Quarter 1, Week 2 Module 2 Solving Quadratic Equations

Mathematics Quarter 1, Week 2 – Module 2 Solving Quadratic Equations

Mathematics - Grade 9

Alternative Delivery Mode Quarter 1, Week 2 - Module 2: Solving Quadratic Equations

First Edition, 2020

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Mathematics Quarter 1 Week 2 – Module 2 Solving Quadratic Equations

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Table of Contents

What This Module is About ....................................................................................................................... i

What I Need to Know .................................................................................................................................. i

How to Learn from this Module .............................................................................................................. .i

Icons of this Module ................................................................................................................................... ii

Lesson 2:

Solving Quadratic Equations ............................................................................................... 1

What I Need to Know..................................................................................................... 1

What I Know ..................................................................................................................... 1

Lesson 2a:

Solving Quadratic Equations by Extracting Square Roots ..................... 4


What’s In ............................................................................................................................ 4

What’s New ................................................................................................................... 5

What Is It ........................................................................................................................... 8

What’s More .................................................................................................................... 12

What I Have Learned..................................................................................................... 12

What I Can Do ................................................................................................................. 13

Summary

Lesson 2b:

Solving Quadratic Equations by Factoring ............................................................ 15


What’s In ............................................................................................................................ 15

What’s New ................................................................................................................... 16

What Is It .......................................................................................................................... 17

What’s More .................................................................................................................... 21


What I Can Do ................................................................................................................. 23

Summary

Lesson 2c:

Solving Quadratic Equations by Completing the Square ........................ 25


What’s In ............................................................................................................................ 25

What’s New ................................................................................................................... 27

What Is It .......................................................................................................................... 29

What’s More .................................................................................................................... 34


What I Can Do ................................................................................................................. 35

Summary

Lesson 2d:

Solving Quadratic Equations by Quadratic Formula .................................... 37


What’s In ............................................................................................................................ 37

What’s New ................................................................................................................... 39

What Is It ........................................................................................................................... 40

What’s More .................................................................................................................... 43


What I Can Do ................................................................................................................. 44

Summary Assessment………………………………………………………………………………………………47

Key to Answers ...................................................................................................................................... 49

References ............................................................................................................................................... 56

i

What This Module is About

This module consists of four lessons on Solving Quadratic Equations.

As you go through each part of this module, you will be able to demonstrate

understanding of the key concepts of solving quadratic equations by (a) extracting

square 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 using

these four methods in solving Quadratic Equations.

What I Need to Know

In this module, you are expected to solve quadratic equations by (a)

extracting square roots; (b) factoring; (c) completing the square; and (d) using the

quadratic 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 formula

2. solve quadratic equations by:

(a) extracting square roots;

(b) factoring;

(c) completing the square; and

(d) using the quadratic formula

3. use available or recyclable resources to perform the tasks set for you.

How to Learn from this Module

To 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 exercises

diligently.

3. Answer all the given tests and exercises.

ii

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.

1

Lesson

Solving Quadratic Equations

2

What I Need to Know

You have already learned the Illustrations of Quadratic Equations in the

previous module. Now, this module; Solving Quadratic Equations will enable you to

find the values of the variable in quadratic equations using the four different methods

presented in each lesson, namely:

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

using the four different methods. Answer all items and take note of the items that you

were not able to answer correctly. Find the right answer as you go along this module.

Pre-Assessment Directions: Find out how much you already know about this module. Choose the

letter of the correct answer.

1. What method can we use to solve a quadratic equation that can be written in

the form x2 = r?

A. Quadratic Formula C. Extracting Square Roots

B. Factoring D. Completing the Square

2

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 Property C. Identity Property

B. Zero Product Property D. Transitive Property

3. In the equation x2 + 5x – 14 = 0, the solutions are _______________.

A. 7 and -2 B. -7 and 2 C. 7 and 2 D. -7 and -2

4. The roots of 4x2 + 12x – 16 = 0 are _______________.

A. 8 and -2 B. -4 and 1 C. 4 and -1 D. -8 and 2

5. In the equation x2 +121 = 22x, the roots are _______________.

A. 9 and -9 B. 12 and -12 C. 11 and 11 D. 8 and 13

6. Find the solutions of the equation x2 - 5x = 14.


7. Find the solutions of the equation x2 - 3x – 40 = 0.

A. -5 and -8 B. 5 and -8 C. -5 and 8 D. 5 and 8

8. Solve for x in the equation x2 + x = 12.

A. 6 and -2 B. -3 and -4 C. - 4 and 3 D. -2 and 6

9. In the equation 2x2 -2x – 12 = 0, the values of x are _______________.

A. -6 and 2 B. -3 and 4 C. - 4 and 3 D. -2 and 3

10. Solve for x in the equation x2 = 256 by extracting square roots.

A. 14 and -14 B. 23 and -23 C. 16 and -16 D. 18 and -18

11. Solve by extracting square roots: 2x2 = 162


12. In the equation x2 + 18x + 81 = 0, the roots are _______________.

A. 8 and -8 B. - 9 and - 9 C. 9 and 8 D. 9 and 9

13. In the equation x2 – 5x – 14 = 0, the solutions are _______________.


14. The roots of 4x2 + 12x – 16 = 0 are?

A. 8 and -2 B. - 4 and 1 C. 4 and -1 D. -8 and 2

15. In the equation x2 + 64 = 16x, the roots are _______________.

A. 9 and -9 B. 12 and -12 C.11 and 11 D. 8 and 8

16. In the equation 4x2 - 16x + 12 = 0, one of its roots is _______________.

A. 3 B. 4 C. -3 D. 2

3

17. In the equation x2 - 8x + 15 = 0, the solutions are_______________.

A. 3 and -3 B. 5 and -3 C. 5 and 3 D. -5 and -3

18. 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 2 B. 2 and -16 C. -9 and -2 D. -2 and 16

20. In the equation x2 - 6x - 11 = 0, the solutions are?

A. 3+ and 3 - C. 3+ and 3-

B. 3+ and 3 - D. 3+ and 3 -

4

Lesson

Solving Quadratic Equations by Extracting Square Roots

2a

What I Need to Know Solving quadratic equations by extracting square roots is one of the four

methods in solving quadratic equations. In this lesson, you are expected to learn the

steps on how to solve quadratic equations by extracting square roots, solve

quadratic equations by extracting square roots and apply its properties.

What’s In

Activity 1: Extract Me Please!

Directions: Determine the square roots of the following radicals. Answer carefully

the 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!!!

5

Directions: Give the square roots of each numbers in the box and answer the

questions below.

1. What kind of numbers do we have in this activity?

2. How did you find the square roots of irrational numbers?

What’s New

Activity 3: A Lot of Square!!!

Directions: Read and analyze the situation given below. Answer the questions that

follow.

Mr. Mariano bought a square - shaped lot that measure 2,500 square meters

for his future dream house. Moreover, he wanted to put his dream house particularly

at the center of his property. The house has a dimension of 30m by 30m based on

the 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 that

represents 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 the

square-shaped lot? Provide a solution.

d. What is the area of the lot used to build the house?

In the next activity, you will be dealing with a situation. You will need to

recall the knowledge you learn in writing mathematical sentences and other

mathematics concepts to satisfy the conditions asked in the problem.

, , , , , and .

6

e. What is the remaining area of the square-shaped lot that is not used to build

the house? How will you obtain its area?

f. Using the values you obtain and the variable s as the length of one side of the

square-shaped lot, write an equation that represents the area of the remaining

lot in terms of s?

Activity 4: I am Quadratic!!!

Directions: Use the quadratic equations below to answer the questions that follow.

1. 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 the

equation?

The activity you just have done shows how a real - life situation can be

represented by a mathematical sentence. Were you able to represent the given

situation by a quadratic equation? To further give you more ideas on solving

quadratic equations. Perform the next activity.

x2 = 81 b2 – 49 = 0 3c

2 – 75 = 0

Were you able to determine the values of the variable that make each

equation true? Let us increase your understanding of quadratic equations and

discover more about their solutions by performing the next activity.

7

Activity 5: Real or Not Real

Directions: Find the solutions of each of the following quadratic equations and

answer the questions that follow.

1. 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 the

obtained solutions?

x2 + 10 = 10 x

2 = 16 x2 + 20 = 16

Were you able to determine the values of the variable that make each

equation true? Were you able to find other ways of solving each equation? Let us

increase your understanding of quadratic equations and discover more about their

solutions by performing the next activity. Before doing these activities, read and

understand first some important notes on solving quadratic equations by extracting

square roots and the examples presented.

8

=

What Is It

Quadratic Equations that can be written in the form x2 = r, where r could be

any 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 square

roots.

Solutions:

Rewrite x2 – 36 = 0 in the form x2 = r

by adding both sides of the equation by

36.

x2 – 36 = 0 x2 – 36 + 36 = 0 +36

x2 = 36

Since r = 36 which is greater than 0

(r > 0), we need to use Property 1 which

states “ If r > 0, then x2 = r has two real

solutions or roots: x = ” to find the

values of x that will make the equation

x2 – 36 = 0 true.

x2 = 36 x2 = 36

x =

x =

x = 6 or x = - 6

To check if the values we obtained is correct, we just substitute the values of

x in the original equation.

Checking:

For x = 6: For x = - 6

x2 – 36 ≟ 0 x2 – 36 ≟ 0

( 6 )2 – 36 ≟ 0 (- 6 )2 – 36 ≟ 0

36 – 36 ≟ 0 36 – 36 ≟ 0

0 0 0 0 =

9

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.

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

Example 2: Solve the equation m2 = 0.

Solutions:

The equation m2 = 0 is already in

the form x2 = r.

m 2 = 0

Since r = 0, we need to use the second

Property 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 ≟ 0

0 0

Answer: The equation m 2 = 0 has only one solution which is x = 0.

=

Note: A quadratic equation can have two or only one real solution(s).

In some cases, it can also have no real solutions.

10

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 form

x2 = r by adding both sides of the

equation by -9.

x2 + 9 = 0 x2 + 9 – 9 = 0 – 9

x2 = - 9

Since r = - 9 which is less than 0

(r < 0), we need to use Property 3 which

states “If r < 0, then x2 = r has no real

solutions or roots”. Because there is no

real number that gives - 9 when squared.

Answer: The equation x2 + 9 = 0 has

no real solutions or roots.

In the next example, other mathematical concepts you previously learned are

used along with the property needed to solve the quadratic equation by extracting

square 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

to both sides of the equation and

simplify.

(x – 3)2 – 81 + 81 = 0 + 81

(x – 3)2 = 81

Extract the square roots of both sides of

the equation.

(x – 3) =

The result gives us two equations:

x – 3 = 9 and x – 3 = - 9

x – 3 = 9 , x – 3 = - 9

11

Solve each equation to find the solutions.

For x – 3 = 9

x – 3 + 3 = 9 + 3

x = 12

For x – 3 = - 9,

x – 3 + 3 = -9 + 3

x = -6

To check, substitute the values of x in the original equation.

Checking:

For x = 12 For 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 ≟ 0

81 – 81 ≟ 0 81 – 81 ≟ 0

0 0 0 0


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 equations

by extracting square roots. Use the mathematical ideas and the examples presented

to answer the next activities.

12

What’s More

Activity 6: Label and Dig Me Out!

Directions: Solve each of the following quadratic equations by extracting square

roots. Label every steps of your solution with the steps of solving by

extracting square roots as presented previously in the examples.

1. x2 – 100 = 0 4. x2 =

2. x2 = 121 5. (x – 2)2 – 4 = 0

3. 2x2 = 50

What I Have Learned Activity 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 the

root with

a. two real solutions

b. one real solution

c. no real solution.

2. Were the steps of solving quadratic equations by extracting square roots

helpful to you? Why?

Were you able to extract the roots of each equation? I’m sure you did!

Now, deepen your understanding of solving quadratic equations by

extracting square roots further by doing the next activities.

13

What I Can Do

Activity 8: You Can Do It!

Directions: Read and analyze each item carefully to answer. Provide solutions if

needed and write your answer in your Mathematics notebook.

1. Write a quadratic equation that represents the area of each square. Then find

the length of its side using the equation formulated. Answer the questions that

follow.

2. 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.

Now that you have deeper understanding of the topic, you are ready to

do a practical task in which you will demonstrate your understanding of

solving quadratic equations by extracting square roots.

S

S

S

Area = 225 cm2

S

14

Summary

This lesson was about solving quadratic equations by extracting square roots.

The lesson provided you with opportunities to describe quadratic equations and

solve these by extracting square roots. You were also able to find out how such

equations are illustrated in real life. Moreover, you were given the chance to

demonstrate your understanding of the lesson by doing practical tasks. Your

understanding of this lesson and other previously learned mathematics concepts and

principles will enable you to learn about the wide applications of quadratic equations

in real life.

15

Lesson

Solving Quadratic Equations by Factoring

2b

What I Need to Know

Start Lesson 2b of this module by assessing your knowledge of the different

mathematics concepts previously studied and your skills in performing mathematical

operations. These knowledge and skills will help you understand solving quadratic equations

by factoring. If you find any difficulty in answering the exercises, seek the assistance of your

teacher or peers or refer to the modules and lessons you have gone over earlier. You may

check your answers with your teacher.

What’s In

Activity 1: Deal with my Factor!

Directions: Factor each of the following polynomial expressions and answer the

questions that follow.

1. 2x2 – 6x 4. 4t2 + 8t + 4

2. -3x2 + 21x 5. 4x2 - 9

3. x2 -10x + 24 6. 2y2 – 3y – 14

Process Questions:

a. How did you factor each polynomial expression?

b. What factoring technique did you use to come up with the factors of each

polynomial 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?

16

What’s New

Activity 2: My Zero Products!

Directions: Use the equations inside the box to answer the questions that follow.

1. 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?

What do you think of the activity? Were you able to recall and apply the

different mathematics concepts or principles in factoring polynomials? I’m sure

you were good at it. The activity was a preparation for the next lesson.

x – 2 = 0 x – 9= 0 ( x- 2) (x – 9) = 0

How did you find the activity? Are you ready to learn about solving

quadratic equations by factoring? I know you are always prepared to

explore new challenges just like in real life. But how does finding solutions

of quadratic equations help in solving real life problems and in making

decisions? You will find this out in the next activity. Before engaging these

activities, read and understand first some important notes on solving

quadratic equations by factoring and the examples presented.

17

What Is It

Some quadratic equations can be solved easily by factoring. These type of

quadratic equations is said to be factorable. To solve such quadratic equations, the

following 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 quadratic

expression equal to 0.

4. Solve each resulting equation to get the value of the variable.

5. Check the values of the variable obtained by substituting each in the

original equation.

Example 1: Find the solutions of x2 + 7x = - 6 by factoring.

Steps Solutions

1. Transform the equation into

standard form ax2 +bx +c = 0.

x2 + 7x = - 6 x2 + 7x +6 = 0


x2 + 7x + 6 = 0 (x + 6) (x +1) = 0

Recall: A quadratic trinomial is a product

of two binomials. Thus, we can check if

the factor (x + 6) (x +1) is the right factor.

If it is, we should get x2 + 7x + 6 after

applying FOIL method.

3. Apply the zero product property by

setting each factor of the quadratic


(x + 6) (x +1) = 0

x + 6 = 0 , x +1 = 0

Zero Property Property

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

18

4. Solve each resulting equation to get

the value of the variable

x + 6 = 0

x + - 6 = 0 – 6

x = - 6

x +1 = 0

x + 1 – 1 = 0 – 1

x = - 1

5. Check to determine if the values are

correct by substituting it from the

original equation.

Checking:

For x = -6:

x2 + 7x ≟ -6

(-6)2 + 7(-6) ≟ -6

36 – 42 ≟ -6

- 6 - 6

For x = -1:

x2 + 7x ≟ -6

(-1)2 + 7(-1) ≟ -6

1 – 7 ≟ -6

- 6 - 6


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.

Steps Solutions



4x2 – 9 = 0

In this case, the quadratic equation

is already in standard form.


4x2 – 9 = 0 (2x – 3)(2x + 3) = 0

Recall: The expression x2 – y2 is a

Difference of Two Squares and its factor

is the expression ( x – y ) ( x + y ).

In this case, we can rewrite 4x2 – 9 to

(2x)2 – 32 which is an example of a

difference of two squares. Hence, its

factor is (2x – 3)(2x + 3).

= =

19




(2x – 3) (2x + 3) = 0

2x – 3 = 0 , 2x + 3 = 0


the value of the variable

2x – 3 = 0

2x – 3 + 3 = 0 + 3

2x = 3

=

x =

2x + 3 = 0

2x + 3 - 3 = 0 - 3

2x = - 3

=

x =

5. Check to determine if the values are


original equation.

Checking:

For x = :

4x2 – 9 ≟ 0

4 – 9 ≟ 0

4 – 9 ≟ 0

9 – 9 ≟ 0

0 0

For x = :

4x2 – 9 ≟ 0

4 – 9 ≟ 0

4 – 9 ≟ 0

9 – 9 ≟ 0

0 0


Thus 4x2 – 9 = 0 is true when x = and x = .

Answer: The equation 4x2 – 9 = 0 has two solutions: x = and x = .

= =

20

Example 3: Solve 4y2 +36 = - 24y.

Steps Solutions



4y2 +36 = - 24y 4y2 + 24y + 36 = 0


4y2 + 24y + 36 = 0 (2y + 6) (2y + 6) = 0

(2y + 6) 2 = 0

In this case, the quadratic expression

4y2 + 24y + 36 is a Perfect Square

Trinomial, therefore its factors are

repeated.

Recall: A Perfect Square Trinomial

x2 ± 2xy + y2 has a factor in the form

(x ± y) (x ± y) or (x ± y)2.

Since 4y2 + 24y + 36 is a Perfect Square

Trinomial, we can rewrite it to

(2y)2 + 2(2y)(6) + 62 and its factor is the

expression (2y + 6) (2y + 6) or (2y + 6) 2




(2y + 6) (2y + 6) = 0

2y + 6 = 0 , 2y + 6 = 0

Note: We can apply extracting square

roots method if we choose to use the

factor (2y – 6) 2.


the value of the variable.

2y + 6 = 0

2y + 6 – 6 = 0 – 6

2y = - 6

y = - 3

2y + 6 = 0

2y + 6 – 6 = 0 – 6

2y = - 6

y = - 3

21

In this case, we can say that the

quadratic equation has only one real

solution since the two equations obtained

the same value which is y = - 3.

5. Check to determine if the value is


original equation.

Checking:

4y2 +36 = - 24y

4( - 3 )2 +36 = - 24( - 3 )

4(9) +36 = 72

36 +36 = 72

72 72

The value of y satisfies the given equation.

Thus 4y2 +36 = - 24y is true when y = - 3.

Answer: The equation 4y2 +36 = - 24y has one solution: x = - 3.

What’s More

Activity 3: Factor Me and Know My Value!

Directions: Solve the following quadratic equations by factoring method. Present

your solution in a step-by-step manner. Make sure to label each step as

you solve. Write your answer in your Mathematics notebook.

1. x2 + 6x = 16 4. 4x2 + 12x – 16 = 0

2. x2 - 49 = 0 5. n2 – 81 = 0

3. x2 +121 = 22x

Your goal in this section is to apply key concepts and principles

in solving quadratic equations by factoring. Use the mathematical

ideas and the examples presented in the preceding section to answer

the activities provided

=

22

What I Have Learned Activity 4: How Much Do I Know?

Directions: Read carefully and answer each of the following items. Provide your

solutions if needed.

1. Which of the following quadratic equations may be solved more appropriately

by factoring? Explain your answer.

a. 3x2 = 108 c. x2 – 169 = 0

b. x2 + 18x + 81 = 0 d. 2x2 - 2x - 12 = 0

2. Were the steps of solving quadratic equations by factoring helpful to you?

Why?

3. Do you agree that x2 + 5x – 14 = 0 and 14 – 5x – x2 = 0 have the same

solutions? Justify your answer by providing the solution.

Was it easy for you to find the solutions of quadratic equations by

factoring? Did you apply the different mathematics concepts and principles

in finding the solutions of each equation? I know you did!

Now that you have a deeper understanding of the topic, you are ready to do the

tasks in the next activity.

23

What I Can Do

Activity 5: Meet the Demands!

Directions: Evaluate the following task.

Mrs. Ester would like to increase her production of mangoes due to its high

demand in the market. She is thinking of extending her 10,000 square meter land

with her adjacent 6,000 square meter lot near a river. Help Mrs. Ester by making a

sketch plan of the possible extension to be made in order for her to maximize her

profit. Out of the given situation and the sketch plan you made, formulate as many

quadratic equations then solve by factoring. You may use the rubric below to rate

your work.

Rubric for Sketch Plan and Equations Formulated

4 3 2 1

The sketch plan is

accurately made,

presentable, and

appropriate

The sketch plan is

accurately made,

and appropriate

The sketch plan is

not accurately made

but appropriate

The sketch plan is

made but not

appropriate

Quadratic Equations

Are accurately

formulated and

solved correctly

Quadratic Equations

Are accurately

formulated but not

all are solved

correctly

Quadratic Equations

Are accurately

formulated but are

not solved correctly

Quadratic Equations

Are accurately

formulated but all

are not solved

correctly

24

Summary

This lesson was about solving quadratic equations by factoring. The lesson

provided you with opportunities to describe quadratic equations and solve these by

factoring. Factoring method is a great tool for solving factorable Quadratic Equations.

Moreover, it is a useful method to solve not only Quadratic binomials but Quadratic

trinomials as well. It is important that you know the factoring techniques taught in

your previous year as it is of great help to this method of solving.

You were also able to find out how such equations are illustrated in real life.

Moreover, you were given the chance to demonstrate your understanding of the

lesson by doing a practical task. Your understanding of this lesson and other

previously learned mathematics concepts and principles will facilitate your learning of

the wide applications of quadratic equations in real life.

25

Lesson

Solving Quadratic Equations by Completing the Square

2c

What I Need to Know

Start Lesson 2c of this module by assessing your knowledge of the different


operations. These knowledge and skills will help you understand Solving Quadratic

Equations by Completing the Square. As you go through this lesson, think of this important

question: “How does finding solutions of quadratic equations facilitate in solving real-life

problems and in making decisions?” To find the answer, perform each activity. If you find

any difficulty in answering the exercises, seek assistance of your peers or teacher or turn to

the modules and lessons you have undergone earlier. You may check your work with your

teacher.

What’s In

Activity 1: How Many Roots Do I Have?

Directions: Find the solutions or roots of each of the following equations. Answer

the questions that follow.

1. x2 + 1 = 50 4. s2 - 25 = - 25

2. r2 + 12 = 61 5. w2 – 12w = -36

3. ( t – 9 )2 = 121 6. m2 + 8m = 48

Process Questions:

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

b. Which of the equations has only one solution? Why?

c. Which of the equations has two solutions? Why?

26

Activity 2: From Perfect Trinomial to Binomial Square!

Directions: Express each of the following perfect square trinomials to binomial

square. Answer the questions that follow. Number 1 is done for you.

1. x2 + 6x + 9

Answer:

x2 + 6x + 9 = ( x + 3 )2

2. t2 - 10t + 25 5. t2 - 24t + 144

3. w2 - 2w +1 6. s2 + s +

4. 4. x2 + 3x +

Process Questions:

a. How do you describe a perfect square trinomial?

b. What mathematics concepts or principles did you apply to come up with your

answer? Explain how you applied these.

c. Observe the terms of each trinomial. How is the third term related to the

coefficient of the middle term?

d. Is there an easy way of expressing a perfect square trinomial as a square of a

binomial? If there is any, explain how.

What do you think of the activity? Were you able to recall and apply the

different mathematics concepts or principles in factoring polynomials? I’m sure

you were good at it. The activity is a preparation for the next lesson, solving

quadratic equations by completing the square.

Were you able to express each perfect square trinomial as a square of a

binomial? I’m sure you did! Let us further strengthen your knowledge and skills in

mathematics particularly in writing perfect square trinomials by doing the next

activity.

27

What’s New

Activity 3: Perfect is the Clue!

Directions: Carefully choose a number that will make each of the equation a perfect

square trinomial. Number 1 is done for you.

1. x2 + 22x +_____

Answer:

x2 + 22x + (?)2

x2 + 2(11)x + (11)2 --- Definition of Perfect Square Trinomial

x2 + 2(11)x + 121

Therefore, the missing number is 121.

2. h2 - 12h + _____ 5. h2 - 2h + _____

3. t2 - 8h + _____ 6. t2 + 3h +_____

4. r2 - 18r + _____

Did you find it easy to determine the number that must be added to the

term(s) of polynomials to make it a perfect square trinomial? Were you able to

realize how it can be easily done? In the next activity, you will be representing

a situation using a mathematical sentence. Such mathematical sentence will be

used to satisfy the conditions of the given situation.

28

How did you find the activities? Are you ready to learn about solving

quadratic equations by completing the squares? I know you are!!! From the

previous activities you were able to solve equations, express a perfect square

trinomial as a square of a binomial, write perfect square trinomials, and represent a

real life situation by a mathematical sentence. But how does finding solutions of

quadratic equations facilitate in solving real life problems and in making decisions?

You will find these out in the next activities.

Activity 4: Paint My Room!

Directions: The shaded region of the diagram shows the portion of a square-shaped

room that is already painted. The area of the painted part is 24m2. Use

the diagram to answer the following questions.

2m

1. If y represents the side length of the

room and x represents the width of

the painted portion,

a. write an equation that represents

the side length of the room in

terms of x.

b. write an equation that represents

the area of the painted portion

in terms of x and y.

2. What equation would represent the

area of the painted part of the

room in terms of x only?

3. Using the equation formulated, solve for the dimension of the room.

A = 24 m2

29

What Is It

Extracting square roots and factoring are two methods commonly used to

solve quadratic equations of the form ax2 - c = 0 . If the factors of the quadratic

expression of ax2 + bx + c = 0 are determined, then it is more convenient to use

factoring to solve it.

Another method of solving quadratic equations is by completing the square.

This method involves transforming the quadratic equation ax2 + bx + c = 0. into the

form (x – h)2 = k, where k ≥ 0. The value of k should be positive to obtain a real

number solution.

To solve the quadratic equation ax2 + bx + c = 0 by completing the square,

the following steps can be followed:

1. Divide both sides of the equation by a then simplify.

2. Write the equation such that the terms with variables are on the left side of

the equation and the constant term is on the right side.

3. Add the square of one-half of the coefficient of x on both sides of the

resulting equation. The left side of the equation becomes a perfect square

trinomial.

4. Express the perfect square trinomial on the left side of the equation as a

square of a binomial.

5. Solve the resulting quadratic equation by extracting the square root.

6. Solve the resulting linear equations.

7. Check the solutions obtained against the original equation.

Example 1: Find the solutions of 2x2 + 12x – 14 = 0 by completing the square.

Steps Solution

1. Divide both sides of the equation by

the coefficient a then simplify.

In the given equation, 2x2 + 12x – 14 = 0,

a = 2.

2x2 + 12x – 14 = 0

=

x2 + 6x – 7 = 0

30

2. Write the equation such that the terms

with variables are on the left side of the

equation and the constant term is on the

right side.

x2 + 6x – 7 = 0 x2 + 6x – 7 + 7 = 0 + 7

x2 + 6x = 7

3. Add the square of one-half the

coefficient b on both sides of the

resulting equation. Then, the left side of

the equation becomes a perfect square

trinomial.

x2 + 6x = 7

Since b = 6,

( b ) ( 6 ) = 3 32 = 9

Thus, 9 will be added on both sides of

the equation

x2 + 6x = 7 x2 + 6x + 9 = 7 + 9

x2 + 6x + 9 = 16

4. Express the perfect square trinomial

on the left side of the equation as a


x2 + 6x + 9 perfect square trinomial

( x + 3)2 square of a binomial

Thus,

x2 + 6x + 9 = 16 (x + 3)2 = 16

5. Solve the resulting quadratic equation

by extracting the square

(x + 3)2 = 16 x + 3 =

x + 3 = 4


x + 3 = 4 x + 3 = - 4

x + 3 – 3 = 4 – 3 x + 3 – 3 = -4 – 3

x = 1 x = - 7

31

7. Check the solutions obtained against

the original equation.

Checking:

For x = 1:

2x2 + 12x – 14 ≟ 0

2(1)2 + 12(1) – 14 ≟ 0

2(1) + 12 – 14 ≟ 0

2 + 12 – 14 ≟ 0

0 0

For x = -7:

2x2 + 12x – 14 ≟ 0

2(-7)2 + 12(-7) – 14 ≟ 0

2(49) – 84 – 14 ≟ 0

98 – 84 – 14 ≟ 0

0 0


Thus 2x2 + 12x – 14 = 0 is true when x = 1 and x = -7.

Answer: The equation 2x2 + 12x – 14 = 0 has two solutions: x = 1 and x = - 7.

Example 2: Find the solutions of x2 - 8x - 9 = 0.

Steps Solution

1. Divide both sides of the equation by

the coefficient a then simplify.

In the given equation, x2 - 8x - 9 = 0,

a = 1.

Since a = 1, we don’t need to divide

both sides of the equations by 1 because

the equation will stay as it is.

x2 - 8x - 9 = 0

=

=

32

2. Write the equation such that the terms

with variables are on the left side of the

equation and the constant term is on the

right side.

x2 - 8x - 9 = 0 x2 - 8x – 9 + 9 = 0 + 9

x2 - 8x = 9

3. Add the square of one-half the

coefficient b on both sides of the

resulting equation. Then, the left side of

the equation becomes a perfect square

trinomial.

x2 - 8x - 9 = 0

Since b = 8,

( b ) ( 8 ) = 4 42 = 16

Thus, 16 will be added on both sides of

the equation

x2 - 8x = 9 x2 - 8x + 16 = 9 + 16

x2 - 8x + 16 = 25

4. Express the perfect square trinomial

on the left side of the equation as a


x2 - 8x + 16 perfect square trinomial

( x – 4 )2 square of a binomial

Thus,

x2 - 8x + 16 = 25 (x - 4)2 = 25

5. Solve the resulting quadratic equation

by extracting the square

(x - 4)2 = 25 x – 4 =

x – 4 = 5


x – 4 = 5 x – 4 = - 5

x – 4 + 4 = 5 + 4 x – 4 + 4 = - 5 + 4

x = 9 x = - 1

33

7. Check the solutions obtained against


Checking:

For x = 9:

x2 - 8x - 9 ≟ 0

(9)2 – 8(9) – 9 ≟ 0

81 – 72 – 9 ≟ 0

0 0

For x = -1:

x2 – 8x – 9 ≟ 0

(-1)2 – 8(-1) – 9 ≟ 0

1 + 8 – 9 ≟ 0

0 0


Thus x2 - 8x – 9 = 0 is true when x = 9 and x = -1.

Answer: The equation x2 - 8x – 9 = 0 has two solutions: x = 9 and

x = -1.

=

Your goal in the next section is to apply the key concepts of solving

quadratic equations by completing the square. Use the mathematical ideas

and the examples presented to answer the activities provided.

=

34

What’s More

Activity 5: Complete Me!

Directions: Find the solutions of each of the following quadratic equations by

completing the square.

1. x2 – 2x = 7 3. m2 + 10m + 9 = 0

2. s2 + 4s – 60 = 0 4. w2 + 3w = 3

What I Have Learned Activity 6: What Does The Equation Means To Me?

Directions: Answer the following problems completely.

1. Do you agree that any quadratic equation can be solved by completing the

square? Explain your answer.

2. If you are going to choose between completing the square and factoring in

finding the solutions in each of the following equations, which would you

chose? Explained and answer the given equation using your preferred

method.

a. 4x2 -16x+ 12 = 0 b. x2 - 8x + 15 = 0

3. Meg wants to use completing the square in solving the quadratic equation

x2 – 25 = 0. Can she use it in finding the solutions of the equation? Explain

why or why not?

How did you find the method of completing the square? Was it easy for you to

find the solutions of a quadratic equation by completing the square? If it is, you did a

good job! You may now proceed to the next activities and test your learning further.

35

What I Can Do

Activity 7: Create Your Own

Directions: Read and analyze the task given below. Perform the task by following

the conditions given and answer the questions that follow.

A. Form an open box out from a rectangular piece of cardboard whose length

is 6 cm longer than its width. To form the box, a square of side 3 cm will

be removed from each corner of the cardboard. Then the edges of the

remaining cardboard will be turned up.

a. Draw a diagram to illustrate the given situation.

b. How would you represent the dimensions of the cardboard?

c. What expressions represent the length, width, and height of the box?

d. If the box is to hold 448 cm3, what mathematical sentence would

represent the given situation?

e. Using the mathematical sentence formulated, how are you going to

find the dimensions of the rectangular piece of cardboard?

f. What are the dimensions of the rectangular piece of cardboard?

g. What is the length of the box? How about its width and height?

Rubric for Sketch Plan and Equations Formulated

4 3 2 1

The sketch plan is

accurately made,

presentable, and

appropriate

The sketch plan is

accurately made,

and appropriate

The sketch plan is

not accurately made

but appropriate

The sketch plan is

made but not

appropriate

Quadratic Equations

Are accurately

formulated and

solved correctly

Quadratic Equations

Are accurately

formulated but not

all are solved

correctly

Quadratic Equations

Are accurately

formulated but are


Quadratic Equations

Are accurately

formulated but all

are not solved

correctly

36

Summary

This lesson was about solving quadratic equations by completing the square.

The lesson provided you with opportunities to describe quadratic equations and

solve these by completing the square. You were able to find out also how such


demonstrate your understanding of the lesson by doing a practical task. Your


principles will facilitate your learning of the wide applications of quadratic equations

in real life.

37

Lesson

Solving Quadratic Equations by Quadratic Formula

2d

What I Need to Know

Start Lesson 2d of this module by assessing your knowledge of the different


operations. These knowledge and skills will help you in understanding Solving Quadratic

Equations using Quadratic Formula. As you go through this lesson, think of this important

question: “How does finding solutions of quadratic equations facilitate in solving real-life

problems and in making decisions?” To find the answer, perform each provided activity. If

you find any difficulty in answering the exercises, seek the assistance of your peers or

teacher or turn to the modules and lessons you have undergone earlier. You may check

your work with your teacher.

What’s In

Activity 1: Can You Simplify Me?

Directions: Simplify each of the following expressions. Answer the questions that

follow.

1. 4.

2. 5.

3. 6.

38

7. 9.

8. 10.

Process Questions:

a. How would you describe the expressions given?

b. How did you simplify each expression?

c. Which expression did you find difficult to simplify? Why?

Activity 2: Go For The Standards!

Directions: Write the following quadratic equations in standard form, ax2 + bx + c = 0.

Then identify the values of a, b, and c. Answer the questions that follow.

1. 3x2 + 12x =18 4. 2x(x – 5) = 9

2. x2 = – 7x + 8 5. (x + 3) (x + 2) = 0

3. 21 + 15x – 3x2 = 0 6. 3(x – 2)2 + 10 = 0

How was the activity for you? Were you able to simplify each

expression? I’m sure you were good at it. Now let us test your memory further.

Have you already recalled how to write quadratic equations in standard

from? I’m sure you did well! Now, recall your learning in the previous lessons

especially in completing the square. It will surely help you as you answer the

next activity.

39

What’s New

Activity 3: Deriving Quadratic Formula

Directions: The quadratic formula can be derived by applying the method of

completing the square. Analyze each step carefully and complete the

table by supplying the reasons as shown below.

Steps Reason

ax2 + bx + c = 0 ax2 + bx = - c

Why?

x2 + =

Why?

) = ; ( )2 =

Why?

x2 + +

Why?

(x + )2

=

Why?

x + x +

Why?

40

How did you find the activities? Are you ready to learn about solving

quadratic equations using quadratic formula? I know you are! Now, read and

understand the important notes on solving quadratic equations using the quadratic

formula and the examples that will be presented in this lesson. It will help you a lot

as you answer the next activities.

X =

Why?

x =

Why?

What Is It

The equation we obtained in activity 3 is what we call the Quadratic Formula

for solving any quadratic equations in the form ax2 + bx + c = 0.

To use it, you must follow these steps:

1. Write the equation to its standard form ax2 + bx + c = 0. If it is already

written in standard form, proceed to the next step.

2. Determine the values of the coefficients a, b, and c.

3. Substitute these values in the Quadratic Formula:

.

4. Evaluate and simplify the result.

5. Check the solutions obtained using the original equation.

41

Study the example that follows to help you understand more.

Example 1: Find the solutions of the equation 2x2 + 3x = 27 using the quadratic

formula.

Steps Solution

1. Write the equation to its standard

form ax2 + bx + c = 0

2x2 + 3x = 27 2x2 + 3x – 27 = 0

2. Determine the values of the

coefficients a, b, and c.

2x2 + 3x – 27 = 0

a = 2, b = 3, and c = - 27

3. Substitute these values in the

Quadratic Formula:

4. Evaluate and simplify the result.

x = 3

We obtained two equations

42

5. Check the solutions obtained using


Checking:

For x = 3:

2x2 + 3x ≟ 27

2(3)2 + 3(3) ≟ 27

2(9) + 9 ≟ 27

18 + 9 ≟ 27

27 27

For x = :

2x2 + 3x ≟ 27

2( )2 + 3( ) ≟ 27

2( ) - ≟ 27

- ≟ 27

27 27


Thus 2x2 + 3x = 27 is true when x = 3 and x = .

Answer: The equation 2x2 + 3x = 27 has two solutions: x = 3 and x = .

=

=

=

Now, let’s test your understanding by doing the activities set for you.

43

What’s More

Activity 4: Use the Formula!

Directions: Find the solutions of the following quadratic equations using the

quadratic formula. Answer the questions that follow.

1. x2 + 5x = 14 4. 2x2 + 7x = -9

2. x2 + 5x + 4 = 0 5. x2 + 4x + 4 = 0

3. 9x2 - 63 = 0

Process Questions:

a. Base on your answers, what is the maximum number of solutions a quadratic

equation can have?

b. Which equation has only one solution? Describe this equation.

c. Which equation has no real solution? Describe this equation.

d. How did the use of the quadratic formula in finding the solution/s of each

equation helped you?

Now, let us do more activities to assess your understanding with the

concepts of Quadratic Equations and its different methods of solving by

answering the next activities set for you.

44

What I Have Learned Activity 5: The Best That It Has!

Directions: Analyze what is asked in each item. Answer all the questions carefully

and write your answer in your Mathematics notebook.

1. The values of a, b, and c of a quadratic equation written in standard form are

3, - 8, and 2, respectively. Another quadratic equation has a = 3, b = 8, and

c = - 2. Will the two equations have the same solutions? Justify your answer.

2. How are you going to use the quadratic formula in determining whether a

quadratic equation has no real solutions? Formulate one example of quadratic

equation with no real solution.

3. Can the quadratic formula be used to solve any quadratic equation? Why or

why not?

4. If you are to solve each of the following quadratic equations, which method

will you use (you can choose among the 4 methods)? Explain why you

choose this method/s for solving the specific equation. You can have as many

answers as you can in each item and answers can be repeated.

a. x2 = 36

b. x2 + 8x +15 = 0

c. 3x2 + 13x + 9 = 0

d. x2 + 4x – 13 = 0

What I Can Do

Activity 6: Cut for the Cake!

Directions: Read and understand the situation below then answer the questions that

follow.

Suppose you own a family bakeshop and you are to cut different sizes

of rectangular box board to be used as a box for your business. The sizes of

the box board you have are listed below:

45

Box 1: The length is twice its width and the area is 338 sq.in.

Box 2: The length is 12 inches less than thrice its width and the area is

96 sq.in.

Box 3: The perimeter of the box board is 80 in. and the area is 384 sq. in

Process Questions:

a. What quadratic equation represents the area of each piece of the box board?

Write the equation in terms of the width of the box board.

b. Write each quadratic equation formulated in item 1 in standard form. Then

determine the values of a, b, and c.

c. Solve each quadratic equation using the quadratic formula.

d. Which of the solutions or roots obtained represents the width of each box

board? Explain your answer.

e. What is the length of each piece of box board? Explain how you arrived at

your answer.

Rubric for Equations Formulated

4 3 2 1

Quadratic Equations

Are accurately

formulated and

solved correctly

Quadratic Equations

Are accurately

formulated but not

all are solved

correctly

Quadratic Equations

Are accurately

formulated but are


Quadratic Equations

Are accurately

formulated but all

are not solved

correctly

46

Summary

This lesson was about solving quadratic equations using the quadratic

formula. The lesson provided you opportunities to describe quadratic equations and

solve these by using the quadratic formula. You were able to find out also how such


demonstrate your understanding of the lesson by doing a practical task. Your


principles will facilitate your learning of the wide applications of quadratic equations

in real life.

47

Assessment

Directions: Read each item carefully and choose the letter of the correct answer.

1. Find the solutions of the following equation x2 - 3x - 40.

A. -5 and -8 B. 5 and -8 C. -5 and 8 D. 5 and 8

2. Solve for x in the equation x2 + x = 12.

A. 6 and -2 B. -3 and -4 C. - 4 and 3 D. -2 and 6

3. In the equation 2x2 -2x – 12 = 0, the values of x are?

A. -6 and 2 B. -3 and 4 C. - 4 and 3 D. -2 and 3

6. Solve for x by extracting square roots, x2 = 256


7. By extracting square roots, 2x2 = 162,


8. In the equation 3x2 – 12 = 0, the roots are?

A. 2 and -2 B. 3 and -3 C. 4 and - 4 D. none of these

9. Solve for x by extracting square roots, 3x2 + 7 = 250.


10. In the equation x2 + 7x -120 = 0, the solutions are?

A. 12 and -10 B. - 20 and 6 C. 15 and -8 D. 8 and -15

11. In the equation x2 + 6x = 16, the roots are?

A. 8 and -2 B. -8 and 2 C. -8 and -2 D. 8 and 2

12. The roots of x2 + 11x – 60 = 0 are?

A. 8 and -2 B. -8 and 2 C. -8 and -2 D. -15 and 4

13. In the equation x2 + 18x + 81 = 0, the roots are?


14. In the equation x2 + 5x – 14 = 0, the solutions are?


15. The roots of 4x2 + 12x – 16 = 0 are?

A. 8 and -2 B. -4 and 1 C. 4 and -1 D. -8 and 2

16. In the equation x2 +121 = 22x, the roots are?


18

48

17. Find the solutions of the following equation x2 - 5x = 14


18. In the equation 4x2 -16x+ 12 = 0, the solution is?

A. 3 B. 4 C. -3 D. 2

19. In the equation x2 - 8x + 15 = 0, the solutions are?

A. 3 and -3 B. 5 and -3 C. 5 and 3 D. -5 and -3

20. In the equation x2 – 2x = 7, the solutions are?

A. 1+ and 1- C. 1+ and 1-

B. 3+ and 3 - D. 2+ and 2-

49

Key to Answers

What I Know

1. C 11. B

2. B 12. B

3. B 13. A

4. B 14. B

5. C 15. C

6. A 16. A

7. C 17. C

8. C 18. A

9. D 19. B

10. C 20. B

Lesson 2a Activity 1

1. ±7

2. –(9) = ±9

3. ±13

4. ±

5. ±

6. ±

Activity 2

2.45

2.83

2.24

3.46

5.48

7.07

8.66

Process Questions:

a) Answers may vary

b) two (the positive and the negative square

roots)

c) No. Squaring a number does not give a

negative result

d) , – , ,

e) and

1. Irrational numbers

2. Answers may vary

50

Activity 3

a. Answers may vary

b. s2 = 2,500

c.

d. 900 sq. M

e.

f.

Activity 6

1. x = ± 10 4. x = ±

2. x = ± 5. x = 0 , x = 4

3. x = ±

Activity 7

1. Answers may vary

2. Answers may vary

Activity 8

1. s x s = 225 s2 = 225

2. Answers may vary

Lesson 2b Activity 1

1. 2x (x-3) 5. (2y – 7)(y + 2)

2. -3x (x – 7) 6. (x – 6) (x -4)

3. ( 2t + 2 ) ( 2t + 2 ) or 4 ( t+1) ( t+1)

4. (2x-3) (2x+ 3)

Activity 4

1. Answers may vary

2. Answers may vary

3. Substitute the values to the

original equation to check

Activity 5

1. Answers may vary

2.

3.

4.

5. Answers may vary

51

Process Questions:

a. Answers may vary

b. Answers may vary

c. By multiplying the expressions obtained to check

d. Answers may vary

Activity 2

1. Linear , linear, quadratic

2. X =2 , x = 9, and (x=2 and x= 9)

3. Check the values of x from the original equation to satisfy the given

equation.

4. Answers may vary

5. No

6. The product of (x – 2) and (x – 9) is 0. Therefore, one of them must be

zero or both.

Activity 3 1. (x + 8) (x – 2 ) = 0 , x = -8 and x = 2

2. x = ±7

3. (x – 11) (x – 11) =0 , x = 11 and x = 11

4. (x + 4)(x – 1 ) = 0 , x = - 4 and x = 1

5. n = ±9

Activity 4

1. b and d

2. Answers may vary

3. Yes

Lesson 2c Activity 1

1. x = 7 and x = -7

2. t = 7 and t = -7

3. t = 20 and t = -2

4. s = 0

5. w = 6

6. m = -12 and m = 4

Process Questions:

52

a. Answers may vary

b. s2 - 25 = - 25 and w2 – 12w = -36

c. all except #4 and #5

Activity 2

1. ( x + 3 )2

2. ( t – 5 )2

3. ( w – 1 )2

4. ( x + 3/2 )2

5. ( t – 12 )2

6. ( s + ½ )2

Process Questions:

a. The quadratic term and the constant are perfect squares and the linear

term is the product of the square root of the quadratic term and the

constant times 2.

b. Special products

c. The 3rd term is the square of one half the coefficient of the middle term.

d. Answers may vary

Activity 3

1. 121

2. 36

3. 16

4. 81

5. 1

6. 9/4

53

Activity 4

1. Let y be the length of the shaded part of the room

X be the width of the shaded part of the room.

(a) y = x + 2

(b) xy = 24

2. x (x+2) = 24

3. x (x+2) = 24 x2 + 2x = 24 x2 + 2x - 24 = 0

4. (x +6) (x – 4) = 0 (x +6 ) = 0 , (x – 4) = 0 x = -6 , x = 4

Using x = 4 to find y:

y = x + 2

y = (4) + 2

y = 6

Activity 5

1. x = 1+ and x = 1-

2. s = 6 and s = -10

3. m = -1 and m = -9

4. w = -3/2 + and w = -3/2 -

Activity 6

1. Yes

2. Methods may vary

a. x = 3 and x = 1

b. x = 5 and x = 3

3. No

Activity 7

(Refer to the rubric)

54

Lesson 2d Activity 1

1. 5/4 Process Questions: Answers may vary

2. 3/4

3. -1/2

4. 2

5. 4

6. -3/2

7. -4

8. 2

9. -2

10. 5/3

Activity 2

1. 3x2 +12x -18 = 0 , a =3 , b= 12 , c = -18

2. x2 +7x - 8 = 0 , a =1 , b= 7 , c = -8

3. -3x2 +15x +21 = 0 , a =-3 , b= 15 , c = 21

4. 2x2 - 10x -9 = 0 , a =2 , b= -10 , c = -9

5. x2 +5x +6 = 0 , a =1 b= 5 , c = 6

6. 3x2 -12x +22 = 0 , a =3 , b= -12 , c = 22

Activity 4

1. x = 2 and x = -7

2. x = -1 and x = - 4

3. x = ±

4. No real solution

5. x = - 2

Process Questions

a. Two

b. x2 +4x + 4 = 0, Perfect square trinomial

c. 2x2 +7x = - 9, The expression b2 – 4ac is

equal to – 23.

d. Answers may vary

55

Activity 5 1. Yes

2. Answers may vary

3. Yes

4. a) Extracting square roots, Factoring and Completing the square

b) Factoring, Completing the square and Quadratic Formula

c) Completing the square and Quadratic Formula

d) Completing the square and Quadratic Formula

Activity 6

(Refer to the rubric)

Assessment

1. C 11. D

2. C 12. B

3. D 13. B

4. C 14. C

5. B 15. A

6. A 16. A

7. B 17. C

8. D 18. A

9. B 19. B

10. D 20. B

56

References:

(Bryant, Merden L.; Bulalayao, Leonides E.; Callanta, Melvin M.; Cruz, Jerry D.; De

Vera, Richard F.; Garcia, Gilda T.; Javier, Sonia E.; Lazaro, Roselle A.; Mesterio,

Bernadeth J.; Saladino, Rommel Hero A.; 2014)

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www.9-IGCSEmathworksheet.com

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www.basic-mathematics.com

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www.premath.com

http://www.premath.com/

http://www.pinterest.com/

http://www.9-igcsemathworksheet.com/

http://www.cism.connect.org/

http://www.basic-mathematics.com/

http://www.map.mathshell.org/

http://www.chilimath.com/

http://www.math-onlymath.com/

http://www.premath.com/

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Documents

Quarter 1, Week 2 Module 2 Solving Quadratic Equations