CLASS-X UNIT-4 QUADRATIC EQUATION CORE … UNIT-4 CLASS X Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India (CORE) The CBSE-International is grateful for permission

QUADRATIC EQUATIONS

MATHEMATICS

UNIT-4

CLASS

XCBSE-i

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 IndiaShiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

(CORE)

CBSE-i

QUADRATIC EQUATIONS

MATHEMATICS

UNIT-4

CLASS

X

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

(CORE)

The CBSE-International is grateful for permission to reproduce

and/or translate copyright material used in this publication. The

acknowledgements have been included wherever appropriate and

sources from where the material may be taken are duly mentioned. In

case any thing has been missed out, the Board will be pleased to rectify

the error at the earliest possible opportunity.

All Rights of these documents are reserved. No part of this publication

may be reproduced, printed or transmitted in any form without the

prior permission of the CBSE-i. This material is meant for the use of

schools who are a part of the CBSE-International only.

The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos.

The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view.

The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary.

The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements.

The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners.

The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve.

The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens.

The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board.

I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr. Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the development and implementation of this material.

The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions are welcome.

Vineet Joshi

Chairman

PREFACEPREFACE

ACKNOWLEDGEMENTSACKNOWLEDGEMENTSAdvisory Conceptual Framework

Ideators

Shri Vineet Joshi, Chairman, CBSE Shri G. Balasubramanian, Former Director (Acad), CBSE

Shri Shashi Bhushan, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step School, Noida

Dr. Sadhana Parashar, Head (I & R),CBSE

Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija

Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty

Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja

Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija

Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry

Coordinators:

Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi, Head (I and R) E O (Com) E O (Maths) O (Science)

Shri R. P. Sharma, Consultant Ms. Ritu Narang, RO (Innovation) Ms. Sindhu Saxena, R O (Tech) Shri Al Hilal Ahmed, AEO

Ms. Seema Lakra, S O Ms. Preeti Hans, Proof Reader

Material Production Group: Classes I-V

Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur

Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary

Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty

Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya

Ms. Ritu Batra Ms. Kalyani Voleti

English :

Geography:

Ms. Sarita Manuja

Ms. Renu Anand

Ms. Gayatri Khanna

Ms. P. Rajeshwary

Ms. Neha Sharma

Ms. Sarabjit Kaur

Ms. Ruchika Sachdev

Ms. Deepa Kapoor

Ms. Bharti Dave Ms. Bhagirathi

Ms. Archana Sagar

Ms. Manjari Rattan

Mathematics :

Political Science:

Dr. K.P. Chinda

Dr. Ram Avtar Gupta

Dr. Mahender Shankar

Mr. J.C. Nijhawan

Ms. Reemu Verma

Ms. Himani Asija

Ms. Sharmila Bakshi

Ms. Archana Soni

Ms. Srilekha

Science :

Economics:

Ms. Charu Maini

Ms. S. Anjum

Ms. Meenambika Menon

Ms. Novita Chopra

Ms. Neeta Rastogi

Ms. Pooja Sareen

Ms. Mridula Pant

Mr. Pankaj Bhanwani

Ms. Ambica Gulati

History :

Ms. Jayshree Srivastava

Ms. M. Bose

Ms. A. Venkatachalam

Ms. Smita Bhattacharya

Material Production Groups: Classes IX-X

English :

Ms. Rachna Pandit

Ms. Neha Sharma

Ms. Sonia Jain

Ms. Dipinder Kaur

Ms. Sarita Ahuja

Science :

Dr. Meena Dhami

Mr. Saroj Kumar

Ms. Rashmi Ramsinghaney

Ms. Seema kapoor

Ms. Priyanka Sen

Dr. Kavita Khanna

Ms. Keya Gupta

Mathematics :

Political Science:

Ms. Seema Rawat

Ms. N. Vidya

Ms. Mamta Goyal

Ms. Chhavi Raheja

Ms. Kanu Chopra

Ms. Shilpi Anand

Geography:

History :

Ms. Suparna Sharma

Ms. Leela Grewal

Ms. Leeza Dutta

Ms. Kalpana Pant

Material Production Groups: Classes VI-VIII

Preface

Acknowledgements

1. Syllabus 1

2. Scope document 2

3. Teacher's Support Material 5

Teacher Note 6

Activity Skill Matrix 13

Warm Up Activity W1 14

Identify the Polynomials of Degree 2

Warm Up Activity W2 15

Recognize Zeroes of a Polynomial

Pre -Content Worksheet P1 15

Quadratic Equations from Quadratic Polynomials

Identify Polynomials which cannot be Factorized

Pre -Content Worksheet P2 16

Revisit Key Concepts

Content Worksheet CW1 17

Zeroes of a Quadratic Polynomial


Roots of a Quadratic Equation


Nature of Roots


Discriminant Method of Finding the Roots


Relations Between Roots and Coefficients of a Quadratic Equation


Formation of a Quadratic Equation.


Application of Quadratic Equation in Real Life Problems.

Post Content Worksheet PCW1 25


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Content



Assessment Plan 26

Study Material 30

Student's Support Material 63

SW1: Warm Up Worksheet (W1) 64


SW2: Warm Up Worksheet (W2) 66


SW3: Pre Content Worksheet (P1) 68


Identify Polynomials which cannot be Factorized

SW4: Pre Content Worksheet (P2) 71


SW5: Content Worksheet (CW1) 73





Nature of Roots

SW8:Content Worksheet (CW4) 83

Discriminant method


Relation Between Roots and Coefficients


Forming a Quadratic Equation

SW11:Content Worksheet (CW7) 95

Application of Quadratic Equation in Real Life Problems

SW12: Post Content Worksheet (PCW1) 103

SW13: Post Content Worksheet (PCW2) 104

SW 14: Post Content Worksheet (PCW3) 107

SW 15: Post Content Worksheet (PCW4) 107

Suggested Videos & Extra Readings. 109

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1

SYLLABUS UNIT 4: QUADRATIC EQUATION (CORE)

Introduction to Quadratic Equation

Quadratic Equations are of the form

ax 2+ b x + c=0, a≠0, a, b, c being real

numbers

Methods to solve quadratic equations Factorisation Method,

Discriminant Method

(D = b2 – 4ac ) formula , x= √

Nature of roots Nature of roots when D=0, D 0, D 0

Sum of roots, product of roots, conjugate

roots

Finding a quadratic equation when relations

between the roots are given.

Application in daily life Number problems, age problems, work ratio

problems, distance time problems, speed‐

time problems, upstream‐downstream

motion problems.

2

SCOPE DOCUMENT

Key Concepts:

1. Quadratic Equation

2. Discriminant

3. Roots of a quadratic equation

4. Nature of roots

5. Sum of roots

6. Product of roots

Learning objectives:

1. To recognise a quadratic equation as equation of the form a x 2+ b x +c = 0, where a, b, c

are real numbers and a ≠0

2. To understand that roots of a quadratic equation are those real numbers which satisfy the

quadratic equation. Roots are also known as solution of quadratic equation.

3. To find the roots of equation by factorisation method.

4. To predict the nature of roots based on the sign of discriminant.

5. To find the roots of equation by discriminant method.

6. To know the relation between sum of the roots of quadratic equation and the coefficients

of x2, x and constant

7. To know the relation between product of the roots of quadratic equation and the

coefficients of x2, x and constant

8. To form a quadratic equation if sum of roots and product of roots are known.

9. To solve the problems from real life situations having application of quadratic equations.

3

Extension Activities

1. Equations reducible to quadratic forms, e.g.

a) x + = 0 b) (2x+ y) 2+ 4(2x+y) = 3

c) 3x + 3‐x = 2 d) + = 6

2. Zeroes of quadratic polynomial determined graphically represent the roots of quadratic

equation obtained by equating the quadratic polynomial to zero. So, the points

representing zeroes represent quadratic polynomial p(x) =0 but rest of the points on

polynomial curve represent quadratic inequality i.e. p(x) <0 or p(x) >0. With the help of

graph show that quadratic inequalities has infinite solutions and quadratic equation has at

the most two solutions.

3. Biquadratic equations are the equation of the form ax4 +bx2 +c =0. Biquadratic equations

can be solved by reducing them to quadratic forms.

Life skill

Today’s world is world of options. In this materialistic era it is very important to make informed

choices. Using knowledge of finding solution to quadratic equations students can compare the

investment plans or insurance plans offered by various banks or financial agencies and

understand that they should not pick up any product from market on the basis of catchy

language or glamorous brand ambassador. Rather they shall workout rationally the long term

effects of any plan.

SEWA

Applying their knowledge of quadratic equations as discussed above (life skill) students can

save their relatives and elders to be misguided by catchy language of advertisements.

4

Cross –curricular link:

1. The study of forces and their effect on the motion of objects traveling through the air is

called aerodynamics. Quadratic equations find its application in aerodynamics.

Look at the following problem:

A model rocket is launched with an initial velocity of 200 ft. /s. The height h, in feet, of the

rocket t seconds after the launch is given by h= −16t2 + 200t. How many seconds after the

launch will the rocket be 300 ft above the ground?

2. Car safety

A car with good tire tread can stop in less distance than a car with poor tread. The formula

for the stopping distance, in feet of a car with good tread on dry cement is approximated

by d= 0.4v2 +0.5v, with v as speed of the car. If the driver must be able to stop within

60 ft., what is the maximum safe speed, to the nearest miles per hour, of the car?

Many more application can be identified in architecture, construction, geometry etc.

5

TTEEAACCHHEERR’’SS

SSUUPPPPOORRTT

MMAATTEERRIIAALL

6

Teacher’s Note

The teaching of Mathematics should enhance the child’s resources to think and reason, to

visualise and handle abstractions, to formulate and solve problems. As per NCF 2005, the vision

for school Mathematics include :

1. Children learn to enjoy mathematics rather than fear it.

2. Children see mathematics as something to talk about, to communicate through, to discuss

among themselves, to work together on.

3. Children pose and solve meaningful problems.

4. Children use abstractions to perceive relationships, to see structures, to reason out things,

to argue the truth or falsity of statements.

5. Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and

trigonometry, the basic content areas of School Mathematics, all offer a methodology for

abstraction, structuration and generalisation.

6. Teachers engage every child in class with the conviction that everyone can learn

mathematics.

Students should be encouraged to solve problems through different methods like

abstraction, quantification, analogy, case analysis, reduction to simpler situations, even

guess‐and‐verify exercises during different stages of school. This will enrich the students

and help them to understand that a problem can be approached by a variety of methods for

solving it. School mathematics should also play an important role in developing the useful

skill of estimation of quantities and approximating solutions. Development of visualisation

and representation skills should be integral part of Mathematics teaching. There is also a

need to make connections between Mathematics and other subjects of study. When

children learn to draw a graph, they should be encouraged to perceive the importance of

graph in the teaching of Science, Social Science and other areas of study. Mathematics

should help in developing the reasoning skills of students. Proof is a process which

encourages systematic way of argumentation. The aim should be to develop arguments, to

evaluate arguments, to make conjunctures and understand that there are various methods

of reasoning. Students should be made to understand that mathematical communication is

7

precise, employs unambiguous use of language and rigour in formulation. Children should

be encouraged to appreciate its significance.

At the secondary stage students begin to perceive the structure of Mathematics as a

discipline. By this stage they should become familiar with the characteristics of

Mathematical communications, various terms and concepts, the use of symbols, precision

of language and systematic arguments in proving the proposition. At this stage a student

should be able to integrate the many concepts and skills that he/she has learnt in solving

problems.

Learning objectives:

1. To recognise quadratic equation as equation of the form a x 2+ b x +c = 0, where a, b, c

are real numbers and a ≠0

2. To understand that roots of a quadratic equation are those real numbers which satisfy the

quadratic equation. Roots are also known as solution of quadratic equation.

3. To find the roots of a quadratic equation by factorisation method.

4. To predict the nature of roots based on the sign of discriminant.

5. To find the roots of a quadratic equation by discriminant method.


of x2, x and constant


coefficients of x2, x and constant



Quadratic Equation is extension of quadratic polynomials learnt earlier. In fact, quadratic

equations are most widely used part of algebra to solve real life problems. Treatment of

8

this subject requires very delicate handling in the class as the students need to

understand the significance of the topic in practical context as well as need to acquire the

skill of solving the problems based on quadratic equations.

Students are already familiar with Quadratic polynomial and their factorization. When a

quadratic polynomial is equated to zero it becomes quadratic equation. Naturally the

factors of quadratic expressions will also be equated to zero while making it quadratic

equation equated to zero and this will fix the values of the variable. For example when a

quadratic polynomial

P(x) = x2 ‐4x +3 with factors x‐1 and x‐3

Is equated to zero i.e. p(x) = 0, it implies x2 ‐4x +3 =0 or (x‐1)(x‐3)=0.

At this point it is necessary to remind that if the product of two real numbers a and b is

zero then at least one of them is zero.

Mathematically, a.b =0 implies either a=0 or b=0 or both a and b are zero.

Applying the concept for, (x‐1)(x‐3)=0 ,we get either x‐1 = 0 or x‐3 =0 .

This implies x=1 or x=3.

A question may come from the students very naturally, “what is the significance of getting

value of x as 1 or 3.”

Or,” In what way the values of x are different from factors x‐1 and x‐3 etc.”

The questions may appear to be vague as the students may not be able to state their

doubts clearly due to lots of confusion with respect to quadratic polynomial and quadratic

equations. It may also bother them that why the equation is required in place of

polynomial.

Teacher must take all queries one by one to clear all doubts by describing the quadratic

equations in historical, geometrical and practical context.

9

Historical introduction of Quadratic Equation

Egyptians came across second degree equations related to land survey.

Babylonians formed the quadratic equations to solve their problems regarding agriculture and

irrigation. They tried to find the amount of crop that can be grown in the square field of side

length x and found that the amount of crop that can be grown will be proportional to the area

of field.

In mathematical terms, if units is the length of the side of the field, is the amount of crop

you can grow on a square field of side length 1 unit, and is the amount of crop that you can

grow, then

This was the first quadratic equation obtained. Babylonians also tried to solve the quadratic

equations by the method of completing the square.

Knowledge built up by the Egyptians and Babylonians was passed to Greeks who in turn, gave

mathematics a scientific form.

Greeks tried to solve the quadratic equation x2+2x=8 by completing the square in the following

manner:

They interpreted x2 as square of side length x units, 2x as 2(1)(x) i.e. as rectangles of length

x units and breadth 1 unit.

x 1

x x

x

1 1

10

To complete the square they added a square of side 1 unit.

Their form of reasoning corresponds to our algebraic method of completing the square

x2 + 2x = 8 or x2 + 2x+ 12 = 8 +1 or (x+1)2 = 9

This implies x+1 = 3 or x+1 = ‐3

Hence x=2 or x= ‐4.

Mathematicians have come across some quadratic equations while solving the geometric

problems.

One of the most famous problems is,

“Is it possible to construct a square whose area is that of a given circle?”

It was about 2300 years later that mathematicians were able to prove that this type of

construction is impossible. (Why?)

For more details visit

http://plus.maths.org/content/os/issue29/features/quadratic/index

In the unit three methods of solving the quadratic equations are discussed:

A. Factorization method

B. Completing the square

C. Discriminant method

Good practice is required to acquire the skill of finding the roots of quadratic equations.

11

Geometrical interpretation of quadratic equations

Some Geogebra activities are already described in the unit of polynomials.

Students are familiar with the curves obtained for quadratic polynomials. They should be given

clear understanding that all points on a curve satisfy the quadratic polynomial written as

equation y= ax2+bx+c

At the most two points satisfy the quadratic expression written as quadratic equation

ax2+bx+c = 0. These points are values of variable x satisfying the quadratic equation ax2+bx+c =0

and are known as roots of quadratic equation. Teacher can illustrate with the help of geogebra

activities examples of quadratic equations with repeated roots, with rational roots, with

irrational conjugate roots and ask the students to verify the conditions on discriminant

simultaneously to help them to internalize the concepts learnt rigorously.

Some thought provoking extension activities as suggested below can be used to further

enhance the interest in the topic. For example teacher can take two different irrational roots

(not conjugate of each other), ask the students to form a quadratic equation using the formula

learnt

x2‐ (sum of roots) x+ product of roots =0

For example if roots are √2 and √3, then equation is

x2 – (√2 +√3)x+ √6 =0

which is a quadratic equation. Then why do we say that irrational roots of a quadratic equation

occur in conjugate pairs.

To answer this question following modification can be made in above statement:

A quadratic equation with rational coefficients can have rational roots or irrational roots in

conjugate pairs.

Quadratic equations with irrational coefficients have different irrational roots (not in conjugate

pairs).

12

Common errors made by students

1. While solving the equations like x2 =16 students do not give the solution as x= ± 4.They

write single solution i.e. x=4

2. When given (x‐a)( x‐b) =c, c ≠ 0; they write x‐a =c and x‐b =c

3. While verifying whether quadratic equation (x‐3)(x‐5)= 0 has solution x=3,x=5.

Students put x=3 and x=5 in equation (3‐3)(5‐5) =0 or 0x0=0.

They fail to understand that x=3 is one solution and in both (x‐3) and (x‐5), x will be replaced

by x=3 in both factors. Similarly x=5 is one solution.

These errors shall be discussed in the class with the students.

Once the students have acquired the skill of solving the quadratic equations, word problems

can be introduced to expose the students with daily life situations.

13

Activity Skill Matrix

Type of Activity Name of Activity Skill to be developed

Warm UP(W1) Recalling quadratic

expressions

Observation, analytical skill

Warm UP(W2) Factors of a

polynomial

Problem solving skill

Pre‐Content (P1) Quadratic Expression

and quadratic

equations

knowledge and creative skill

Pre‐Content (P2) Basics of quadratic

equation

Knowledge, thinking skill,

Content (CW 1) Zeroes of a quadratic

polynomial

Understanding, verification, application

Content (CW 2) Roots of a quadratic

equation

Application, Problem Solving Skill

Content (CW 3) Solving a quadratic

equation

Observation, Analytical Skill

Content (CW 4) Discriminant Method

of Finding the roots

Observation and analytical skills

Content (CW 5) Relation between sum

and product of roots

and coefficients of

quadratic equations

Analytical and synthesizing skills

Content (CW 6) Forming a quadratic

equation

Observation, application

Content (CW 7) Application of

quadratic equation in

real life problems

Application of knowledge, analytical skills,

Problem Solving Skill

14

Post ‐ Content

(PCW 1)

Identifying Quadratic

Equation

Knowledge, understanding analytical skill

Post ‐ Content

(PCW 2)

Assignment based

Basics of Quadratic

equations and nature

of roots

Analytical skills and computational skill

Post ‐ Content

(PCW 3)

Assignment based on

method of solving

Quadratic Equations

Problem solving skills, Computational Skill

Warm up Activity (W1)


Specific Objective

To review and recall the concept of quadratic expressions

Description In the previous chapter, the students have learnt about quadratic

expressions. They know how to differentiate between a

polynomial and an algebraic expression. They know that a

polynomial of degree 2 is called a quadratic expression.

Execution The teacher may give printouts of the worksheet and ask the

students to filter down the quadratic expressions from the

algebraic expressions in the funnel and write the quadratic

expressions in the box provided.

Parameters for

assessment

Understanding of the difference between a polynomial and an

algebraic expression

Finding out the degree of a given polynomial

15

Warm up Activity (W2)


Specific Objective: To recall the formation of a polynomial when the factors of the polynomial

are given. To identify the zeroes of the polynomial from the given factors.

Description: The task calls for forming a polynomial when the factors of the polynomial

are given. The students shall form the polynomials by multiplying the

factors given in the table. They shall also read the zeroes of the polynomial

from the given factors.

Execution: The teacher may write the factors on the board. The teacher may

encourage the students to form polynomials of degree >2 by giving them

more than two factors.

Parameters for Assessment:

Students will be able to:

• Understand that a polynomial is a product of its factors

• Reading the zeroes of a polynomial from the given factors

Pre Content Worksheet (P1)


Specific Objective

To recall a polynomial equation and form a quadratic equation

from the given quadratic expression. To review the factorization

of the quadratic expression by splitting of the middle term

method.

Description In task 1, the students are expected to give answers to the given

questions and form quadratic equations. In task 2, the students

16

shall factorize the given quadratic expressions by splitting the

middle term. They would not be able to factorize two of them viz.

3x2+4x‐2 and √2x2‐3x‐5. The teacher shall ask the students to list

them down in the provided space and discuss them later while

using discriminant of a quadratic equation.

Execution Teacher may take printouts of the sheet or ask the questions of

task 1 orally and by using the worksheet done in warm up

worksheet for task 2.

Parameters for

assessment

Understanding the concept of an equation

Formation of an equation

Factorization of a quadratic polynomials using the splitting of

middle term method


Revisit Key Concepts Specific Objective

To review, recall a polynomial equation, quadratic equation, zero of a polynomial and other concepts covered in the chapter on Polynomials. To review the factorization of the quadratic expression by splitting of the middle term method.

Description The task describes some questions designed to test the basic understanding of the students on the Polynomials chapter.

Execution Teacher may take printouts of the sheet or ask the questions orally.

Parameters for assessment

Understanding the concept of a quadratic polynomial and equation Understanding the concept of number of zeroes of a quadratic polynomial General understanding of the concepts covered in the Polynomials chapter

17

Activity 5: Content Worksheet (CW1)


Specific Objective

Recall the zeroes of a quadratic and match them with a given

quadratic expression

Description The students shall match the zeroes given in the form of flowers

in the flower vase with the quadratic expressions given in the

vase. Each quadratic expression corresponds to a pair of zeroes of

a quadratic expression. They would then complete the table given

in the task by writing the quadratic expression with the

corresponding pair of zeroes.

Execution Teacher may take printouts of the sheet and ask the students to

write in the given space

Parameters for

assessment

Understanding of the concept of zeroes of a polynomial

Finding out the zeroes of the polynomial

18


Roots of a Quadratic Equations

Specific Objective To understand the concept of roots and solution of a given

quadratic equation and correlate it with the zeroes of a

quadratic expression

To check whether a given value of x is a solution of a given

quadratic equation

Description of task The students use their previous knowledge of factorization of

a quadratic polynomial by splitting of the middle term method

and find the zeroes of the quadratic. They shall then write that

the found values of x are the roots and hence the solutions of

the given quadratic equation. They shall complete the task on

similar lines as the example.

In task 2, the students shall verify whether a given value of x is

a solution of the given quadratic equation by substituting the

value of x in the quadratic equation.

Execution The teacher may give printouts of the given worksheet or write

the questions on the board and discuss them with the class.

Parameters for assessment Understanding of the concept quadratic equation

Writing the zeroes of a polynomial and the roots and solutions

of the quadratic equation

Checking if the given value of x is a solution of the given

quadratic equation

Extra reading

http://www.mathsisfun.com/algebra/factoring‐quadratics.html

19

Activity7: Content Worksheet (CW3)

Nature of Roots

Specific Objective (s) To predict the nature of the roots by using the sign of the

discriminant.

Description of task The teacher may ask the students orally about the conditions

of the roots to be real and distinct, real and equal and non real

roots. For task 1, the teacher may discuss with the students.

In task 2, the teacher may give printed worksheets and ask the

students to find out the discriminants in each case and colour

the grid following ther instructions.

The teacher may draw the final answer figure on the board and

may discuss the figure so obtained ‘swastik’ («) with the

students and may, if feasible, ask them to explore more about

it through net surfing.

Execution Discussions in task 1 and printed worksheets in task 2

Parameters of assessment Is able to state the conditions on the discriminant of a

quadratic equation for the existing roots

Is able to calculate the discriminant

Is able to categorize the quadratic equations into 3 types

according to the types of roots of the quadratic equation by

finding the discriminant of the quadratic equation.

Is able to complete the design correctly.

Extra readings:

http://www.mathsisfun.com/algebra/completing‐square.html

Answe

er grid to the colouuring wo

20

rksheet

21


Discriminant Method of Finding the Roots

Specific Objective (s) To find the value of an unknown constant in a quadratic

equation when the nature of the roots is given. To find the

solution of a given quadratic equation by the discriminant

method in particular and by any of the two methods in general.

Description of task In task 1, the student shall complete the table by finding out the

discriminat and categorizing the given equation to have real and

distinct roots, real and equal roots or no real roots according as

D>0, D=0 or D<0.

In task 2, the student shall find the roots of the given equation.

The algorithm to proceed is provided in the worksheet.

In task 3, the student may be free to choose one of the two

methods: factorization or discriminant method to solve the

equation. The teacher here would need to make it clear that

not all equations can be solved by the method of factorization,

but the discriminant method is universally true for all the

quadratic equations.

Execution The teacher may either write the equations on the board or

give printout of the worksheets

Parameters of assessment Is able to find the value of an unknown constant when the

nature of the roots is given

Is able to find the roots of a general quadratic equation

Is able to identify the situation when the equation has no real

roots.

Is able understand that while the method of factorization is not

universally applicable for all quadratic equations, the method of

discriminant is universally applicable.

Extra readings: http://www.mathsisfun.com/algebra/quadratic‐equation.html

http://www.mathsisfun.com/algebra/quadratic‐equation‐graph.html

22

Activity 9‐ Content Worksheet (CW5)

Relations between Roots and Co‐efficients of a Quadratic Equations

Specific Objective:


of x2, x and constant.


coefficients of x2, x and constant.

Description: In this task students will first solve the equation and then recognize and write the

coefficients of x2, x and constant term. They will find –b/a and c/a. Students will

write sum of roots & product of roots. They will generalize the relationship

between a, b & c with the sum of roots & product of roots for any quadratic

equation ax2+bx+c=0.

Execution: Teacher may distribute the photocopies of worksheets and students will solve the

questions on their worksheets individually.



• Recognize coefficients of x2, x and constant terms.

• Find roots of a quadratic equation.

• Calculate sum of roots and products of roots correctly.

• Relate sum & product of roots with the coefficients of x2, x and constant terms.

Extra Reading:

Sum of roots, product of roots and discriminant formula

http://www.hitxp.com/zone/tutorials/mathematics/world‐of‐quadratic‐equations/

Solving quadratic equation

http://library.thinkquest.org/20991/alg2/quad.html

23


Formation of a Quadratic Equation

Specific Objective:


Description: The task calls for forming a quadratic equation when the sum & product of roots

are already given. Reeta, Ranjeeta and Saleem each has a bag. In Reeta’s bag,

there are 5 equations, Ranjeeta’s and Saleem’s bags contain one root of each

equation of Reeta. Reeta lost his equations. In this task students are supposed to

trace Reeta’s equations by making use of their roots.

Execution: Teacher may show the dialogue script to the class on projector. Students will

solve and write their equations in their notebooks.



• Find sum of roots of a quadratic equation.

• Find product of roots of a quadratic equation.

• Form a quadratic equation with given roots.

24



Specific Objective:


Description: The task is designed to make students apply quadratic equations to solve real life

situations. Students will frame the quadratic equation after comprehending the

word problem. They will then solve the equations to get the answers of the

problems.

Execution: Teacher may distribute the photocopies of worksheets and students will solve the

questions on their worksheets individually.



• Comprehend the word problem.

• Express the word problem in the form of quadratic equation.

• Solve the quadratic equation.

• Write the answer of the given word problem.

Extra Reading:

http://www.youtube.com/watch?v=EhPPci8shA8

http://www.youtube.com/watch?v=Vu3px08WX_8

http://www.youtube.com/watch?v=lS9S1iEjlPI

25

Activity 12‐ Post Content Worksheet (PCW1)

Students will be assessed on the worksheet containing questions based of recognizing

quadratic polynomials & quadratic equations.


Students will be assessed on the worksheet containing questions based on nature of roots of a

quadratic equation.


Students will be assessed on the worksheet containing questions based on finding the roots of

a quadratic equation.


Students will be assessed on the worksheet containing questions based on the complete

chapter including application of quadratic equations in solving real life problems.

• Read about the applications of Quadratic equations in real life at


26

Assessment Plan

Assessment guidance plan for teachers

2.22 Assessment Plan

Assessment guidance plan for teachers

With each task in student support material a self –assessment rubric is attached for students.

Discuss with the students how each rubric can help them to keep in tune their own progress.

These rubrics are meant to develop the learner as the self motivated learner.

To assess the students’ progress by teacher two types of rubrics are suggested below, one is for

formative assessment and one is for summative assessment.

Suggestive Rubric for Formative Assessment (exemplary)

Parameter Mastered Developing Needs motivation Needs personal

attention

Method of

solving

quadratic

equations .

Able to apply

factorization

method to solve

the quadratic

equations , get

correct values of x,

,can verify the

correctness of

solution

Able to apply

factorization

method to

solve the pair of

linear

equations ,get

correct value of

x, cannot verify

the correctness

of solution

Able to apply

factorization

method to solve

the quadratic

equations ,not able

to get the correct

value of x

Not Able to apply

factorization

method to solve

the quadratic

equations.

Able to apply

discriminant

method to solve

Able to apply

discriminant

method to

Able to find

discriminant to

solve the quadratic

Not Able to apply

discriminant

method to solve

27

the quadratic

equations ,get

correct values of x,

can verify the

correctness of

solution

solve the

quadratic

equations ,get

one correct

value of x,

cannot verify

the correctness

of solution

equations , but

cannot get correct

values of x, cannot

verify the

correctness of

solution

the quadratic

equations .

From above rubric it is very clear that

• Learner requiring personal attention is poor in concepts and requires the training of basic

concepts before moving further.

• Learner requiring motivation has basic concepts but face problem in calculations or in

making decision about suitable substitution etc. He can be provided with remedial

worksheets containing methods of solving the given problems in the form of fill‐ups.

• Learner who is developing is able to choose suitable method of solving the problem and is

able to get the required answer too. May have the habit of doing things to the stage where

the desired targets can be achieved, but avoid going into finer details or to work further to

improve the results. Learner at this stage may not have any mathematical problem but may

have attitudinal problem. Mathematics teachers can avail the occasion to bring positive

attitudinal changes in students’ personality.

• Learner who has mastered has acquired all types of skills required to solve the pair of linear

equations in two variables.

28

Teachers’ Rubric for Summative Assessment of the Unit

Parameter 5 4 3 2 1

Solving the

quadratic

equation

• Able to solve quadratic

equation by factorization

method

• Able to identify which

quadratic equation cannot be

solved by factorization.

• Able to apply discrminant

method to solve the

quadratic equation

• Not able to solve quadratic

equation by factorization

method

• Not able to identify which

quadratic equation cannot be

solved by factorization.

• Not able to apply discrminant

method to solve the quadratic

equation

Nature of

roots

• Can find the discriminant

accurately

• Can predict the nature of

roots on the basis of

discriminant

• Can state the conditions for

existence of real roots

• Can find the unknown

coefficient for prescribed

nature of roots

• Cannot find the discriminant

accurately

• Cannot predict the nature of

roots on the basis of

discriminant

• Cannot state the conditions for

existence of real roots

• Cannot find the unknown

coefficient for prescribed

nature of roots

Forming

quadratic

equation

• Can state the formula for

forming quadratic equation

• Can form correct quadratic

equation when both roots are

known

• Cannot state the formula for

forming quadratic equation

• Cannot form correct quadratic

equation when both roots are

known

29

• Can form quadratic equation

with rational real

co‐effecients when one

irrational root is given

• Cannot form quadratic

equation with rational

co‐effecients when one

irrational root is given

Application

in word

problems

• Able to identify the variables

from given statement

• Able to form quadratic

equations correctly from

given statement

• Able to solve the equations by

any of the above methods

• Able to verify the solution.

• Not able to identify the

variables in given statement

• Not able to form quadratic

equation correctly from given

statement

30

SSTTUUDDYY

MMAATTEERRIIAALL

31

Quadratic Equations

Introduction

You have already studied about linear equations in one and two variables. You have also

learnt how to solve a number of daily life problems by converting them into the form of

equations. Let us try to solve the following problem by converting it into the form of an

equation. “A motor boat goes 30km downstream and comes back to the same positions in 4

hours 30 minutes. If the speed of the stream is 5 km / h, find the speed of the boat in still

water.”

Let the speed of the boat in still water be x km /h

So, speed of the boat downstream = (x +5) Km/h

And speed of the boat upstream = ( x‐5) Km/h

Now, time taken to cover 30 Km downstream = hours

And time taken to cover 30km upstream = hours

So, total time= + hours

Therefore, according to the given condition,

+ = 4

or + =

or

=

or =

32

Or 120x = 9x2 −225

Or 9x2 −120x −225=0

Or 3x2 −40x −75=0 (I)

Thus, we have obtained an equation representing the given daily life problem. Can we solve

such problems? Definitely not, because this equation is different from those equations which

we have solved so far. That is, it is not a linear equation.

However, this equation appears to have some resemblance with the polynomials of the type

ax2 +bx + c. Can you recall that polynomials of the type ax2 +bx + c are called quadratic

polynomials?

Keeping in view this rememblance, equations of the type 3x2 −40x −75=0 are called quadratic

equations in one variable . In general, an equation of the type

ax2 + bx +c =0 is called a quadratic equation in one variable, where a, b, and c are real numbers

and a ≠ o.

In this chapter, we shall make a beginning of the study of quadratic equations and solve them

by different methods. We shall also learn about the nature of the roots, relationship between

the roots and coefficients of given quadratic equation and solve some daily life problems,

with the help of quadratic equations.

(1) Introductions to Quadratic Equations

We have seen above that while solving some daily life problems, we may come across

equations of the type 3x2 – 40x −75=0 which are of the form ax2+bx+c=0, where a, b and c

are real numbers and a ≠0. These types of equations are called quadratic equations in one

variable. ax2+bx+c=0 is sometimes also referred to as the general form or the standard

form of a quadratic equation; a is called the coefficient of x2 , b the coefficient of x and c is

the constant term. In the case of equation obtained by us, a = 3,b= −40 and c= −75.

33

Some more examples of quadratic equations are 3x2 − 5x+7=0 , −7x2 +2x + 8 = 0,

2y2 + 3y −8 = 0, x2 −2x +3=0, x2 − 4 = 0, 2y2 −3y = 0, 8z2 + 4z + 9 = 0.

Let us now consider an example to identify the quadratic equations in one variable.

Example 1:‐

Which of the following are quadratic equations in one variable and which are not?

(i) 3x2 +7x−2=0

(ii) (x+2)2 =x (x+1) +2

(ii) 2x3 +5x2 −6=2x2 (x−2) +4x

(IV) 3x2 −5=0

(v) 5y2 −12y−9=0

(vi) (2z+5)2 ‐5z = z (4z+9)

(vii) 2p −7 =9p2

(viii) (2p −7)2 = 4p (p2 – 7)

Solutions:‐

(i) 3x2 + 7x − 2 = 0 is Quadratic equation as it is of the form ax2 +bx +c=0

(ii) (x+2)² = x (x+1) +2 gives

x2 + 4x + 4 = x2 + x + 2

Or 3x + 2 = 0. So, it is not a quadratic equation

(iii) From 2 + 5x −6 = 2x2 (x−2) + 4x, we have :

2x3 + 5x −6 = 2x3 − 4x2 + 4x

Or 2x3 −2x3 + 4x2 + 5x − 4x − 6 = 0

or 4x2 + x−6=0, so it is a quadratic equation.

(iv) 3x2 − 5 = 0. It is of the form ax2 + bx + c = 0,

Where a = 3, b = 0and c = −5.

34

So, it is a Quadratic equation.

(v) 5y2 −12y − 9 = 0. It is of the form ax2 + bx + c = 0.

So, it is a quadratic equation.

(vi) From (2z+5)2 −5z = z (4z+9), we have

4z2 +20z + 25−5z = 4z2 + 9z

or 4z2 +15z +25 −4z2 − 9z = 0

or 6z + 25 = 0. So it is not a quadratic equation.

(vii) From 2p −7 = 9p2 , We have

−9p2 + 2p −7 = 0. So, it is a quadratic equation.

(viii) From (2p −7)2 = 4p (p2 −7), we have:

4p2 − 28p + 49 = 4p3 −28p

or 4p3 − 4p2 − 49 = 0.

It is not of the type ax2 + bx + c=0 (though it has three terms in L H S). So, it is not a

quadratic equation.

(2) Solving a quadratic Equation

You have already learnt how to solve linear equations in one or two variable (s). Recall that

the value (s) of the variable (s) which satisfies a given equation are called its solution (s). By

satisfying an equation, we mean that when the value (s) of one or two variable (s) are

substituted in the equation we get L H S = R H S.

Let us now examine how we can solve a quadratic equation. For example, let us again

consider the equation 3x2 − 40x −75=0

If we substitute x=15 in the L H S of this equation, we have:

L H S = 3 x152 − 40 x 15 − 75

=675 − 600 −75 = 0 = R H S.

Thus, we can say that x = 15 is a solution of the above equation.

35

Again, let us substitute x = in the L H S of above equation. We have:

L H S = 3 x 2 −40 x 53 −75

= + − 75

= – 75 = 75 −75 = 0 = R H S.

Thus, we can also say that x = is also a solution of the equation 3x2 − 40x −75=0

Let us check whether x = 2 is a solution of this equation or not

L H S = 3x2 – 40x −75

= 3(4) – 40(2) – 75

= 12 − 80 −75

= −143

R H S = 0

So, L H S ≠ R H S

Hence x = 2, is not a solution of the given equation.

Thus, we have seen that x = 5 and x= are the solutions of the given equation, while x = 2

is not a solution. It is a matter of chance that we got two solutions of the equation

3x2 − 40x −75 =0. But the problem before us is how to find such solutions. Let us discuss

method of factorization of solving a quadratic equation.

Factorization Method

You are already familiar with the factorization of the trinomials of the type ax2 + bx + c, by

splitting the middle term. In the factorization method of solving a quadratic equation, we

first write the equation in the standard form ax2 + bx + c = 0 and factorize the L H S of the

equation by splitting the middle term. We explain the process by taking again quadratic

equation 3x2 − 40x −75=0,

We have 3x2 − 40x −75 = 0 splitting − 40x as − 45x + 5x

36

Or 3x2 − 45x + 5x −75=0

Why? Because −45 x 5 = 3 x (‐75)

Or 3x (x−15) + 5 (x−15) = 0

Or (x−15) (3x+5) =0

(when product of two numbers a and b is zero then either a=0 or b=0 or both a and b are

zero.)

So , (x−15)=0 or (3x+5)=0

i.e. x=15 or x =

So, the required solution of two quadratic equations3 — 40 75 0

15 .

Note: .

: Solve the following quadratic equation by factorization method:

i x 6x 5 0

ii 5x 3x 2 0

iii 8x 22x 21 0

How many solutions are possible for a quadratic equation?

Are there always two solutions?

37

iv 6x x 2 0

v z 6 0

vi y ‐ 2√3x +3=0

vii abx +(b ac x bc 0

viii 2y ay a 0

ix 5z z 0

x 3√2 x 3 2 √3x 3 0

: i we have:

x 6x 5 0

Or x 5x x 5 0 splitting the middle term 6 x

Or x x 5 1 x 5 0

Or x 5 x 1 0

so, x 5 0 or x 1 0

x 5 or x 1

Thus x 5 and x 1 are solution of the given equations

Check; LHS 5 6 5 5 0 RHS, when x 5

LHS 1 6 1 5 0 RHS, when x = −1

ii 5x 3x 2 0

Or 5x 5x 2x 2 0 (splitting the middle term)

Or 5x x 1 2 x 1 0

Or x 1 5x 2 0

so, x 1 0 or 5x 2 0

i. e., x 1 or x

so, 1 and x are solutions of the equation

Check: LHS 5 1 3 1 2 0 RHS

38

and LHS 52 5 3

25 2

= 2

= 0 RHS

iii 8x 22x 21 0

Or 8x 28x 6x 21 0

Or 4x 2x 7 3 2x 7 0

Or 2x 7 4x 3 0

so, 2x 7 0 or 4x 3 0

i. e, x or x

so, x and x are solution of the equation.

iv 6x x 2 0

Or 6x 4x 3x 2 0

Or 2x 3x 2 1 3x 2 0

Or 3x 2 2x 1 0

so, 3x 2 0 or 2x 1 0

i. e. x23 or x

12

so, x 23 and x

12 are solution of the equation.

v z 6 0

Or z 6 z 6 0

so, z 6 0 or z 6 0

i. e z √6 or z √6

39

so, z √6 and z √6 are solution of the equation

vi y 2√3 y 3 0

Or y √3y √3y 3 0

Or y y 3 − √3 (y − 3 =0

Or y 3 (y − 3 = 0

so, y √3 0 or y √3 0

i. e. y √3 or y √3

Note In this case the two solutions are the same.

vii abx b ac x bc 0

Or abx b x ac x bc 0

Or bx ax b c ax b 0

so, ax b 0 or bx c 0

i. e. x or x

so, x and x are solution of the equation.

Check for x ,

LHS ab ² b ac – bc

³ ³

bc bc

0 RHS

For x= ,

40

LHS ab ( )2 + (b2 –ac) ( – bc

acb bc

acb bc 0

RHS

x = and x = both are solutions of the equation.

viii 2y ay a 0

Or 2y 2ay ay a 0

Or 2y y a a y a 0

Or y a 2y a 0

so, y a and y are solution of the equation

ix 5z z 0

Or 5z z 0

Or 5z z

2z 3 0

Or 5z z 32 0

so, 5z 0 or z 0

i. e, z 110 or z

so, z 110 and z

32 are solutions of the equation

x 3√2 x 3 √2 √3 x −3 =0

Or 3√2 x 3√3x √6x 3 0

Or 3x(√2x √3 √3 √2x √3 0

41

Or (√2x √3 (3x −√3)=0

so, √2x √3 0 or 3x √3 =0

i. e., x or x1

√3

so, x and x1

√3 are solutions of the eqution.

You are advised to check the solutions you obtain, with the given original equation.

:

You are already familiar with the identities

² 2 2

We use these identities in solving a quadratic equation by the method of completing the

square. Let us explain it through an example:

Consider the equation

3 40 75 0

or 403 25 0 Divide the equation by 3

403 40

2 x 3 ² 402 x 3 ² 25 0 (Adding and Subtracting (½ of the

coefficient of x)² in L H S

or 203 ²

400 2259

6259

253

or 203

253

, 203

253 20

3253

15

We have again, got two solutions as obtained earlier.

42

Let us now consider the general or standard form of equation 0

We have:

0

Or 0

Or x2 + 2b

2a⎛ ⎞⎜ ⎟⎝ ⎠

2b

2a⎛ ⎞⎜ ⎟⎝ ⎠

= 0 (Adding and subtracting the square of of the

coefficient of 2x) in LHS)

Or 2 2b b +

2a 2ax⎛ ⎞ ⎛ ⎞−⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠+ = 0 (we obtain a complete & same a ( )2

2

2

b b² cor + = 2a 4a a

x⎛ ⎞ −⎜ ⎟⎝ ⎠

Or 2 2

2

b b 4ac + = 2a 4a

x −⎛ ⎞⎜ ⎟⎝ ⎠

22 2b b 4acor + = 2a 2a

x⎛ ⎞± −⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Or 2 2b b 4ac b b 4ac + = or x + =

2a 2a 2a 2x

a− −

−

Or 2 2b b 4ac b + b 4ac = or x =

2a 2x

a− − − − −

Note that

√ 42

√ 42

.

43

x = √ √ are the two roots of the equation.

, 5 5

3 3 40 75 0

Note : It is interesting to note that this method was first given in by an ancient Mathematician

Sridhar (around 1029AD)

Example 3: Solve the equation

6 5 0 .

Solution: 6 2 − 2 + 5=0

Or 6 9 9 5 0

Or 3 2 = 4

Or 3 √4 2

, 3 2 or 3 2

1 5 [See example 2 (i)]

1, 5

1, 5 .

: 5 −3x−2=0 by the method completing the square.

Solution: 5 3 2 0

or + − = (why?)

or 0

or 100 0

or =

or 10 =1

44

or

1

1 ,2

5

Example 5: Solve the equation 8 22 21 0 .

Solution: Comparing the given equation with , 8, 22, 21.

We have, √

(Quadratic Formula )

²

22 484 67216

√

So, =

,72

34

7 2

34

Example 6: Solve the equation 3√2 ² 3 √2) √3 3 0 By using the quadratic

formula

Solution : Here a = 3√2 , √3 3 2 =3 √3 √6 , 3

45

√ (Quadratic formula)

= √ √ √ ²

√

= √ 27 √ √

√

= √ √

√

= √ √ ²

√

= √ √

√

, √√

√√

√√

thus, √

and are the roots of the given equation.

Example 7 : Solve the equation 2√3 3 0 .

Solution: here 1, 2√3, c=3

√ 4

2

So roots are

√ 42 ,

√ 42

46

. 2 3 2 3 ² 4 1 3

2 1 ,2 3 2 3 ² 4 1 3

2 1

. 2√3 0

2 ,2√3 0

2

. . √3 , √3 2

(4) Nature of Roots

You have seen that the roots of the quadratic equation 0 ,

, :

√ 42 ,

√ 42

Observe that:

In Example 5, the value of b 4ac 1156 0

In Example 6, the value of 4 √6 3√3 2 >o and roots are √

√

In Example 7, the value of 4 0 √3 , √3

(i) We can say that roots of a quadratic equation are real and unequal or distinct, if

4 0

(ii) roots of a quadratic equation are real and equal (coincident) if 4 0

What will happen, if 4 0, , 36? Can you find √ 4 i.e. √ 36 .

Can you find a real number whose square is −36?

There is no real number whose square is −36.

So,

(iii) Roots of a quadratic equation are not real when 4 0.

47

Note that the nature of roots of a quadratic equation are depending on the value of the

expression 4 . That is why we call as the discriminant of the quadratic

equation , where a, b,c are real numbers with a 0.

We denote the expression 4 .

So, D is the discriminant of the equation.

Example 8: Find the discriminant of the following quadratic equations:

(i) 3 5 2 0

(ii) 2 6 3 0

(iii) 5 27 0

(iv) 2 0

Solution:

(i) Here, 3 , 5 , 2.

iscriminant b 4 5 ² 4 3 2 25 24 1

(ii) Here , a 2 , 6 , 3

b 4 6 2 4 2 3 36 24 12

(iii) 5 27 0

Here 5 , 0 ? . 27

Let 0. a 0

if D > 0, then the equation has two distinct and real roots

if D=0, then the equation has two coincident real roots

if D < 0, then the equation has no real roots

A quadratic equation can have at the most two roots

48

b 4 0 4 5 27 540

(iv) 2 0

Here, 1 , 2 , 0 ?

b 4 4 4 1 0 4

Example 9: State the nature of roots of the following equations:

(i) 2 1 0

(ii) 1 0

(iii) 9 12 4 0

(iv) 2 5 5 0

(v) 4 1 0

(vi) 16 40 25 0

Solution: (i) 2 1 0

Here 2 , 1 , 1

b 4 1 4 2 1 1 8 9 0

So, the equation has two distinct real roots

(ii) 1 0

Here , 1 , 1 , 1

b 4 1 4 1 1 3 0,

So, the equation does not have real roots

(iii) 9 12 4 0

Here, 9 , 12 , 4

b 4 12 2 4(9)(4) = 144−144=0

So, the equation has two coincident real roots

(iv) 2x2 +5x + 5=0

Here a=2, b=5, c=5

49

D=b2−4ac = 25 74 (2) (5) = 15 <0

So, the equation does not have real roots

(v) 4 1 0

Here, 1 , 4 , 1

b 4 16 4 20 0

So, the equation has two distinct real roots.

(vi) 16 ² 40 25 0

Here, 16 , 40 , 25

b 4 1600 4 16 25 0

So, the equation has two coincident real roots .

Example: 10 Find the value of k in each of the following equation:

(i) 9 1 0 , if its roots are coincident and real

(ii) 4 1 0 ,

(iii) 9 12 0,

(iv) √ 8 0 ,

Solution: (i) 9 1 0

Here 9 , , 1

b 4 4 9 1 = 36.

As, the roots are coincident, 0 . . 36 0

or 36 6)2

or 6

(ii) 4 1 0

, 4, 1

b 4 16 4

50

As, the roots are distinct and real, or 0

. . 16 4 0 16 4 4 16 4

(iii) 9 12 0 .

9, 12,

144 36

Since the roots are not real, so

0

or 144 36 0

or 144<36k

or 36k>144 or k>4

(v) √kz 8 0

, √ 2 4 1 8

= k 32

Since, roots are real, D 0

Therefore 32 0 32 0

i.e. k>32 or k=32

, 32

(5) Relation between the Roots and Coefficients of a Quadratic Equation

You know that roots of the equation

0 0

are √ and √

We see that

Sum of the roots = √ √

51

= √ √

or C f

Coefficients

Product of the roots = √

√

= √

√

= 2b

2a⎛ ⎞−⎜ ⎟⎝ ⎠

–

22b 4ac

2a

⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠

=

= . . C f

• In Example 5, the roots are and

Sum of the roots =

Also C f

C f =

Thus, if pand q are roots of 0, 0

Then i.e. C f

C f

And pq . . C f

52

Thus, we have verified that

Sum of roots = C f

C f

Similarly, Product of roots = = C

C f

Example 11: Find the roots of the equation

9 3 2 0

And verify the relations between the roots and coefficients.

Solution:

Roots are given by

x √

=

= √

= ,

Sum of roots =

Also, sum of roots = C f

Coefficients =

Hence, Sum of roots =Coefficients

C f

53

Similarly,

Product of roots =

Also

C f

So, product of roots =

C f

Example 12: Find the roots of the equation

4 1 0

And hence verify the relations between the roots and coefficients.

Solution: Roots are given by

√

√

√

2 √3

So, the roots are 2 √3 and 2 √3

Sum of roots 2 √3 2 √3 =4

Also Coefficients

C f = 4

Hence, sum of roots =C f

C f

Similarly the product of roots

2 √3 2 √3 4 3 1 Constant termCoefficients of x2

• Note that roots of equation are

2 √3 2 √3

Such roots are called conjugate roots.

Conjugate roots always occur in pairs when coefficients a, b and c are rational numbers.

54

Example 13: If a quadratic equation with rational coefficients has one root 3 √15, then

What will be the other root?

Solution: Since, irrational roots occur in conjugate pairs when coefficients are rational

number, so the other root will be 3 √15

(5) Forming a Quadratic Equation with given roots

We know that for the quadratic equation

0 , 0,

Now 0 , written as

0 , 0

Or 0

Or

Example 14: Form the quadratic equation whose roots are 6 and ‐3

Solution:

6 3 3

6 3 18

So, required quadratic equation will be

3 18 0

0

55

Alternatively, the equation with roots 6 and ‐3 can also be found as

6 3 =0

or 3 18 0

Note that the equations

2 3 18 0 ,

3 3 18 0

3 18 0

2 3 18 0 etc. all have the same roots 6 and −3 (check!!)

In fact, k 3 18 0 6 3 .

Example 15: Find the quadratic equation whose roots are 1 √5 1 √5

Solution: Sum of roots = 1 √5 1 √5 2

1 √5 1 √5 1 5 4

So, the required equation is

2 4 0

Alternatively, the quadratic equation with

1 √5 1 √5

1 √5 1 √5 0

. 2 4 0

Note that the roots 1 √5 1 √5 are called conjugate roots and they always occur in

pairs if the coefficients of the quadratic equation are rational numbers. In other words, if the

56

coefficients of the quadratic equation are rationals and one of the roots is a √ , then the

other roots must be √ .

Example 16: Find the quadratic equation the sum of whose roots is and the Product of the

roots is 59

Solution: the required equation is 67

59 0

or 63 54 35 0

(6) Applications of Quadratic Equations

Now, we will discuss applications of quadratic equations in solving problems related to daily

life.

Example: 17 The sum of squares of three consecutive natural number is 110. Determine the

numbers.

Solution: Let the three consecution natural numbers be , 1, 2.

According to given condition,

1 2 2 2 =110

Or 2 35 0

Or 7 5 35 0

Or 7 5 7 0

Or 7 5 0

So, 7 5

It cannot be ‐7, (why) ?.

Thus x=5,

And the numbers are 5, 5 1, 5 2, . 5,6,7,

57

Example 18: In 5 hours, a person travelled 12 km down the river in his motorboat and then

returned. If the rate of the river’s current is 2km/hour, find the speed of

motorboat in still water.

Solution: Let r be the speed of the boat in still water.

Distance Speed Time

Down the river (with

the current)

12 km r+2 122

Upriver (against the

current)

12 km r−2 122

Thus

12

212

2 5

Or =5

Or 24 5 2 2

Or 24 5 4

Or 5 24 20 0

Therefore ²

√

5.5 0.7

Since speed cannot be negative, so

5.5 km/hr

58

Example: 19 Two pumps are used to empty a tank full of water. When put together, they

can empty it in 6 hours. One pump alone can do this work by itself in 2 hours less time than the

other could do it alone. How long would it take each pump to complete the job alone?

Solution:

Let t be the number of hours for slower pump to complete the work itself.

Then 2 would be the number of hours for the faster pump to complete the job

by itself.

So, 1

or 1

or 2 1

or 12 12 2

or ² 14 12 0

t ² √

13.1 0.9 .

If we take t= 0.9, then the faster pump will take t − 2 hrs. i.e.(0.9 − 2) which is negative. So, we

reject the value of t=0.9.

Hence, slower pump will take 13.1 hours and faster will take 13.1 − 2 = 11.2 hours to complete

the job.

Example 20: A two digit number is such that the product of the digits is 12. When 36 is

added to the number the digits interchange their places. Find the numbers.

Solution: Let digit at ten’s place be x and at unit’s place be y.

So, the number = 10

When digits are interchanged, the new number 10

According to the problem,

xy 12 1

59

and 10 36 10

Or 9 9 36

Or 4 2

Putting the value of from (2) in (1).

4 12

Or 4 12 0 )

∴ √

=√64

= 6, 2

Rejecting 2 ? . , we get

6

So, 2 [(using (1)]

Thus, the number is 26.

Check : Product of digits =2 6 12

Also

26+36=62

Example 21: The sum of ages of a father and daughter is 45 years. Five year ago, the Product

of their ages was four times the father’s age at that time. Find their present ages.

Solution: Let father’s age be years, then daughter’s age = 45 years

5 Years ago

Father’s age = 5 .

Daughter’s age = 45 5 .

60

= (40 x) years

According to the problem,

5 40 = 4 5

40 200 5 4 20

45 200 4 20

41 180 0

41 180 0

So, 41

= 36,5

x can not be 5 (why ?)

Thus, 36 i.e father’s present age =36 years and daughter’s presents age= 9 years.

Check :

(i) Sum of Ages = 36 + 9 = 45

(ii ) 5 years ago, father’s age = 36‐5=31 years

Daughter’s age = 9‐5 =4 years.

So, 31 4 =4x

= 4 31

Example:22 A model rocket is shot straight up. Its height y , (in metres) , above the ground

level, after seconds is given by

5 200

Determine, in how many seconds will the rocket be 1875 metres above the ground.

Product of their ages (in years)

4 times the father’s age (in years)

61

Solution: Here y=1875

So, we have

1875 5 200

Or 40 375 0

Or 15 25 0

i.e. 15 25.

So, the rocket will be at the height of 1857 metres after 15 or 25 seconds.

Check: 5 15 200 15 1875

5 25 ² 200 25 1875

Note that the rockets will be at the same height (other than maximum) once while going up

and other while coming down.

Example:23 Triangles ABC and DEF in the following figure are similar. Find the length of

sides AB and EF.

Solution: As triangles ABC and CEF are similar

∴ we have

62

= 8

Or x (x −3) = 40

Or x² − 3x − 40 =0

Or (x−8) (x+5) = 0

∴ x −8 =0 or x + 5 = 0

∴ x = 8 or x = − 5

x cannot be equal to −5 (why?).

So, x = 8.

Hence AB = 8 − 3 = 5 units

EF = 8 units.

(Note: example 23 may be done after completing the unit on similar triangles)

63

STUDENT’S SUPPORT MATERIAL

64

Student’s Worksheet 1 (SW1)

Warm up Worksheet (W1)


Name of Student___________ Date________

Send the balloons to their respective homes

x2‐3x‐40x2‐3x+4 x2‐x‐72

2x‐7/4‐5x+7

x2‐4x+42√2x2‐3x‐5

5x1/2‐3x+4

3x2+4x‐2

x2‐5x +6

5x2‐7x3+2x+2

x2‐2x‐3

65

Self Assessment Rubric

Parameters of assessment

Understanding of the difference

between a polynomial and an

algebraic expression

Finding out the degree of a given

polynomial

NON POLYNOMIALS

POLYNOMIALS OF DEGREE2

POLYNOMIALS OF DEGREE OTHER

THAN2

66


Warm up Worksheet (W2)

Recognize Zeroes of Polynomials


Given below is a table where factors of a polynomial are given. Form the polynomials and write

the zeroes of the polynomials in the table below. The first one is done for you

Factors Polynomials Zeroes of polynomials

x‐2, x+3 (x‐2)(x+3)=x2+x‐6

2,‐3

2x‐3, x+2

5x+4, 4x+1

x‐3, 7x‐1/2

x+5/2, 3x+2

7x‐1, 3x‐2

3x‐2,3x+2,2x+1

67



Understanding that a polynomial

is formed as a product of the given

factors

Reading the zeroes of a

polynomial from the given factors

68





Task 1

Referring to the activity done in the warm up worksheet 1, answer the following questions:

1. What are the expressions in the balloons called?

___________________________________________________________________

2. What are the polynomials with degree 2 called?

___________________________________________________________________

Write the quadratic expression of your choice in the table below.

Now form the quadratic equations from the quadratic expressions.

Quadratic Expression, Q(x) Quadratic Equation, Q(x)=0

69

Task 2

Identify Quadratic Polynomials which cannot be factorized

Referring to the warm up worksheet, factorize the quadratic expressions by using the splitting

of middle term method. Is there any quadratic polynomial which cannot be factorized? List

them down here. We shall talk about them later.

List the quadratics which could not be factorized:

______________________________________________________________________________

______________________________________________________________________________

______________________________________________________________________________

70



Understanding of the concept of

a polynomial and an equation

Formation of equation

Factorization of quadratic

polynomials using the splitting of

middle term method

71





Answer the following questions

1. The general form of a quadratic equation is ________________

2. The conditions on the coefficient of x2 is__________

3. The coefficients and the constant terms are ___________ numbers.

4. The number of zeroes of a polynomial of degree 2 is ____________

5. What is the degree of the expression√2 9 13?

_________________. Does it represent a Quadratic equation? (Y/N)

Why?__________________

6. Is (x‐3)(2x+1) = x(x+5) a quadratic equation? Why?

_________________________________________________________

7. The product of two linear equations is always a _______________ equation

8. The equation =3 is a _____________ equation

9. One of the zeroes of the quadratic polynomial ax2+bx is always ________ irrespective of the

values of a and b .

10. While factorizing a quadratic polynomial by the splitting of the middle term method, the

product of the numbers chosen should be equal to the product of coefficients of

____________ and ____________________________________the sum of the numbers

chosen should be equal to the coefficient of _________.

72



Understanding the concept of a

quadratic polynomial and a

quadratic equation

Understanding the concept of

number of zeroes of a quadratic

polynomial

General understanding of the

concepts covered in the chapter on

Polynomials

73


Content Worksheet (CW1)



The flower vase below has flowers with 2 numbers each, depicting the zeroes of the quadratics

inside the vase. If each quadratic corresponds to a stem of the flower, match the stems with

their flowers.

2x2‐5x+2, x2‐6x+8, x2+x‐2, x2+5x+4, x2‐5x+6,2x2+5x+3, 2x2‐x‐1, x2+5x+6, x2‐12x+35, x2‐x+3

2, 4

½,2

‐2,1

‐4,‐1

2,3

‐2,‐3

5,7

‐½,1

3/2,1

1,3

74

Quadratic polynomial Zeroes

75



Understanding of the concept of

zeroes of a polynomial

Identifying the zeroes of the

polynomial

76





Task 1

As zeroes are for polynomials, roots are for equations. So if x = a is a zero for polynomial p(x),

x = a is a root of the equation p(x) = 0.

Now choose any 5 of the quadratics from CW1 and write the above statement in the following

manner e.g.

Quadratic polynomial: p(x) = x2 − 6x + 8

Quadratic equation: x2− 6x + 8 = 0

(x − 2)(x − 4) = 0

x = 2 and x = 4 are the zeroes of the polynomial p(x) = x2 − 6x + 8

x = 2 and x = 4 are the roots of the given equation x2−6x + 8 = 0

1.

2.

77

3.

4.

5.

Roots of a quadratic equation are also called the solutions of a quadratic equation as they

always satisfy the equation.

78

Task 2

Check if the following values of x are the solutions of the given quadratic equations. Complete

the table below. The first one is done for you

Quadratic equations Value of x Your working Conclusion

2x2‐x‐2=0 2 2.22‐2‐2 0 x=2 is not a solution

of the given equation

x2‐x‐2=0 ‐1

x2+3x‐20=0 4

x2‐8x‐12=0 ‐3

x2‐3x‐10=0 5

x2‐4x‐12=0 ‐2

79



Is able to find the roots of the given

quadratic equation by factorization

method.

Writing the zeroes of a polynomial

and the roots or solutions of the

quadratic equation

Checking if the given value of x is a

solution of the given quadratic

equation

Name of

For a give

discrimin

Task 1

Now com

D=0 tta

f Student___

en quadratic

nant of the e

Re

ha

mplete the ta

then the rohe equatioare_______

Stu

_________

c equation a

equation as D

member: Fo

s to be a no

able for a giv

oots of on __

udent’s W

Content W

Natu

x2+bx+c=0 w

D = b2‐4ac

or any quadr

n negative r

ven quadrat

D=b

D>0 then tthe eqare___

80

Worksheet

Worksheet

ure of Root

where a 0 a

ratic equatio

real number

ic equation a

b2‐4ac

the roots oquation ______

t 7 (SW7)

(CW3)

ts

and a, b, c ar

on to have r

r.

ax2+bx+c=0

of D<0 tthe_

)

re real numb

real roots, th

then the roe equation_________

Date_____

bers, we defi

he discrimin

oots of n are __

____

ine

nant

Task 2

Given be

• Red i

• Mauv

• Leave

elow is a grid

f the quadra

ve if the qua

e them whit

d of 25 squar

atic equation

adratic equat

e if the quad

res with poly

n has real an

tion has rea

dratic equat

81

ynomials. Co

nd distinct ro

l and equal r

ion has no r

olour the gri

oots

roots

eal roots.

id as indicated

82



Is able to state the conditions on

the discriminant of a quadratic

equation for the existence of real

roots

Is able to calculate the discriminant

Is able to predict the nature of

roots using discriminant

83



Discriminant Method


Task 1

To find the value of an unknown constant when the nature of the roots of a quadratic equation

and the equation are given.

Complete the following table. The first one is done for you

Polynomial Nature of roots Your working Value of unknown

constant

kx2−2√5x+4=0

Real and equal roots D=(‐2√5)2 −4.k.4

= 20−16k

D=0 gives k=5/4

k=5/4

x2−4x+k=0 Real and equal roots

kx2−2x−1=0 Real and unequal roots

(k+1)x2+2x+1=0 Real and equal roots

x2−2(k−1)x+1=0 Real and equal roots

3x2−5x−k=0 Real roots

84

Task 2

Solving a quadratic equation by the discriminant method (or method of completing the

squares):

Recall that D=b2−4ac

The solution of a quadratic equation ax2+bx+c=0, a 0 and a,b,c are real numbers,

The solution of the quadratic equation is given by

x= √

i.e. x= √ √

Recall that in the pre content worksheet P1, you were left with a few quadratic expressions

which could not be factorized by the splitting of middle term. Use this method for finding the

solution of those equations.

Now to solve the following equations by the method of completing the squares (or discriminant

method); fill up the blank spaces.

1. Solve the equation √5x2‐3x‐√5=0

Soln.: a=________, b=________, c=__________

D=_______________

x= √

i.e. x= √ √

x or x

85

2. Solve the equation x2+3√5x+6=0

Soln.: a=________, b=________, c=__________

D=_______________

x= √

i.e. x= √ √

x or x

3. Solve the equation x2+2√7x−5=0

Soln.: a=________, b=________, c=__________

D=_______________

x= √

ie. x=√ √

x or x

4. Solve the equation x2+5x+5=0

Soln.: a=________, b=________, c=__________

D=_______________

x= √

ie. x=√ √

86

x or x

5. Solve the equation x2−5x+2=0

Soln.: a=________, b=________, c=__________

D=_______________

x= √

ie. x=√ √

x or x

Task 3

Solve the following quadratic equations using discriminant method and complete the table

given below. If no real roots exist, please mention that.

1. 100x2−20x+1=0

2. 2x2+14x+9=0

3. 9x2−30x+25=0

4. 4x2−4x+1=0

5. X2+2x+4=0

6. 4x2+4√3x+3=0

7. 3x2+2√5x+5=0

8. 25x2+20x+7=0

9. 6x2+23x+20=0

10. 2x2+5x+5=0

Can you solve all equations by factorization method?

87

Now complete the table below

Quadratic equation Discriminant Nature of roots Roots (if they exist)

88

Self Assessment Rubric ‐ Content Worksheet (CW4)


Is able to find the value of an

unknown constant when the

nature of the roots is given

Is able to find the roots of a given

quadratic equation

Is able to identify the situation

when the equation has no real

roots

Is able to understand that while the

method of factorization is not

universally applicable for all

quadratic equations, the method of

discriminant is universally

applicable.

89



Relation Between Sum & Product of Roots and Coefficients of Quadratic Equation


Solve the quadratic equation given in column I in the space provided below it. Fill all other

columns considering the general form of quadratic equation as ax2+bx +c =0.

I II III IV V VI VII VIII IX

Equa

tion

&

Solution

Coefficients of x

2 , x and

constant te

rms (a, b

& c)

Roots

(α, β

)

Sum of R

oots

(α + β)

Prod

uct o

f Roo

ts

(αβ)

Relation

between Co

lumn III

& VI

Relation

between Co

lumn IV

& VII

a =

……….

b =

……….

c =

……….

α = ………

β = ………

90

a =

……….

b =

……….

c =

……….

α = ………

β = ………

a =

……….

b =

……….

c =

……….

α = ………

β = ………

a =

……….

b =

……….

c =

……….

α = ………

β = ………

91

a =

……….

b =

……….

c =

……….

α = ………

β = ………

Is there any relationship between the coefficients of x2, x and the constant terms with the

roots of the equations? Reflect.

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

92

Self Assessment Rubric 1 – Content Worksheet (CW5)

Parameter

Able to recognize coefficients of x2,

x and constant terms.

Able to find roots of a quadratic

equation.

Able to calculate sum of roots and

products of roots correctly.

Able to relate sum & product of

roots with the coefficients of x2, x

and constant terms.

Name of

Reeta los

for Reet

Equation

f Student___

st her bag co

ta’s equatio

ns.

Stud

Fo

_________

ontaining 5 e

n in their

dent’s Wo

Content W

orming a Q

equations. F

bags. Work

93

orksheet

Worksheet

Quadratic E

Fortunately,

k with Ranj

10 (SW10

(CW6)

Equation

Ranjeeta an

eeta and S

0)

nd Saleem ha

Saleem to g

Date_____

ad one root

get back Re

____

each

eeta’s

94

The following table may help you trace Reeta’s Equations.

First Root

(α)

Second Root

(β)

Sum of Roots

(S = α + β)

Product of

Roots

(P = αβ)

Reeta’s Equation

x2 ‐ (S)x + P = 0

2 4 2 + 4=6 2 X 4 = 8 x2 ‐ 6x + 8 = 0

‐7 2

3 ‐5

1 6

‐0.5 9

Self Assessment Rubric – Content Worksheet (CW6)

Parameter

Able to find sum of roots.

Able to find product of roots.

Able to form an equation with

given roots.

95





Represent the following in the language of mathematics:

1. Manav and Rahul together have 45 marbles. Both of them lost 5 marbles each, and the

product of the number of marbles they now have is 124.

2. The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than

twice its breadth.

3. Kareena’s mother is 26 years older than her. The product of their ages (in years) 3 years

from now will be 360.

Solve the following:

1. The numerator of a factor is 4 less than the denominator. If 30 is added to the denominator,

or if 10 be subtracted from the numerator, the resulting fractions will be equal. What is the

original fraction?

___________________________________________________________________________

___________________________________________________________________________

96

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

2. Find two consecutive positive integers whose product is 306.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

97

3. Sazi’s mother is 26 years older than him. The product of their ages (in years) 3 years from

now will be 360. Find Sazi’s present age.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

4. A rectangle has a perimeter of 23 cm and an area of 33 cm2. Find the dimensions.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

98

5. The sum of the reciprocals of two consecutive even integers is . What are the integers?

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

_________________________________________

6. A square piece of cardboard was used to construct a tray by cutting 2 units squares out of

each corner and turning up the flaps. Find the size of the original square if the resulting tray

has a volume of 128 cu units.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

99

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

7. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the

other two sides.

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

___________________________________________________________________________

__________________________________________________________________________

8. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

___________________________________________________________________________

___________________________________________________________________________

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9. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer

side is 30 metres more than the shorter side, find the sides of the field.

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10. If Zoriba were younger by 5 years than what she really is, then the square of her age (in

years) would have been 11 more than five times her actual age. What is her age now?

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Self Assessment Rubric – Content Worksheet (CW7)

Parameter

Able to comprehend word

problems.

Able to express a word problem in

the form of quadratic equation.

Able to solve quadratic equations.

Able to write the answer of the

given word problem.

Name o

1. Unde

Sepa

boxes

of Student_

erstand the

rate quadra

s given belo

Qua

2x2 +

x2 +

Stud

Pos

___________

difference

tic equation

w:

e.g.

adratic Polyn

+ 3x – 4

3/4 x – 4

dent’s Wo

st Content

_

e between

ns and quad

nomial

F

264

‐4x2+3x+2

6x2‐2x‐1=4x2‐3x‐3

103

orksheet

t Workshee

quadratic

dratic polyno

Quadra

2x2 + 3

x2 + 3/4

Funnel

2x2‐10x=7 6x2‐14x‐7 4x2‐12x+3

2

=6 3

3xx2√7

12 (SW12

et (PCW1)

polynomial

omials from

atic Equatio

3x – 4 = 0

4 x – 4 = 0

x2+4x‐7=0 2+3x‐7=0 7x2+4x+7

2)

and quad

the given b

ons

Date_____

dratic equat

box into the

____

tions.

e two

104

2. Which of the following are Quadratic Equations?


Post Content (PCW2)


1. What is a quadratic equation? What is the degree of quadratic equation?

2. What do you understand by root of an equation? How many roots will a quadratic equation

have?

3. What is discriminant? Does it help to predict upon the nature of roots of a quadratic

equation? Explain.

4. Write the nature of roots for each quadratic equation (ax2+bx+c=0) given below:

Quadratic Polynomial

Quadratic Equations

105

Equation a, b

& c

D=b2‐4ac D>0 D=0 D<0 Nature of

roots.

x2+4x+5=0 a=…

b=…

c=…

3x2‐5x+2=0 a=…

b=…

c=…

x2+2x‐143=0 a=…

b=…

c=…

x2‐5x+6=0 a=…

b=…

c=…

9x2+3x+5=0 a=…

b=…

106

c=…

5x2‐6x+2=0 a=…

b=…

c=…

5x2‐6x‐2=0 a=…

b=…

c=…

3x2‐5x+2=0 a=…

b=…

c=…

107


Post Content (PCW3)


Find the roots of the following quadratic equations using:

• Factorization.

• Completion of Squares.

• Quadratic Formula.


Post Content (PCW4)


Do as directed:

1. If x2 +5x+1=0, find the value of .

2. Solve for x : √3x2 − 2√2x − 2√3=0

3. Solve for x : 3 (x 1, 2)

4. Find the roots of the following equations:

a) x 3 x 18 0

b) 4 x 4 x 24 0

c) 5 x 25 x 30 0

108

d) 2 x 4 x 16 0

e) 8 x 24 x 32 0

f) 4 x 12 x 40 0

g) 2 x 2 x 4 0

h) 2 x 2 x 40 0

5. Verify the relationship between the sum of roots, products of roots and coefficient of x2, x

and constant term for the equations given in question 4.

6. Solve for x:

2 3 ; x −3, x

7. Rita rows 12 km upstream and 12 km downstream in 3 hours. The speed of her boat in still

water is 9 km/hr. Find the speed of the stream.

8. The equation x2‐9x+2k=0 has roots ‘a’ and ‘b’. If a = 2b, find the value of k.

9. Is 0.3 a root of the equation x2 – 0.9 = 0? Justify.

10. Had Karan scored 10 more marks in her science test out of 30 marks, 9 times these marks

would have been the square of his actual marks. How many marks did he get in the test?

11. A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel

the same distance if its speed were 5 km/h more. Find the original speed of the train.

12. Is it possible to design a rectangular garden grove whose length is twice its breadth, and the

area is 800m² ? If so, find its length and breadth.

109

Useful Online Links

http://www.purplemath.com/modules/quadform.htm

http://mathworld.wolfram.com/QuadraticEquation.html

History of Quadratic Equation

http://www.mytutoronline.com/history‐of‐quadratic‐equation

Sum of roots, product of roots and discriminate formula

http://www.hitxp.com/zone/tutorials/mathematics/world‐of‐quadratic‐equations/

Solving quadratic equation

http://library.thinkquest.org/20991/alg2/quad.html

101 uses of quadratic equation


Introduction to Quadratic equation

http://www.mathsisfun.com/algebra/quadratic‐equation.html

Derivation of quadratic formula

http://www.mathsisfun.com/algebra/quadratic‐equation‐derivation.html

Online Quadratic Equation solver

http://www.mathsisfun.com/quadratic‐equation‐solver.html

http://www.math.com/students/calculators/source/quadratic.htm

http://kselva.tripod.com/quad.html

Test on quadratic equation

http://www.alexmaths.com/cbse10/quadratic/quadratic.html

110

Videos

5 ways to solve quadratic equation:

http://www.youtube.com/watch?v=zAjeVyUFaSc&feature=fvwrel

Solving quadratic equation by Square root method

http://www.youtube.com/watch?v=zAjeVyUFaSc&feature=fvwrel

Solving quadratic equation by factoring

http://www.youtube.com/watch?v=lMU5wMDcJNg&feature=related

Introduction to quadratic equation

http://www.youtube.com/watch?v=IWigvJcCAJ0&feature=related

Quadratic Formula

http://www.youtube.com/watch?v=IvXgFLV2gOk&feature=related

Using UnFOIL to Factor Quadratic Equations

http://www.youtube.com/watch?v=z57PKs3Bm4U&feature=related

Solving Word Problems

http://www.youtube.com/watch?v=EhPPci8shA8

http://www.youtube.com/watch?v=Vu3px08WX_8

http://www.youtube.com/watch?v=lS9S1iEjlPI

Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India

CENTRAL BOARD OF SECONDARY EDUCATION

Documents

CLASS-X UNIT-4 QUADRATIC EQUATION CORE … UNIT-4 CLASS X Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India (CORE) The CBSE-International is grateful for permission