

3. Suppose the length of the tarpaulin is 5 ft. longer than its width. What equation represents the given situation?

4. How would you write the equation to represent the situation?

5. Do you think you can use the equation formulated to find the length and

the width of the tarpaulin? Justify your answer.

What is It

Polynomials are classified according to the highest power of its variable. A

first degree polynomial, like 2x + 3 is linear, the second degree polynomial, like x2

+3x – 2 is quadratic; a third degree polynomial, like x3 + 2x2+ x + 10 is cubic, the fourth

degree of polynomial like x4-5x3+x2_ x -1 is quartic and the degree of 5, like 5x5- 2x2 +

2x3 +4x4-2x- 2 is quintic.

The equation is in standard form if its term are arranged from the term with

the highest degree, up to the term with the lowest degree

The standard form of 3x2 + 4x5 – 2x3 + 3x4 +5x-10 is 4x5 +3x4-2x3 +3x2 + 5x -

10.

Let us answer the questions posed in the “What’s New” Activity 1 Lead Me to

Quadratic Equation.

Peter was asked by his Filipino teacher to lay out a tarpaulin to be used as

back draft decoration during the Culmination Program of Buwan ng Wika. He

told Peter that the tarpaulin’s area must be 18 square feet.

1. Draw a diagram to illustrate the tarpaulin?

7

2. What are the possible dimensions of the tarpaulin? And how did you

determine such dimensions?

Possible dimensions of the tarpaulin: 2 ft. by 9 ft. and 3 ft. by 6 ft. To determine such dimensions just find two possible numbers whose product is equal to 18

3. Suppose the length of the tarpaulin is 5 ft. longer than its width. What equation represent the given situation?

Let w be the width ( in ft.) then the length is w + 5 . Since the area is 18 sq.ft. and the formula in finding area is A= l x w, then 18 = (w + 5) w.

4. How would you write the equation to represent the situation?

Taking the product on the left side of the equation formulated in item number 4

resulted to w 2 + 5w =18

The equation 𝑤2 + 5w = 18 is formed of only one variable (unknown quantity)

x. Here, the highest power of x is 2 (two). This type of equation is called Quadratic Equation.

Quadratic Equation in one variable is a mathematical sentence of degree 2

that can be written in the standard form a𝒙𝟐 + bx +c = 0, where a, b,and c are real

numbers and a ≠ 0.

In the equation, a𝒙𝟐 is the quadratic term, bx is the linear term, and c is the

constant term.

Illustrative Example 1:

2𝑥2 + 6x -3 = 0 is the quadratic equation in standard form with a=2, b=6, and c=-3

Discuss: Why do you think “a” must not be equal to zero in the equation

a𝑥2 + bx +c = 0

Substituting a = 0 in the equation a𝑥2 + bx +c = 0 will yield a linear equation, so a must not be equal to zero

Illustration: 0𝑥2 + bx +c = 0

bx +c = 0 the derived equation is in first degree

Illustrative Example 2:

3x (x - 4) = 10 is a quadratic equation, however, it is not written in standard form.

To write it in standard form, expand the product and make one side of

the equation zero.

3x (x -4) = 10 3𝑥2-12x = 10 multiply 3x to (x-4)

3𝑥2-12x - 10 = 10 – 10 Apply APE by adding -10

3𝑥2-12x - 10 = 0 both sides of the equation

8

The equation becomes 3𝑥2-12x - 10 = 0, which is in standard form.

In the equation 3𝑥2 −12x - 10 = 0, a = 3, b= -12, and c=-10

• When b =0 in the equation a𝑥2+ bx +c=0, it resulted to a quadratic equation

of the form a𝑥2 + c = 0.

Examples: Equations such as 𝑥2 +7 =0, -2 +𝑥2 + 5 = 0, and 15𝑥2 -19=0 are

quadratic equations of the form a𝑥2+ c =0. In the equation, the value of b=0.

• When c=0 in the equation a𝑥2+ bx +c=0, it resulted to a quadratic equation

of the form a𝑥2 + bx = 0.

Examples: Equations such as 𝑥2-81x =0, 2𝑥2+4x=0, 14𝑥2 +6x= 0 are quadratic

equations of the form a𝑥2+ bx =0. In the equation, the value of c=0

• All answers are reported in the form a𝑥2 + bx + c = 0 with a >0, and where the

greatest common factor of all nonzero coefficients is 1.

What’s More

Activity 3: Illustrate My Situation!

Direction: Identify whether or not the given situations illustrate quadratic equations. Justify your answer by representing each situation with a mathematical sentence.

1. Mario is looking for the dimensions of the rectangular garden that has

an area of 14m2 and a perimeter of 18 meters.

2. Mrs. Salome charged Php. 3,655.00 worth of groceries on her credit card for

the victims of earthquake last December. The balance of her credit card after she

made a payment was Php2450.00.

9

What I Have Learned

A. Supply the ideas to the given sentences below.

1. One will know if the equation is quadratic if

2. I learned that the equation is in standard form if its terms are

What I Can Do

Activity 4: Dig Deeper! (LM, Mathematics 9, p.16, no.5)

1. Read the paragraph carefully and answer the question that follows:

The members of the school Mathematics Club shared equal amounts for a new Digital Light Processing (DLP) projector amounting to Php25,000. If there had been

25 members more in the club, each would have contributed Php50 less.

What mathematical sentence would represent the given situation? Write this in

standard form then identify the a, b, and c.

10

Lesson

2 Solving Quadratic Equations

A number is a 𝑟𝑜𝑜𝑡 of a quadratic equation if, when the number is substituted

for the variable, the equation becomes a true statement. For quadratic

equations, there are at most two real roots. There are several ways by which

roots of a quadratic equation can be found. A recall of algebraic techniques

involving polynomials is needed to fully understand the procedures.

What’s In

Activity 1: Am I Standard?

Encircle the number if its shows a quadratic equation. Write it in standard

form and determine the values of 𝑎, 𝑏 and 𝑐. (J. Ulpina, L. Tizon, E. Fernando; Math

Builders 9; JO-ES Publishing House, Inc.; 2014)

Equation Standard Form 𝒂 𝒃 𝒄

1. 𝑥 = 2𝑥2 + 5

2. 𝑥2 + 4𝑥 − 5 = 0

3. 9𝑥 = 3𝑥2

4. 𝑦2 + 𝑦 + 5 = 0

5. 5𝑥2 − 7𝑥 = 0

6. 4 − 2𝑥 + 3𝑥2 = 0

7. 𝑥(𝑥 + 5) − 2𝑥 = 0

8. 𝑥2 − 2𝑥(𝑥 − 2) + 5 = 0

9. 2𝑥 + 8 = −𝑥2

10. 2𝑥 − 5(𝑥 + 3) = 𝑥(𝑥 − 4)

11

What’s New

Activity 2: Cardboard Box!

Use the situation below to answer the questions that follow.

A square piece of cardboard is to be used to form a box without a top by cutting off squares, 3𝑐𝑚 on a side from each corner and then folding up the sides. The volume of the box must be

3000𝑐𝑚3.

1. Draw a diagram to illustrate the given situation.

2. How are you going to represent the length of the side of a square piece

cardboard? How about its volume 3. What will be the dimension of the box with a square base?

4. How will you find the length of each side of the box?

What is It

SOLVING A QUADRATIC EQUATION

Addition, subtraction, multiplication, and division of real numbers are

the key to your success in solving linear and quadratic equations.

In solving quadratic equation, you must know how to get the square

root of a number, how to factor expressions, and how to apply properties of

real numbers. The different methods of solving quadratic equations are

extracting square roots, factoring, completing the square and using the

quadratic formula.

The solution of a quadratic equation is called roots of the equation. A

quadratic equation has at most two roots.

12

𝑥 − 6

3cm 3cm

3c

3c

m 3

m 3

cm

cm

3cm 3cm

𝑥 − 6 𝑥

Based on the activity Cardboard Box:

𝑥

Let 𝑥 be the length of a side of the cardboard. When a 3𝑐𝑚 square is cut

off from each corner and then folded up, a box with square base 𝑥 − 6 𝑐𝑚 on

each side, and a height of 3𝑐𝑚 will be formed. Therefore, its volume is 3(𝑥 − 6)2.

Since the volume must be 300𝑐𝑚3, we have

𝑥 − 6 𝑐𝑚 – length of the box

𝑥 − 6 𝑐𝑚 – width of the box

3𝑐𝑚 – height of the box

300𝑐𝑚3 – volume of the box

𝑉 = 𝑙𝑤ℎ

300 = (𝑥 − 6)(𝑥 − 6)(3)

3(𝑥 − 6)2 = 300

(𝑥 − 6)2 = 100

𝑥 − 6 = ±10

Solve for at: 𝑥 − 6 = 10 Solve for at: 𝑥 − 6 = −10

𝑥 − 6 = 10 𝑥 − 6 = −10

𝑥 − 6 + 6 = 10 + 6 𝑥 − 6 + 6 = −10 + 6

𝑥 = 16 𝑥 = −4 ---- reject this value

Therefore the square cardboard must be 16𝑐𝑚 on each side.

𝑥 − 6

𝑥 − 6 3cm

13

A. Extracting Square Roots.

Quadratic equations that can be transformed in the form 𝑥2 = 𝑘 can be

solved by applying the properties:

1. If 𝑘 > 0, then 𝑥2 = 𝑘 has two real solutions or roots: 𝑥 = ±√𝑘.

2. If 𝑘 = 0, then 𝑥2 = 𝑘 has one real solution or root: 𝑥 = 0.

3. If 𝑘 < 0, then 𝑥2 = 𝑘 has no real solutions or roots.

The method of solving a quadratic equation 𝑥2 = 𝑘 is called extracting

square roots. (Math 9 LM, p. 21)

The square root method is used in solving incomplete quadratic

equations of the form x2 = c, when c is a non-negative number.

A shorter way of writing the two solutions u = √𝑑 and u = −√𝑑 -is to write

using double sign notation: u = √𝑑. (Project EASE, Module I)

Example 1: Find the roots of 4𝑥2 − 81 = 0.

Solution:

4𝑥2 − 81 = 0

4𝑥2 − 81 + 81 = 0 + 81

4𝑥2 = 81 Write the equation in the form 𝑥2 = 𝑘.

4𝑥2

4=

81

4 Apply multiplicative inverse property of equality.

𝑥2 = 81

4

√𝑥2 = √81

4 Get the square root of both sides of the equation.

𝑥 = ± √81

4

𝑥 = ± 9

2

𝑥1 = 9

2

𝑥2 = − 9

2

The Square Root Property of Real Numbers

If u2 = d, then u = d or u = - d for d 0.

14

Check against the original equation.

B. Factoring

Factoring is a method used to solve a quadratic equation in the form

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 or 𝑎𝑥2 + 𝑏𝑥 = 0.

Example 2: Find the roots of (3𝑥 − 18) = 21 by factoring.

Solution:

(3𝑥 − 18) = 21 3𝑥2 − 18𝑥 = 21 3𝑥2 − 18𝑥 − 21 = 0 Transform the equation into standard form 𝑎𝑥2 +

𝑏𝑥 + 𝑐 = 0 or 𝑎𝑥2 + 𝑏𝑥 = 0. (3𝑥 + 3)(𝑥 − 7) = 0 Factor the quadratic expression.

3𝑥 + 3 = 0; 𝑥 − 7 = 0 Apply the zero product property by setting each

factor of the quadratic expression equal to 0.

3𝑥 + 3 = 0 Solve each resulting equation. 3𝑥 + 3 − 3 = 0 − 3 3𝑥 = −3 3𝑥

3 =

−3

3

𝒙𝟏 = −𝟏

𝑥 − 7 = 0 Solve each resulting equation.

𝑥 − 7 + 7 = 0 + 7 𝒙𝟐 = 𝟕

Zero Product Property

If 𝑎 and 𝑏 are real numbers and 𝑎𝑏 = 0 then 𝑎 = 0 or 𝑏 = 0. This is aso true for three or more factors, For any factor equal to zero, the product is zero.

15

Check against the original equation.

C. Completing the Square

There are quadratic equations that are not factorable. If an equation is

not factorable, you can apply another method to solve for the roots of the quadratic equation called completing the square. Using this method

means transforming one side of the equation into a perfect square trinomial. (J. N. Ulpina, L. Tizon, E. Fernando, Math Builders 9;JO-ES Publishing House, Inc. 2014)

Example 3: For what values of is the equation 2𝑥2 − 5𝑥 + 1 = 0 Solution:

2𝑥2 − 5𝑥 + 1 = 0 2𝑥2

2−

5𝑥

2+

1

2= 0 Since is not equal to 1, divide the equation by the

value of 𝑎.

𝑥2 −5𝑥

2+

1

2= 0

𝑥2 −5𝑥

2+

1

2−

1

2= 0 −

1

2 Group all variable terms on one side of the equation and

constant on the other side, 𝑥2 + 𝑏𝑥 = 𝑐.

𝑥2 −5𝑥

2= −

1

2 Complete the square of the resulting binomial by adding on

both sides of the equation the square of half of b.

𝑥2 −5𝑥

2+

25

16= −

1

2+

25

16 𝑏 =

5

2;

5

2÷ 2 =

5

4

𝑥2 −5𝑥

2+

25

16=

−8+25

16 (

5

2)2 =

25

16

𝑥2 −5𝑥

2+

25

16=

17

16 Simplify.

(𝑥 −5

2) (𝑥 −

5

2) =

17

16 Factor the resulting perfect square trinomial and write

it as square of binomial.

(𝑥 −5

2)2 =

17

16

16

√(𝑥 −5

2)2 =√

17

16 Use the square root property to solve for 𝑥.

𝑥 − 5 =±√17

16

𝑥1 = √17

4 +

5

4 or 𝑥1 =

5+√17

4 Solve each resulting equation.

𝑥2 = −17

4 +

5

4 or 𝑥2 =

5−√17

4

Check against the original equation.

D. Quadratic Formula

To solve any quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 using the quadratic formula,

determine the values of 𝑎, 𝑏, and 𝑐 then substitute these in the equation

𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎. Simplify the result if possible, then check the solutions obtained against

the original equation. (Math 9 LM, p.50)

Example 4: Find the roots of the quadratic equation 4𝑥2 + 12𝑥 + 9 = 0 using

quadratic formula.

Solution:

4𝑥2 + 12𝑥 + 9 = 0 Write the equation in standard form.

𝑎 = 4; 𝑏 = 12; = 9 Determine the values of 𝑎, b, and 𝑐.

𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎

𝑥 =−12±√122−4(4)(9)

2(4) Substitute the values of 𝑎, 𝑏, and 𝑐 in the quadratic

formula.

𝑥 = −12±√144−144

8 Simplify.

𝑥 = −12±0

8

𝑥 = −12

8

𝑥 = −3

2

17

Check against the original equation.

For 𝒙 = −𝟑

𝟐

What’s More

Activity 3: Extract My Roots!

Find the roots of each quadratic equation. Simplify your answer and check the result. Write your answer on a separate sheet.

1. 𝑥2 − 8 = 0 3. 2𝑥2 − 10 = 0

2. 4𝑥2 − 5 = 0 4. 2𝑥2 = 100

Questions:

a. How did you find the roots of each equation?

b. Which equation did you find difficult to solve by extracting square

roots? Why?

Activity 4: Factor Then Solve!

Solve the following quadratic equation by factoring and check the result.

Write your answer on a separate sheet.

1. 2𝑥2 − 6𝑥 = 0 3. 𝑥2 + 22𝑥 + 121 = 0

2. 𝑥2 − 5𝑥 = 24 4. 2𝑥2 − 3𝑥 + 1 = 0

Questions:

a. How did you find the solutions of each equation?

b. What mathematical concepts or principles did you use in finding the solutions? Explain how you used these?

18

Activity 5: Make Me Complete!

Solve for the roots by completing the square and check your answer. Write

your answer on a separate sheet.

1. 𝑥2 + 4𝑥 = 96 3. 𝑝2 + 2𝑥 − 48 = 0

2. 𝑥2 + 4𝑥 − 45 = 0 4. 3𝑥2 − 21𝑥 − 6 = 0

Questions:

a. How did you find the solution of each equation?

b. What mathematical concepts or principles did you use in finding the solution? Explain how did you use these?

Activity 6: Is the Formula Effective?

Solve for the roots of the following quadratic equations and check. Write your answer on a separate sheet.

1. 𝑥2 + 3𝑥 − 10 = 0 3. 2𝑥2 − 4𝑥 = 2

2. 𝑥2 − 3𝑥 − 27 = 0 4. 𝑥2 + 9𝑥 + 14 = 0

Questions:

a. How did you use the quadratic formula in finding the solution/s of

each equation?

b. How many solutions does each equation have?

What I Have Learned

Answer the following questions.

1. Extracting square roots is used to solve a quadratic equation in the

form .

2. How do you know that a given trinomial is a perfect square trinomial?

3. How will you determine the constant to be added in completing the

square if the given trinomial is not a perfect square trinomial?

19

4. Can you use the quadratic formula to solve quadratic equation of any

form? Why?

What I Can Do

Problem 1: The area of a rectangular garden is 220 𝑠𝑞. 𝑚. The length of the

garden is 12𝑚 more than its width. What are the dimensions of the rectangular garden?

Solution:

Step 1: Represent the given information in an equation.

Let represents the width of the garden. 𝑤 + 12 represents the length of the garden.

(𝑤 + 12)𝑤 = 220 Why?

Step 2: Solve the equation

𝑤2 + 12𝑤 − 220 = 0 Why? (𝑤 + 22)(𝑤 − 10) = 0 Why?

𝑤 = −22 𝑜𝑟 = 10 Why? Step 3: Interpret the answer

The length cannot be −22𝑚. Why?

The width of the garden is 10𝑚. Why?

The length of the garden is 10 + 12 = 22𝑚. Why?

20

Assessment

POST ASSESSMENT:

Directions: Let us check how much you have learned from this module. Read each question carefully. Write the letter of your answer on a separate sheet of paper.

1. The length of the garden is 4 more than twice its width and its area is 28𝑚2. Which

of the following represent the given situation? A. 2 + 2x = 14 C. 𝑥2 + 2x = 28

B. 2+ 4x = 14 D. 𝑥2+4x = 28

2. It is a polynomial equation of degree two 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 C. Quadratic Equation B. Linear Inequality D. Quadratic Inequality

3. What is 3x ( x - 5) = 12 in standard form of quadratic equation?

A. 3𝑥2 +15x +12 = 0 C. 3𝑥2 -15x +12 = 0

B. 3𝑥2 -15x - 12 = 0 D. 3𝑥2 +15x -12 = 0

4. Which of the following is a quadratic equation?

A. 2x2 + 6r – 1 C. x2 + 4x -6 = 0

B. 8t – 8 = 10 D. 5x2 – 7x ≤ 2

5. What is c in the quadratic equation 2x2 + 8x + 10 = 0?

A. 10 B. 8x C. 2x2 D. 2x

6. In the quadratic equation 4x2 + 6x – 3 = 0, which is the linear term?

A. x2 B. 6x C. 4x2 D. -3

7. The two consecutive positive even numbers whose sum of their squares is

460. Which of the following equations represents the given situation?

A. 2x2 + 4x - 456 = 0 C. 2x2 + 2x + 456 = 0 B. 2x2 - 4x + 456 = 0 D. 2x2 + 2x - 456 = 0

8. If 𝑥2 − 16𝑥 + 64 = 0, then 𝑥 = . A. ±8 B. 8 C. −8 D. 0

9. If 𝑦2 − 49 = 0, then 𝑦 = _ _. A. ±7 B. 7 C. −7 D. 0

10. If 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, then which correctly states the possible values for 𝑥?

A. −𝑏±√𝑏2−4𝑎𝑐

2𝑎 C.

−𝑏±√𝑏2+4𝑎𝑐

2𝑎

B. −𝑏±√𝑏2−4𝑎𝑐

2𝑎 D.

−𝑏±√𝑏2+4𝑎𝑐

2𝑎

21

11. The sum of two positive integers is 24. Which of the following represents their largest product?

A. 140 B. 240 C. 154 D. 144 12. The length of the garden is 5𝑚 longer than its width and the area is

36𝑚2. How long is the garden?

A. 4𝑚 B. 5𝑚 C. 9𝑚 D. 13𝑚

13. Arvin is 5 years older than Prince. The product of their ages after 10

years is 2 750. How old is Prince now? A. 55 B. 50 C. 45 D. 40

14. Alyssa and Valerie can finish cleaning the house in 2 hours. If it takes Alyssa working alone 3 hours longer that it takes Valerie working alone, how many hours will Alyssa finish the work alone?

A. 3 hours B. 6 hours C. 8 hours D. 10 hours 15. Mark is planning to enlarge his graduation picture. His original picture

is 7𝑐𝑚 long by 3𝑐𝑚 wide. He asked the photographer to enlarge it by increasing its length and width by the same amount. If he wants the

area of the enlarged picture is 96 𝑠𝑞. 𝑐𝑚, what is its new dimension?

A. 16𝑐𝑚 𝑏𝑦 6 𝑐𝑚 C. 14𝑐𝑚 𝑏𝑦 8𝑐𝑚 B. 12𝑐𝑚 𝑏𝑦 8 𝑐𝑚 D. 12𝑐𝑚 𝑏𝑦 10𝑐𝑚

Additional Activities

Squaring digits

Select a natural number between 1 to 50. Square the digits and add. Repeat this process until you see a pattern.

Example:

The number is 23. Observe the pattern.

23: 22 + 32 = 4 + 9 = 13 → 12 + 32 = 1 + 9 = 10 → 12 + 0 = 1 What other numbers ends in 1?

What happens if you do the process to number 37?

What conclusion can you give?

22

Answer Key

References Https://www.mathisfun.com

Https://practice-questions.wizako.com DepEd Materials: Mathematics 9 Learners Material

Bettye C. Hall and Mona Fabricant (1999). Prentice Hall Algebra 2 with

Trigonometry, Prentice Hall, Inc. Englewood Cliffs, New Jersey 07632 Jisela N. Ulpina (2014). Math Builders 9, Mega-Jesta Prints, Inc.,

Valenzuela City Orlando A. Oronce and Marilyn O. Mendoza (2014), E Math 9, Rex

Printing Company, Inc. Soledad Jose-Dilao, Ed.D. and Julieta G. Bernabe(2009), Intermediate

Algebra, SD Publications, Inc.

Activity 5: Make Me Complete!

1. -12, 8

2. -9, 5

3. -8, 6

4. 7±√57

2

Activity 6:

Is the Formula Effective?

1. -5, 2

2. 3±3√3

2

3. 1±√2

4. -7, -2 Pre-Assessment 1.C 6. C 11.B 2.C 7. B 12.B 3.C 8. A 13.B 4.B 9. C 14.B 5.B 10. D 15. C Post-Assessment 1. A 6. B 11.D 2. C 7. A 12.C 3. B 8. B 13.D 4. C 9. A 14.B

5. A 10. D 15. B

What I Have Learned

1. The equation is quadratic if the highest exponent of the

variable in the mathematical sentence is 2. 2. The equation is in the

standard form if its terms are arranged in descending order. What’s I Can Do 𝑚

2+25𝑤−12500= 0

a = 1, b= 25, c= -12500

Activity 1: Am I standard? Standard Form a b c

−𝟐𝒙𝟐

+𝒙−𝟓=𝟎 -2 1 -5

𝒙𝟐

+𝟒𝒙−𝟓=𝟎 1 4 -5

−𝟑𝒙𝟐

+𝟗𝒙=𝟎 -3 9 0

𝒙𝟐

+𝒙+𝟓=𝟎 1 1 5

𝟓𝒙𝟐

−𝟕𝒙=𝟎 5 -7 0

𝟑𝒙𝟐

−𝟐𝒙+𝟒=𝟎 3 -2 4

Activity 3: Extract my Roots!

1. ±2√2

2. ±2√5

2

3. ±√5

4. ±5√2

What I Know 1.C 6. C

2.A 7. B 3.D 8. A 4.D 9. A 5.D 10. C

Lesson 1 What’s In

1. 4𝑥2

+20

2. 3𝑚2

−12 3. 𝑝

2+8𝑝+15

What’s New 1. Possible dimensions are 2 ft. by 9 ft and 3 ft. by 6 ft. 2. length= w + 5, width = w 3. 18 = (w+5)w 4. 𝑤

2+5𝑤=18

5. yes What’s More

1. 𝑤2

−9𝑤+14= 0 Quadratic 2. x -3,655 = 245 Non Quadratic

Activity 4: Factor then

Solve!

1. 0, 3 2. -3, 8

3. -11

4. 1

2, 1

23

DISCLAIMER

This Self-Learning Module (SLM) was developed by DepEd SOCCSKSARGEN

with the primary objective of preparing for and addressing the new normal.

Contents of this module were based on DepEd’s Most Essential Learning

Competencies (MELC). This is a supplementary material to be used by all

learners of Region XII in all public schools beginning SY 2020-2021. The

process of LR development was observed in the production of this module.

This is version 1.0. We highly encourage feedback, comments, and

recommendations.

For inquiries or feedback, please write or call:

Department of Education – SOCCSKSARGEN Learning Resource Management System (LRMS)

Regional Center, Brgy. Carpenter Hill, City of Koronadal

Telefax No.: (083) 2288825/ (083) 2281893

Email Address: region12@deped.gov.ph