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Mathematics

Quarter 1 - Module 3: Nature of Roots of

Quadratic Equations

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Department of Education ● Republic of the Philippines

9

Mathematics- Grade 9

Alternative Delivery Mode Quarter 1, 3 - Module 1: Nature of Roots of Quadratic Equations

First Edition, 2020

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Mathematics

Quarter 1 - Module 3: Nature of Roots of

Quadratic Equations

This instructional material was collaboratively developed and reviewed

by educators from public and private schools, colleges, and or/universities.

We encourage teachers and other education stakeholders to email their

feedback, comments, and recommendations to the Department of Education-

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Department of Education ● Republic of the Philippines

9

Table of Contents

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

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

How to Learn from this Module .............................................................................................................. ii

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

Lesson 1:

Nature of Roots of Quadratic Equation ..................................................................... 1

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

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

What’s New ................................................................................................................... 2

What Is It ........................................................................................................................... 3

What’s More .................................................................................................................... 10

What I Have Learned..................................................................................................... 11

What I Can Do ................................................................................................................. 11

Lesson 2:

Sum and Product of Roots of Quadratic Equation ......................................... 12

What I Need to Know ................................................................................ 12

What I Know ……………………………………………………………………..12

What’s In ………………………………………………………………………… 13

What’s New ............................................................................................. . 14

What Is It ................................................................................................ . 15

What’s More ............................................................................................ 18

What I Have Learned …………………………………………………………. 19

What I Can Do ........................................................................................ 19

Summary ………………………………………………………………………………………. 20

Assessment: (Post-Test)………………………………………………………………………21 Key to Answers ............................................................................................................ 23 References ................................................................................................................... 29

What This Module is About

In everyday operations or daily life activities, we come across situations where

quantities such as fare, cost of daily commodities, salary, gains and losses, prizes of things

we usually buy, time, tuition, rent, and many others that affects our daily routines and

sometimes asked ourselves what can we do to make things easier for us? Would it be lighter

for problems to be easily solve? Did some of us think that these quantities can be

mathematically represented? And that could beneficially help us in making decisions?

Let us figure out the answers to these questions and determine the different ways of

using quadratic equations in daily life.

What I Need to Know

In this module, we will explore those questions and learn the following lesson:

Characterize the roots of a quadratic equation using the discriminant

Describe the relationship between the coefficients and the roots of a quadratic

equation

I

How to Learn from this Module

To achieve the objectives cited above, you are to do the following:

• Take your time reading the lessons carefully.

• Follow the directions and/or instructions in the activities and exercises diligently.

• Answer all the given tests and exercises.

Icons of this Module

What I Need to Know

This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know

This part includes an activity that aims to check

what you already know about the lesson to take.

If you get all the answers correct (100%), you

may decide to skip this module.

What’s In

This is a brief drill or review to help you link the

current lesson with the previous one.

What’s New

In this portion, the new lesson will be introduced

to you in various ways such as a story, a song, a

poem, a problem opener, an activity or a

situation.

What is It

This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More

This comprises activities for independent practice

to solidify your understanding and skills of the

topic. You may check the answers to the

exercises using the Answer Key at the end of the

module

What I Have Learned

This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Can Do

This section provides an activity which will help

you transfer your new knowledge or skill into real

life situations or concerns.

Assessment

This is a task which aims to evaluate your level of

mastery in achieving the learning competency.

Additional Activities

In this portion, another activity will be given to

you to enrich your knowledge or skill of the

lesson learned. This also tends retention of

learned concepts.

Answer Key

This contains answers to all activities in the

module.

ii

At the end of this module you will also find:

The following are some reminders in using this module:

1. Use the module with care. Do not put unnecessary mark/s on any part of the module.

Use a separate sheet of paper in answering the exercises.

2. Do not forget to answer What I Know before moving on to the other activities

included in the module.

3. Read the instruction carefully before doing each task.

4. Observe honesty and integrity in doing the tasks and checking your answers.

5. Finish the task at hand before proceeding to the next.

6. Return this module to your teacher/facilitator once you are through with it.

If you encounter any difficulty in answering the tasks in this module, do not hesitate to

consult your teacher or facilitator. Always bear in mind that you are not alone.

We hope that through this material, you will experience meaningful learning and gain

deep understanding of the relevant competencies. You can do it.

iii

References This is a list of all sources used in developing

this module.

1

Lesson

The Nature of Roots of Quadratic Equations

1

What I Need to Know

In this module we will start with assessing your knowledge of the different

mathematics concepts previously studied and your skills in performing mathematical

operations. These knowledge and skills will help you in understanding the nature of roots

of quadratic equations. As you go through this lesson think on how you could

characterize the roots of a quadratic equation using the discriminant.

What I Know

Pre-Assessment

Directions: Find out how much you already know about this module. Solve the following problems and write your answer on the space provided. Please answer all the items. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module.

1. Determine the discriminant and nature of roots of each quadratic equation.

a.) x2 –6x 9 0

b.) x2 – 4x 3 0

c.) x2 –7x – 4 0

d.) 2x2 3x 5 0

2. Describe the nature of roots of a quadratic equation given the values of the discriminant. Write your answer on the space provided. a) 36 f. -49

b) -17 g. 676

c) 0 h. -100

d) 196 i. 1

e) 143 j. 2025

3. Find the sum and product of the roots of the given quadratic equation.

a. 3x2 5x 6 0

Sum of Roots : ___________________________

Product of Roots : ___________________________

2

b. 4x2 6x 15 0

Sum of Roots : ___________________________

Product of Roots : ___________________________

4. Use the sum and product rule to determine if the two given values are the roots of the quadratic equation.

a. Are

and -2 the roots of 3x

2 2x – 5 0

b. Are -1

6 and

the roots of 3x

2 2x – 5 0

c. Are

and -

3

4 the roots of 3x

2 2x – 5 0

What’s In

What’s New

Activity 1: Which are Real? Which are Not?

Direction: Put a check (✓) on the corresponding box that best describes the given

numbers. Answer the questions that follow.

Real Numbers Not Real Numbers

1. √

2.

3. 21.5

4.

5. √-9

6. √12

9

7. √-4

8

8. 169

Process Questions: 1. Which of the following numbers above are familiar to you? Why? Describe

these numbers. 2. Which of the numbers are rational? Irrational? Explain your answer. 3. Which of the numbers are perfect squares? Not perfect squares? 4. How do you describe numbers that are perfect squares?

Comment [WU1]: Please present your review here in

activity form or whichever you like it.

3

Activity 2: Do I know My A,B,C?

Direction: Tell whether the given quadratic equations are in standard form or Not. If Not,

rewrite the equation in the form ax2 x 0 then identify the values of a, b, and c. Answer the questions that follow.

1. 6x² + 11x – 35 = 0. ________________ a= ____ b= ____ c= ____

2. 2x² – 2 = 4x. ________________ a= ____ b= ____ c= ____

3. -7x +12 = 4x2. ________________ a= ____ b= ____ c= ____

4. 5x (4x – 3) – 10 = 0. ________________ a= ____ b= ____ c= ____

5. 5x² – 2x – 9 = 0. ________________ a= ____ b= ____ c= ____

6. 3x² + 4x = -2. ________________ a= ____ b= ____ c= ____

Process Questions:

a. Where you able to write the equations in standard form? How? b. Is there another way of writing each quadratic equation in standard form? If

yes, show and determine the values of a, b and c.

Activity 3: Find the Value of b – ac

Direction: Evaluate the expression 2 –4a given the following values of a, b, c.

1. a = 6 b = -2 c = -3

2. a = 1 b = 5 c = 2

3. a = 5 b = 1 c = -2

4. a = -4 b = -4 c = 5

5. a = 2 b = 5 c = -4

Process Questions:

a. Where you able to evaluate the expression 2 –4a given the values of a, b,

and c?

b. What do you think is the importance of the expression 2 –4a in determining

the nature of the roots of quadratic equation?

What Is It

We have already studied the quadratic formula,

x √ 2 – 4a

2a

The binomial inside the radical sign is called the discriminant. It is used to

determine the nature of the roots of a quadratic equation. We can also determine the number of real roots for a quadratic equation with this number. The following table will give us the relation between the discriminant and the nature of the roots.

4

Discriminant Nature of the Roots Number of real

roots

b2 – 4ac = 0 Real and Equal 1

b2 – 4ac > 0 and a perfect square Rational and Unequal 2

b2 – 4ac < 0 but not a perfect square Irrational and Unequal 2

b2- 4ac < 0 Imaginary/No Real

Roots None

We will discuss here about the different cases of discriminant to understand the nature of the roots of a quadratic equation.

We know that x1 and x2 are the roots of the general form of the quadratic equation

ax2 x 0 where (a ≠ 0) then we get

x1 √ 2 – 4a

2a an x2

– √ 2 –4a

2a

Here a, b and c are real and rational.

Then, the nature of the roots x1 and x2 of equation depends on the quantity or expression i.e., b2 – 4ac under the radical sign.

Thus, the expression b2 – 4ac is called the discriminant of

the quadratic equation .

Generally we denote discriminant of the quadratic equation y ‘∆‘ or ‘D’.

Therefore, Dis riminant ∆ 2 4ac.

Depending on the discriminant we shall discuss the following cases about the

nature of roots x1 and x2 of the quadratic equation .

When a, b and c are real numbers, a ≠ 0

Case I: b2 – 4ac = 0

When a, b and c are real numbers, a ≠ 0 an discriminant is zero (i.e., b2 4ac =

0), then the roots x1 and x2 of the quadratic equation are real and equal. Example 1:

Find the discriminant value of x2 –12x + 36 = 0 and determine the number of real roots. Solution:

Step 1: Identify the values of a, b and c.

The given equation is x2 – 12x + 36 = 0. The equation is in the form ax2 + bx + c = 0 where,

a = 1 b = -12 and c = 36

5

Step 2: Substitute the values of a, b and c to the Discriminant = b2 4ac Discriminant = b2 4ac

= (-12)2 4(1)(36) = 144 144 = 0 Step 3: Describe the nature of the roots.

Since the discriminant value of the equation is zero then the equation x2 – 12x + 36 = 0 has a double root and the roots are real and are equal.

This can be checked by determining the roots of x2 – 12x + 36 = 0 using

any of the methods of solving quadratic equations. If factoring is used, the roots that can be obtained are the following: The roots of the quadratic equation x2 – 12x + 36 = 0 are real numbers and are equal.

Example 2: Find the nature of the roots of the equation x2 – 18x + 81 = 0.

Solution:

The coefficients of the equation x2 – 18x + 81 = 0 are rational.

The discriminant of the given equation is Discriminant = b2 4ac

= (-18)2 4(1)(81) = 324 – 324 = 0

Clearly, the discriminant of the given quadratic equation is zero and coefficient of x2 and x are rational.

Therefore, the roots of the given quadratic equation are real and equal.

To check, solve for the roots of x2 – 18x + 81 = 0.

Case II: b2 - 4ac > 0 and perfect square

When a, b and c are real numbers, a ≠ 0 an is riminant is positive an perfe t

square, then the roots x1 and x2 of the quadratic equation ax2 + bx + c = 0 are real, rational, unequal.

x2 – 12x + 36 = 0

(x – 6) (x – 6) = 0

x – 6 = 0 x – 6 = 0

x = 6 x = 6

(a double root)

x2 – 12x + 36 = 0

(x – 9) (x – 9) = 0

x – 9 = 0 x – 9 = 0

x = 9 x = 9

6

Example 3: Find the discriminant value for the equation x2 + 5x + 6 = 0 and determine the

number of real roots. Solution:

Step 1: Identify the values of a, b and c.

The given equation is x2 + 5x + 6 = 0. The equation is in the form ax2 + bx + c = 0 where,

a = 1 b = 5 and c = 6

Step 2: Substitute the values of a, b and c to the Discriminant ∆ = b2 4ac

Discriminant = b2 4ac

= (5)2 4(1)(6)

= 25 24 = 1 Step 3: Describe the nature of the roots.

Since the discriminant value of the equation is greater than 0 and a perfect square, then there are two real roots of the equation x2 + 5x + 6 = 0 and the roots are rational numbers but not equal.

This can be checked by determining the roots x2 + 5x + 6 = 0 using any of the methods of solving quadratic equations. If factoring is used, the roots that can be obtained are the following:

The roots of the quadratic equation x2 +5x + 6 = 0 are -3 and -2.

Example 4: Find the nature of the roots of the equation 3x2 – 10x + 3 = 0 without actually solving them.

Solution:

The coefficients of the equation 3x2 – 10x + 3 = 0 are rational.

The discriminant of the given equation is Discriminant = b2 4ac

= (-10)2 4(3)(3) = 100 – 36 = 64

Clearly, the discriminant of the given quadratic equation is positive and a perfect square.

Therefore, the roots of the given quadratic equation are rational and unequal.

To check, solve for the roots of 3x2 – 10x + 3 = 0.

x2 + 5x + 6 = 0

(x + 3) (x + 2) = 0

x + 3 = 0 x + 2 = 0

x = -3 x = -2

7

Case III: b2 – 4ac > 0 and not perfect square

When a, b and c are real numbers, a ≠ 0 an is riminant is positive (i.e., b2 – 4ac>0) but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal. Here the roots x1 and x2 form a pair of irrational conjugates.

Example 5: Describe the nature of the roots of the quadratic equation 2x2 – 8x + 3 = 0.

Solution:

The coefficients of the equation 2x2 – 8x + 3 = 0 are rational.

The discriminant of the given equation is

Discriminant = b2 4ac = (-8)2 4(2)(3)

= 64 – 24 = 40

Clearly, the discriminant of the given quadratic equation is positive but not a perfect square.

Therefore, the roots of the given quadratic equation are irrational and unequal.

To check, solve for the roots of 2x2 – 8x + 3 = 0 using the quadratic equation.

x √ 2 – 4a

2a

x ( 8) √( 8)2 – 4(2)(3)

2(2)

x 8 √64 – 24

4

x 8 √40

4

x √

2 √

2 –

√

3x2 – 10x + 3 = 0

(3x – 1) (x – 3) = 0

3x – 1 = 0 x – 3 = 0

3x = 1 x = 3

x 1

3

8

Case IV: b2 - 4ac < 0

When a, b and c are real numbers, a ≠ 0 and discriminant is negative (b2 - 4ac < 0), then the roots x1 and x2 of the quadratic equation ax2 + bx + c = 0 are unequal and imaginary. Here the roots x1 and x2 are a pair of the complex conjugates. Example 6:

Find the discriminant value 2x2 + x + 3 = 0 and determine the number of real roots. Solution:

Step 1: Identify the values of a, b and c.

The given equation is 2x2 + x + 3 = 0. The equation is in the form ax2 + bx + c = 0 where,

a = 2 b = 1 and c = 3

Step 2: Substitute the values of a, b and c to the Discriminant = b2 4ac Discriminant = b2 4ac

= (1)2 4(2)(3) = 1 24

= 23 Step 3: Describe the nature of the roots.

Since the discriminant value of the equation is less than zero then the equation 2x2 + x + 3 = 0 has no real roots or imaginary. Also, the graph of this equation does not touch the x-axis. To check, solve for the roots of 2x2 + x + 3 = 0 using the quadratic formula.

x √ 2 – 4a

2a

x (1) √(1)2 – 4(2)(3)

2(2)

x 1 √1 – 24

4

x 1 √ 23

4

1 √ 23

4

1 – √ 23

4

Example 7: Describe the nature of the roots of the quadratic equation x2 + x + 1 = 0.

Solution:

The coefficients of the equation x2 + x + 1 = 0 are rational.

9

The discriminant of the given equation is Discriminant = b2 4ac

= (1)2 4(1)(1) = 1 – 4 = -3 Clearly, the discriminant of the given quadratic equation is negative.

Therefore, the roots of the given quadratic equation are imaginary and unequal. Thus, the roots of the given equation are a pair of complex conjugates.

To check, solve for the roots of x2 + x + 1 = 0 using the quadratic equation.

x √ 2 – 4a

2a

x (1) √(1)2 – 4( )(1)

2(2)

x 1 √1 – 4

4

x 1 √ 3

4

1 √ 3

4

1 – √ 3

4

Learn more about the nature of roots of a quadratic equation through the web. You may open the following links.

https://mymathszone.weebly.com/uploads/5/4/2/1/54214975/lesson_7_discriminant_and_nature_of_roots_of_quadratic_equations.pdf

https://www.youtube.com/watch?v=WcDuVQ_k1K0

https://www.math-only-math.com/nature-of-the-roots-of-a-quadratic-equation.html

10

What’s More

Activity 4: Find My Nature Directions: Describe the nature of the roots of the quadratic equation using its

discriminant. Answer the questions that follow. Example:

3x2 – 2x – 5 = 0 Discriminant 64 Nature of Roots rational and unequal

1. –6x2 7x 3 0 Discriminant ______ Nature of Roots __________

2. 9x2 – 3n 2 0 Discriminant ______ Nature of Roots __________

3. –2x2– 8x – 8 0 Discriminant ______ Nature of Roots __________

4. 2x2 5x – 4 0 Discriminant ______ Nature of Roots __________

5. 9x2 – 6x 1 0 Discriminant ______ Nature of Roots __________

Process Questions: 1. How did you solve for the discriminant of the quadratic equations? 2. Where you able to describe the nature of the roots? Explain. 3. In what way does your knowledge on discriminant help you in describing the

nature of the roots? Activity 5: Let’s Do Some Framing Directions: Study the situation below the answer the questions that follow.

A rectangular frame has an area of 14 square inches and a perimeter of 18 inches. Find the dimensions of the rectangular frame. The figure below shows how to set up the problem.

11

Process Questions: 1. Form a quadratic equation that represents the situation. 2. Without actually computing for the roots, determine whether the dimensions

of the table are rational numbers. Explain. 3. Give the dimension of the rectangular frame.

What I Have Learned

Activity 5: Is it Possible? Directions: Answer the following.

1. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m²? If so, find its length and breadth.

2. Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of ages in years was 48.

3. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.

What I Can Do

Activity 6: Using Discriminant in Real Life

Directions: Answer the following problem.

You and a friend are planning to have a camping after COVID19

pandemic. You want to hang your food pack from a branch 20 feet from the

ground. You will attach a rope to a stick and throw it over the branch. Your friend

can throw the stick upward with an initial velocity of 29 feet per second. The

distance of the stick after t seconds from an initial height of 6 feet. Will the stick

reach the branch when it is thrown?

Use the vertical motion model h = -16t 2+ vt + s where h represents the

height you are trying to reach, t the time in motion, v the initial velocity, and s the

initial height.

Were you able to determine the nature of the roots of the quadratic equation? Try to compare your answers with your classmate, did you have the same ideas? How much of your ideas where similar to your classmates? Which ideas were different?

12

Lesson

The Sum and Product of Roots of Quadratic Equations

2

What I Need to Know

In this lesson, you will recall and apply previously learned mathematical concepts and principles in performing the sum and product of roots. To be able to answer the following activities, you will need to understand the relationship between the coefficients and the roots of a quadratic equation. If you have some difficulties along the way, you may seek help from your teacher and refer your answers to them.

What I Know

Activity 1: Relate Me to My Roots! Direction: Consider and carefully analyze the table below.

Quadratic Equation Roots Sum of the

Roots Product of the Roots

Values of a, b and c

(r1, r2) r1 + r2 r1 ● r2 a b c

x2 + 7x + 12 = 0 (-3, -4) -7 12 1 7 12

2x2 – 3x – 20 = 0 (4, -5/2) 3/2 -10 2 -3 -20

Process Questions: 1. What do you observe about the sum and the product of the roots of each

quadratic equation in relation to the values of a, b, and c? 2. Do you think a quadratic equation can be determined given its roots or solutions?

Justify your answer by giving 3 examples. 3. Do you think a quadratic equation can be determined given the sum and product

of its roots? Justify your answer by giving 3 examples.

13

What’s In Activity 2: Sum and Product Puzzles

Direction: Observe the following puzzle and complete the given problem using your skills on the basic mathematical operations. Write your answer in the box. Example: Find two factors whose product is 10 and sum is 7.

.

1. 2. 3.

4. 5. 6.

7. 8. 9.

PRODUCT

SUM

Factor Factor

10

7

2 5

The factors are 5 and 2. Since (5)(2) = 10 and 5 + 2 = 7

Were you able to complete the puzzle correctly? In the next activity you will enhance your mathematical skills in finding the roots of the quadratic equation.

-3

5 -

1

-

3

1

3

6

7

16

8

-

5 -

3

8

12

7

-8

2

-18

-7

1

3

14

Activity 3: Find My Roots Direction: Find the roots of the following quadratic equation using any of the method

(Extracting Square Roots, Factoring, Completing the Square, Quadratic Formula).

1. x2 - 14x - 40 0 6. 4x2 2x - 12 0

2. x2 4x - 12 0 7. 9x2 7x - 4 0

3. 4x2 17x - 15 0 8. 3x2 9x - 6 0

4. -8x2 - 15x 2 0 9. x2 6x - 40 0

5. x2 14x 45 0 10. x2 8x 5

Process Questions: 1. How did you find the roots of each quadratic equation? Which method of

solving quadratic equation did you use in finding the roots?

2. Which of the given quadratic equation did you find difficult to solve? Why?

3. Compare your answers with your classmates. Do you have the same

answers? If NOT, explain why?

What’s New

In this next Activity, try to find the roots of the quadratic equation using any method then evaluate their sum and product and observe its relevance to the coefficients of the quadratic equation. Activity 4: Are We Related? Directions: Given the following quadratic equation, complete the table below, then

answer the following questions. You may work in groups of 4. The first one is done for you.

Quadratic Equation Coefficients Roots Sum of

Roots Product of

Roots

a b c x1 x2 x1 x2 x1)( x2

x2 4x 12 0 1 8 -12 6 -2 4 -12

x2 5x 14 0

3x2 3x 6 0

Process Questions:

1. What do you observe about the sum and product of the roots of each

quadratic equation in relation to the values of the coefficients a, b, and c?

2. Can you solve for the quadratic equation given its roots? Explain.

3. How about if the sum and product of the roots are given? Can you determine

the quadratic equation? Justify your answer. Give at least 3 examples.

15

What Is It

Activity 5: Stretch Me Out! Directions: Study the given problem and answer the questions that follow.

A picture has a width that is 4/3 its height. If it has an area of 192 square inches, what will be the dimension of the picture? What is the perimeter of the picture?

https:/ /www.shutterstock.com/blog/take-pictures-moon

Process Questions:

1. What equation would best describe the area of the picture? Write the equation in term of the width of the picture.

2. What can you say about the equation formulated in item 1? 3. What are the roots of the equation in number 1? What do the roots represent? 4. What is the perimeter of the picture? How is it related to the sum of the roots of

the equation in number 1? We will now discuss on how the sum and product of roots of the quadratic

equation ax2 x 0 can be determined using the coefficients a, b, and c.

We have seen that the b2 – 4ac is the radicand of the quadratic formula, called the discriminant, can tell us the type of roots of a quadratic equation. The quadratic formula can also give us information about the relationship between the roots and the coefficient of the second term and the constant of the equation itself. Consider the following:

How did you find the values of the coefficients a, b, and c helpful in finding the sum and product of roots? Were you able to relate them? For example, you are solving for the quadratic equation given the sum and product of roots, will you be able to give the equation? To answer the following activities, you will now read and understand the importance of the sum and product of roots of the quadratic equations and the examples presented.

16

Given a quadratic equation: ax2 x 0. By the quadratic formulas, the two roots can be represented as

r1 √ 2 – 4a

2a an r2

– √ 2 –4a

2a

Sum of the Roots, r1 + r2:

Product of the Roots, r1 • r2:

The sum of the roots of a quadratic equation is equal to the inverse of the

coefficient of the second term, divided by the leading coefficient.

r1 r2

a

The product of the roots of a quadratic equation is equal to the constant term, divided by the leading coefficient.

r1 r2

a

Example 1: Find the sum and product of roots of the quadratic equation x2 - 5x + 6 = 0. Solutions: Given the equation x2 - 5x + 6 = 0, we get a = 1, b = -5 and c = 6.

um of the roots

a

a

5

1 5

r1 r2 √ 2 – 4a

2a

– √ 2 – 4a

2a

√ 2 – 4a – – √ 2 – 4a

2a

2

2a

r1 r2

a

r1 r2 √ 2 – 4a

2a

– √ 2 – 4a

2a

√ 2 – 4a – √ 2 – 4a (

2 – 4a )

4a2

2 –

2 4a

4a2

r1 r2

a

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ro u t of the roots

a

a

6

1 6

The roots of the equation x2 - 5x + 6 = 0 are 3 and 2 (using factoring or any method). To check, find the sum and product of these roots.

et r1 3 an r2 2

r1 r2 3 2 5

r1 r2 3 2 6

Therefore, the sum and product of roots of x2 - 5x + 6 = 0 are 5 and 6, respectively.

Example 2: Given the values a 1, 4, an -21. What is the quadratic equation?

Solve for the sum and product of roots.

Since a 1, 4, an -21 thus the equation is x2 4x - 21 0.

um of the roots

a

4

a

4

1 4

ro u t of the roots

a

a

21

1 21

15 By inspection, the two numbers that give a sum of -4 and a product of -21 are -7 and 3.

et r1 -7 an r2 3

r1 r2 -7 3 -4

r1 r2 (-7) 3 -21

Therefore, the quadratic equation is x2 4x - 21 0 and its sum and

product of roots of are -4 and -21, respectively.

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What’s More

Activity 6: Find My Roots

Direction: Using the values of a, b, and c of each of the following quadratic equation solve for the sum and product of roots. Check your answer by using the roots of the quadratic equation. Then answer the question below.

Quadratic Equation

Sum of the Roots Product of the

Roots Roots

x2 3x 10 0

x2 4x 21 0

x2 6x 7 0

2x2 8x 10 0

6x2 7x 2 0

Process Questions:

1. How did you solve for the sum and product of roots?

2. Are the values of a, b, and c helpful in determining the roots of the quadratic

equation? Why? Why Not?

3. What do you think is the importance of knowing the sum and product of roots of

the quadratic equation?

Activity 7: Think of These Further!

Direction: Answer the following.

1. Suppose the product of the roots of a quadratic equation is given, do you think you can determine the equation? Justify your answer.

2. The sum of the roots of the quadratic equation is -5. If one of the roots is 7, how would you determine the equation? Write the equation.

3. The product of the roots of a quadratic equation is 51. If one of the roots is -17, what could be the equation?

4. The perimeter of a rectangular bulletin board is 20 ft. If the area of the board is 21 ft2, what its length and width?

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What I Have Learned

Activity 8: Find My Match

Directions: Match column A (roots) with column B (Quadratic Equation).

Column A Column B

1. 5 and -14 a.) 2x2 14x 24 0

2. 3 and 4 .) 6x2 x 2 0

3. -5

3 and 1 .) x

2 9x 70 0

4. 7 and - 2 .) 3x2 2x 5 0

5. 1

2 and

-2

3 e.) x

2 5x 14 0

Process Questions:

1. How did you determine the quadratic equation given its roots?

2. Which roots did you find it difficult to determine the quadratic equation? Why?

3. Compare your answer with your classmate. Did you have the same answer? If

NOT, why? What are the differences in your solutions?

4. Where you able to solve the problem using other methods? Explain then give

examples.

What I Can Do

Activity 9: Let’s Frame a Collage! Directions: You want to frame a collage of pictures with a 9-ft strip of wood. What

dimensions will help you maximize the area? Make a design or sketch plan of frame. Using the design or sketch plan, formulate problems that involves nature of roots of quadratic equations and then solve them.

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

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 are

not solved.

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Summary

This lesson was about the nature of roots of quadratic equations. The lesson supply you with chances of describing the nature of the roots of quadratic equation without solving the equation. Thus, providing you with ample time in solving related problems in life.

Moreover, you’ve learne that the is riminant ( 2 - 4a ) of quadratic equation can be

exhibited in real-life circumstances. Your discernment of this lesson and other previously learned mathematical ideas and principles will make easier understanding of the succeeding lessons.

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Assessment: (Post-Test)

Directions: Find out how much you already know about this module. Choose the letter of the correct answer.

1. Which equation has irrational and unequal roots? A. x2 – 4x + 5=0 C. x2 + 16x + 15= 0

B. x2 + 10x + 25=0 D. x2 + 12x – 7= 0

2. The coefficients of a quadratic equation are all integers. The discriminant is 0. Which statement best describes its roots?

A. Two irrational roots C. One rational root B. No Real roots D. Two rational roots

3. How many roots are there if the discriminant of a quadratic equation is greater than zero? A. 1 real root C. 3 real roots

B. 2 real roots D. No Solutions

4. Find the value of the discriminant. How will you describe the number and type of

roots for 3x2- 6x 2 0 A. Since the discriminant is greater than 0 and is perfect square, the roots are real and irrational. B. Since the discriminant is greater than 0 and is not a perfect square, the roots are real and irrational.

C. Since the discriminant is less than 0, the roots are non-real D. Since the discriminant is equal to 0, the roots are equal and real.

5. How many real roots does the quadratic equation x2 + 5x + 7 = 0 have?

A. 0 B. 1 C. 2 D. 3

6. The roots of a quadratic equation are -5 and 3. Which of the following quadratic equations has these roots?

A. C.

B. D.

7. Which of the following quadratic equations has no real roots?

A. C.

B. D.

8. What is the nature of the roots of the quadratic equation if the value of its discriminant is zero?

A. The roots are not real C. The roots are rational and not equal. B. The roots are irrational and not equal. D. The roots are rational and equal.

9. One of the roots of is 4. What is the other root?

A.

B.

C.

D.

10. What are the roots of the quadratic equation A. 12 and -1 B. 12 and 1 C. -12 and 1 D. -12 and -1

11. What is the sum of the roots of the quadratic equation ? A. -7 B. -6 C. 6 D. 14

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12. The length of a garden is 5m longer than its width and the area is 14m2. How long is the garden?

A. 9m B. 7m C. 5m D. 2m

13. What is the sum and product of roots for this equation: A. Sum of Roots = 2 , Product of Roots = 3 B. Sum of Roots = -2 , Product of Roots = 1/3 C. Sum of Roots = -2 , Product of Roots = 3 D. Sum of Roots = -2 , Product of Roots = -1/3

14. For the equation , what is the product of the roots?

A.

B.

C.

D.

15. The quadratic equation has only one root. Use the discriminant to determine the value of d. A. -2 B. 2 C. 3 D.4

16. The sum of the roots of quadratic equation is:

A.

B.

C.

D.

17. The product of the roots of quadratic equation is:

A.

B.

C.

D.

18. What is an equation whose roots are √ √ ?

A. C.

D.

19. If p > 0, and x2 - 11x + p = 0 has integer roots, how many integer values can 'p' take? A. 5 B. 6 C. 10 D. 11

20. The sum of two numbers is 12 and their product is 35. What are the two numbers? A. 5 and 7 B. -5 and -7 C. -5 and 7 D. 5 and -7

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Key to Answers

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References

Mathematics – Grade 9 Learner’s Material First Edition, 2014, Department of Education

DepEd link: http://www.depednegor.net/uploads/8/3/5/2/8352879/math_9_lm_draft_3.24.2014.pdf

http://www.findglocal.com/PH/Iligan-City/1412523548964696/Ms-Di-Collection

https://www.flickr.com/photos/eromligg77/5256544947/

https://asweknowitlife.wordpress.com/2012/08/12/gaisano-iligan-mall/

https://mymathszone.weebly.com/uploads/5/4/2/1/54214975/lesson_7_discriminant_and_nature_

of_roots_of_quadratic_equations.pdf

https://www.mathwarehouse.com/downloads/sheets/algebra-2/quadratic-equation-

worksheets/sum-and-product-of-roots-worksheet.pdf

https://calcworkshop.com/intro-algebra/real-numbers/

https://cdn.kutasoftware.com/Worksheets/Alg2/The%20Discriminant.pdf

http://www.mathocean.com/2010/03/discriminant.html

https://www.onlinemath4all.com/solving-word-problems-with-nature-of-roots-of-quadratic-

equation.html

https://pixabay.com/vectors/search/arrow/

https://www.teacherspayteachers.com/Product/Sum-and-Product-Puzzles-positive-numbers-

1989333

https://math.uiowa.edu/sites/math.uiowa.edu/files/FactoringWorksheet.pdf

http://sites.millersville.edu/bikenaga/basic-algebra/quadratic-word-problems/quadratic-word-problems.html

https://www.purplemath.com/modules/quadprob2.htm

https://mathbitsnotebook.com/Algebra2/Quadratics/QDSumProduct.html

https://www.onlinemath4all.com/sum-and-product-of-the-roots-of-a-quadratic-equation-

examples.html

https://www.basic-mathematics.com/word-problems-involving-quadratic-equations.html

http://lzinnick.weebly.com/uploads/1/3/4/2/13428779/2015_quad_eq_wks_nature_of_roots.pdf

https://www.mathsisfun.com/algebra/polynomials-sums-products-roots.html

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For inquiries and feedback, please write or call: Department of Education – Bureau of Learning Resources (DepEd-BLR) DepEd Division of Cagayan de Oro City Fr. William F. Masterson Ave Upper Balulang Cagayan de Oro Telefax: ((08822)855-0048

E-mail Address: cagayandeoro.city@deped.gov.ph