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Unit-4: Quadratic Equations (Core) - New Indian Model School, Dubai

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CBSE-i<br />

CLASS<br />

X<br />

UNIT-4<br />

QUADRATIC<br />

EQUATIONS<br />

MATHEMATICS<br />

(CORE)<br />

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


CBSE-i<br />

QUADRATIC<br />

EQUATIONS<br />

MATHEMATICS<br />

CLASS<br />

X<br />

UNIT-4<br />

(CORE)<br />

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


The CBSE-International is grateful for permission to reproduce<br />

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

acknowledgements have been included wherever appropriate and<br />

sources from where the material may be taken are duly mentioned. In<br />

case any thing has been missed out, the Board will be pleased to rectify<br />

the error at the earliest possible opportunity.<br />

All Rights of these documents are reserved. No part of this publication<br />

may be reproduced, printed or transmitted in any form without the<br />

prior permission of the CBSE-i. This material is meant for the use of<br />

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


PREFACE<br />

The Curriculum initiated by Central Board of Secondary Education -International (CBSE-i) is a progressive step in making<br />

the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a<br />

fresh thought process in imparting a curriculum which would restore the independence of the learner to pursue the<br />

learning process in harmony with the existing personal, social and cultural ethos.<br />

The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It<br />

has about 11500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been<br />

conscious of the varying needs of the learners in countries abroad and has been working towards contextualizing certain<br />

elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged.<br />

The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in<br />

view.<br />

The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to<br />

nurture the independence of the learner, given the fact that every learner is unique. The learner has to understand,<br />

appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations<br />

and additions wherever and whenever necessary.<br />

The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The<br />

speed and methods of assimilating knowledge have put forth many challenges to the educators, forcing them to rethink<br />

their approaches for knowledge processing by their learners. In this context, it has become imperative for them to<br />

incorporate those skills which will enable the young learners to become 'life long learners'. The ability to stay current, to<br />

upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant<br />

life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of<br />

these requirements.<br />

The CBSE-i aims to carry forward the basic strength of the <strong>Indian</strong> system of education while promoting critical and<br />

creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and<br />

media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all<br />

subject areas to cater to the different pace of learners.<br />

The CBSE has introduced the CBSE-i curriculum in schools affiliated to CBSE at the international level in 2010 and is now<br />

introducing it to other affiliated schools who meet the requirements for introducing this curriculum. The focus of CBSE-i is<br />

to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous<br />

and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some nonevaluative<br />

components in the curriculum which would be commented upon by the teachers and the school. The objective<br />

of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal<br />

knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives,<br />

SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this '<strong>Core</strong>'.<br />

The <strong>Core</strong> skills are the most significant aspects of a learner's holistic growth and learning curve.<br />

The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework<br />

(NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to<br />

millions of learners, many of whom are now global citizens.<br />

The Board does not interpret this development as an alternative to other curricula existing at the international level, but as<br />

an exercise in providing the much needed <strong>Indian</strong> leadership for global education at the school level. The International<br />

Curriculum would evolve on its own, building on learning experiences inside the classroom over a period of time. The<br />

Board while addressing the issues of empowerment with the help of the schools' administering this system strongly<br />

recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to<br />

their peers through the interactive platforms provided by the Board.<br />

I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Ms. Abha Adams and her team and Dr.<br />

Sadhana Parashar, Head (Innovations and Research) CBSE along with other Education Officers involved in the<br />

development and implementation of this material.<br />

The CBSE-i website has already started enabling all stakeholders to participate in this initiative through the discussion<br />

forums provided on the portal. Any further suggestions are welcome.<br />

Vineet Joshi<br />

Chairman


ACKNOWLEDGEMENTS<br />

Advisory<br />

Conceptual Framework<br />

Shri Vineet Joshi, Chairman, CBSE<br />

Shri G. Balasubramanian, Former Director (Acad), CBSE<br />

Shri Shashi Bhushan, Director(Academic), CBSE Ms. Abha Adams, Consultant, Step-by-Step <strong>School</strong>, Noida<br />

Dr. Sadhana Parashar, Head (I & R),CBSE<br />

Ideators<br />

Ms. Aditi Misra Ms. Anuradha Sen Ms. Jaishree Srivastava Dr. Rajesh Hassija<br />

Ms. Amita Mishra Ms. Archana Sagar Dr. Kamla Menon Ms. Rupa Chakravarty<br />

Ms. Anita Sharma Ms. Geeta Varshney Dr. Meena Dhami Ms. Sarita Manuja<br />

Ms. Anita Makkar Ms. Guneet Ohri Ms. Neelima Sharma Ms. Himani Asija<br />

Dr. Anju Srivastava Dr. Indu Khetrapal Dr. N. K. Sehgal Dr. Uma Chaudhry<br />

Material Production Groups: Classes IX-X<br />

English :<br />

Ms. Sarita Manuja<br />

Ms. Renu Anand<br />

Ms. Gayatri Khanna<br />

Ms. P. Rajeshwary<br />

Ms. Neha Sharma<br />

Ms. Sarabjit Kaur<br />

Ms. Ruchika Sachdev<br />

Geography:<br />

Ms. Deepa Kapoor<br />

Ms. Bharti Dave<br />

Ms. Bhagirathi<br />

Ms. Archana Sagar<br />

Ms. Manjari Rattan<br />

English :<br />

Ms. Rachna Pandit<br />

Ms. Neha Sharma<br />

Ms. Sonia Jain<br />

Ms. Dipinder Kaur<br />

Ms. Sarita Ahuja<br />

Material Production Group: Classes I-V<br />

Dr. Indu Khetarpal Ms. Rupa Chakravarty Ms. Anita Makkar Ms. Nandita Mathur<br />

Ms. Vandana Kumar Ms. Anuradha Mathur Ms. Kalpana Mattoo Ms. Seema Chowdhary<br />

Ms. Anju Chauhan Ms. Savinder Kaur Rooprai Ms. Monika Thakur Ms. Ruba Chakarvarty<br />

Ms. Deepti Verma Ms. Seema Choudhary Mr. Bijo Thomas Ms. Mahua Bhattacharya<br />

Ms. Ritu Batra<br />

Ms. Kalyani Voleti<br />

Coordinators:<br />

Dr. Sadhana Parashar, Ms. Sugandh Sharma, Dr. Srijata Das, Dr. Rashmi Sethi,<br />

Head (I and R) E O (Com) E O (Maths) O (Science)<br />

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

Ms. Seema Lakra, S O<br />

Mathematics :<br />

Dr. K.P. Chinda<br />

Dr. Ram Avtar Gupta<br />

Dr. Mahender Shankar<br />

Mr. J.C. Nijhawan<br />

Ms. Reemu Verma<br />

Ms. Himani Asija<br />

Political Science:<br />

Ms. Sharmila Bakshi<br />

Ms. Archana Soni<br />

Ms. Srilekha<br />

Science :<br />

Dr. Meena Dhami<br />

Mr. Saroj Kumar<br />

Ms. Rashmi Ramsinghaney<br />

Ms. Seema kapoor<br />

Ms. Priyanka Sen<br />

Dr. Kavita Khanna<br />

Ms. Keya Gupta<br />

Ms. Preeti Hans, Proof Reader<br />

Science :<br />

Ms. Charu Maini<br />

Ms. S. Anjum<br />

Ms. Meenambika Menon<br />

Ms. Novita Chopra<br />

Ms. Neeta Rastogi<br />

Ms. Pooja Sareen<br />

Economics:<br />

Ms. Mridula Pant<br />

Mr. Pankaj Bhanwani<br />

Ms. Ambica Gulati<br />

Material Production Groups: Classes VI-VIII<br />

Mathematics :<br />

Ms. Seema Rawat<br />

Ms. N. Vidya<br />

Ms. Mamta Goyal<br />

Ms. Chhavi Raheja<br />

Political Science:<br />

Ms. Kanu Chopra<br />

Ms. Shilpi Anand<br />

History :<br />

Ms. Jayshree Srivastava<br />

Ms. M. Bose<br />

Ms. A. Venkatachalam<br />

Ms. Smita Bhattacharya<br />

Geography:<br />

Ms. Suparna Sharma<br />

Ms. Leela Grewal<br />

History :<br />

Ms. Leeza Dutta<br />

Ms. Kalpana Pant


Preface<br />

Acknowledgements<br />

1. Syllabus 1<br />

2. Scope document 2<br />

3. Teacher's Support Material 5<br />

vTeacher Note 6<br />

vActivity Skill Matrix 13<br />

vWarm Up Activity W1 14<br />

Identify the Polynomials of Degree 2<br />

vWarm Up Activity W2 15<br />

Recognize Zeroes of a Polynomial<br />

vPre -Content Worksheet P1 15<br />

<strong>Quadratic</strong> <strong>Equations</strong> from <strong>Quadratic</strong> Polynomials<br />

Identify Polynomials which cannot be Factorized<br />

vPre -Content Worksheet P2 16<br />

Revisit Key Concepts<br />

vContent Worksheet CW1 17<br />

Zeroes of a <strong>Quadratic</strong> Polynomial<br />

vContent Worksheet CW2 18<br />

Roots of a <strong>Quadratic</strong> Equation<br />

vContent Worksheet CW3 19<br />

Nature of Roots<br />

Content<br />

vContent Worksheet CW4 21<br />

Discriminant Method of Finding the Roots<br />

vContent Worksheet CW5 22<br />

Relations Between Roots and Coefficients of a <strong>Quadratic</strong> Equation<br />

vContent Worksheet CW6 23<br />

Formation of a <strong>Quadratic</strong> Equation.<br />

vContent Worksheet CW7 24<br />

Application of <strong>Quadratic</strong> Equation in Real Life Problems.<br />

vPost Content Worksheet PCW1 25<br />

vPost Content Worksheet PCW2 25


vPost Content Worksheet PCW3 25<br />

vPost Content Worksheet PCW4 25<br />

vAssessment Plan 26<br />

vStudy Material 30<br />

vStudent's Support Material 63<br />

vSW1: Warm Up Worksheet (W1) 64<br />

Identify the Polynomials of Degree 2<br />

vSW2: Warm Up Worksheet (W2) 66<br />

Recognize Zeroes of a Polynomial<br />

vSW3: Pre Content Worksheet (P1) 68<br />

<strong>Quadratic</strong> <strong>Equations</strong> from <strong>Quadratic</strong> Polynomials<br />

Identify Polynomials which cannot be Factorized<br />

vSW4: Pre Content Worksheet (P2) 71<br />

Revisit Key Concepts<br />

vSW5: Content Worksheet (CW1) 73<br />

Zeroes of a <strong>Quadratic</strong> Polynomial<br />

vSW6: Content Worksheet (CW2) 76<br />

Roots of a <strong>Quadratic</strong> Equation<br />

vSW7: Content Worksheet (CW3) 80<br />

Nature of Roots<br />

vSW8:Content Worksheet (CW4) 83<br />

Discriminant method<br />

vSW9: Content Worksheet (CW5) 89<br />

Relation Between Roots and Coefficients<br />

vSW10: Content Worksheet (CW6) 93<br />

Forming a <strong>Quadratic</strong> Equation<br />

vSW11:Content Worksheet (CW7) 95<br />

Application of <strong>Quadratic</strong> Equation in Real Life Problems<br />

vSW12: Post Content Worksheet (PCW1) 103<br />

vSW13: Post Content Worksheet (PCW2) 104<br />

vSW 14: Post Content Worksheet (PCW3) 107<br />

vSW 15: Post Content Worksheet (PCW4) 107<br />

vSuggested<br />

Videos & Extra Readings. 109


SYLLABUS UNIT 4: QUADRATIC EQUATION (CORE)<br />

Introduction to <strong>Quadratic</strong> Equation<br />

<strong>Quadratic</strong> <strong>Equations</strong> are of the form<br />

ax 2 + b x + c=0, a≠0, a, b, c being real<br />

numbers<br />

Methods to solve quadratic equations<br />

Nature of roots<br />

Application in daily life<br />

Factorisation Method,<br />

Discriminant Method<br />

(D = b 2 – 4ac ) formula , x= √ <br />

<br />

Nature of roots when D=0, D0, D0<br />

Sum of roots, product of roots, conjugate<br />

roots<br />

Finding a quadratic equation when relations<br />

between the roots are given.<br />

Number problems, age problems, work ratio<br />

problems, distance time problems, speedtime<br />

problems, upstream‐downstream<br />

motion problems.<br />

1


SCOPE DOCUMENT<br />

Key Concepts:<br />

1. <strong>Quadratic</strong> Equation<br />

2. Discriminant<br />

3. Roots of a quadratic equation<br />

4. Nature of roots<br />

5. Sum of roots<br />

6. Product of roots<br />

Learning objectives:<br />

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

are real numbers and a ≠0<br />

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

quadratic equation. Roots are also known as solution of quadratic equation.<br />

3. To find the roots of equation by factorisation method.<br />

4. To predict the nature of roots based on the sign of discriminant.<br />

5. To find the roots of equation by discriminant method.<br />

6. To know the relation between sum of the roots of quadratic equation and the coefficients<br />

of x 2 , x and constant<br />

7. To know the relation between product of the roots of quadratic equation and the<br />

coefficients of x 2 , x and constant<br />

8. To form a quadratic equation if sum of roots and product of roots are known.<br />

9. To solve the problems from real life situations having application of quadratic equations.<br />

2


Extension Activities<br />

1. <strong>Equations</strong> reducible to quadratic forms, e.g.<br />

a) x + = 0 b) (2x+ y) 2 + 4(2x+y) = 3<br />

c) 3 x + 3 ‐x = 2 d)<br />

<br />

+ <br />

= 6<br />

2. Zeroes of quadratic polynomial determined graphically represent the roots of quadratic<br />

equation obtained by equating the quadratic polynomial to zero. So, the points<br />

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

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

graph show that quadratic inequalities has infinite solutions and quadratic equation has at<br />

the most two solutions.<br />

3. Biquadratic equations are the equation of the form ax 4 +bx 2 +c =0. Biquadratic equations<br />

can be solved by reducing them to quadratic forms.<br />

Life skill<br />

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

choices. Using knowledge of finding solution to quadratic equations students can compare the<br />

investment plans or insurance plans offered by various banks or financial agencies and<br />

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

language or glamorous brand ambassador. Rather they shall workout rationally the long term<br />

effects of any plan.<br />

SEWA<br />

Applying their knowledge of quadratic equations as discussed above (life skill) students can<br />

save their relatives and elders to be misguided by catchy language of advertisements.<br />

3


Cross –curricular link:<br />

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

called aerodynamics. <strong>Quadratic</strong> equations find its application in aerodynamics.<br />

Look at the following problem:<br />

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

rocket t seconds after the launch is given by h= −16t 2 + 200t. How many seconds after the<br />

launch will the rocket be 300 ft above the ground<br />

2. Car safety<br />

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

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

by d= 0.4v 2 +0.5v, with v as speed of the car. If the driver must be able to stop within<br />

60 ft., what is the maximum safe speed, to the nearest miles per hour, of the car<br />

Many more application can be identified in architecture, construction, geometry etc.<br />

4


TEACHER’S<br />

SUPPORT<br />

MATERIAL<br />

5


Teacher’s Note<br />

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

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

for school Mathematics include :<br />

1. Children learn to enjoy mathematics rather than fear it.<br />

2. Children see mathematics as something to talk about, to communicate through, to discuss<br />

among themselves, to work together on.<br />

3. Children pose and solve meaningful problems.<br />

4. Children use abstractions to perceive relationships, to see structures, to reason out things,<br />

to argue the truth or falsity of statements.<br />

5. Children understand the basic structure of Mathematics: Arithmetic, algebra, geometry and<br />

trigonometry, the basic content areas of <strong>School</strong> Mathematics, all offer a methodology for<br />

abstraction, structuration and generalisation.<br />

6. Teachers engage every child in class with the conviction that everyone can learn<br />

mathematics.<br />

Students should be encouraged to solve problems through different methods like<br />

abstraction, quantification, analogy, case analysis, reduction to simpler situations, even<br />

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

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

solving it. <strong>School</strong> mathematics should also play an important role in developing the useful<br />

skill of estimation of quantities and approximating solutions. Development of visualisation<br />

and representation skills should be integral part of Mathematics teaching. There is also a<br />

need to make connections between Mathematics and other subjects of study. When<br />

children learn to draw a graph, they should be encouraged to perceive the importance of<br />

graph in the teaching of Science, Social Science and other areas of study. Mathematics<br />

should help in developing the reasoning skills of students. Proof is a process which<br />

encourages systematic way of argumentation. The aim should be to develop arguments, to<br />

evaluate arguments, to make conjunctures and understand that there are various methods<br />

of reasoning. Students should be made to understand that mathematical communication is<br />

6


precise, employs unambiguous use of language and rigour in formulation. Children should<br />

be encouraged to appreciate its significance.<br />

At the secondary stage students begin to perceive the structure of Mathematics as a<br />

discipline. By this stage they should become familiar with the characteristics of<br />

Mathematical communications, various terms and concepts, the use of symbols, precision<br />

of language and systematic arguments in proving the proposition. At this stage a student<br />

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

problems.<br />

Learning objectives:<br />

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

are real numbers and a ≠0<br />

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

quadratic equation. Roots are also known as solution of quadratic equation.<br />

3. To find the roots of a quadratic equation by factorisation method.<br />

4. To predict the nature of roots based on the sign of discriminant.<br />

5. To find the roots of a quadratic equation by discriminant method.<br />

6. To know the relation between sum of the roots of quadratic equation and the coefficients<br />

of x 2 , x and constant<br />

7. To know the relation between product of the roots of quadratic equation and the<br />

coefficients of x 2 , x and constant<br />

8. To form a quadratic equation if sum of roots and product of roots are known.<br />

9. To solve the problems from real life situations having application of quadratic equations.<br />

<strong>Quadratic</strong> Equation is extension of quadratic polynomials learnt earlier. In fact, quadratic<br />

equations are most widely used part of algebra to solve real life problems. Treatment of<br />

7


this subject requires very delicate handling in the class as the students need to<br />

understand the significance of the topic in practical context as well as need to acquire the<br />

skill of solving the problems based on quadratic equations.<br />

Students are already familiar with <strong>Quadratic</strong> polynomial and their factorization. When a<br />

quadratic polynomial is equated to zero it becomes quadratic equation. Naturally the<br />

factors of quadratic expressions will also be equated to zero while making it quadratic<br />

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

quadratic polynomial<br />

P(x) = x 2 ‐4x +3 with factors x‐1 and x‐3<br />

Is equated to zero i.e. p(x) = 0, it implies x 2 ‐4x +3 =0 or (x‐1)(x‐3)=0.<br />

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

zero then at least one of them is zero.<br />

Mathematically, a.b =0 implies either a=0 or b=0 or both a and b are zero.<br />

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

This implies x=1 or x=3.<br />

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

value of x as 1 or 3.”<br />

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

The questions may appear to be vague as the students may not be able to state their<br />

doubts clearly due to lots of confusion with respect to quadratic polynomial and quadratic<br />

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

polynomial.<br />

Teacher must take all queries one by one to clear all doubts by describing the quadratic<br />

equations in historical, geometrical and practical context.<br />

8


Historical introduction of <strong>Quadratic</strong> Equation<br />

Egyptians came across second degree equations related to land survey.<br />

Babylonians formed the quadratic equations to solve their problems regarding agriculture and<br />

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

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

of field.<br />

In mathematical terms, if units is the length of the side of the field, is the amount of crop<br />

you can grow on a square field of side length 1 unit, and<br />

grow, then<br />

is the amount of crop that you can<br />

This was the first quadratic equation obtained. Babylonians also tried to solve the quadratic<br />

equations by the method of completing the square.<br />

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

mathematics a scientific form.<br />

Greeks tried to solve the quadratic equation x 2 +2x=8 by completing the square in the following<br />

manner:<br />

They interpreted x 2 as square of side length x units, 2x as 2(1)(x) i.e. as rectangles of length<br />

x units and breadth 1 unit.<br />

x 1<br />

x<br />

x<br />

1<br />

1<br />

x<br />

9


To complete the square they added a square of side 1 unit.<br />

Their form of reasoning corresponds to our algebraic method of completing the square<br />

x 2 + 2x = 8 or x 2 + 2x+ 1 2 = 8 +1 or (x+1) 2 = 9<br />

This implies x+1 = 3 or x+1 = ‐3<br />

Hence x=2 or x= ‐4.<br />

Mathematicians have come across some quadratic equations while solving the geometric<br />

problems.<br />

One of the most famous problems is,<br />

“Is it possible to construct a square whose area is that of a given circle”<br />

It was about 2300 years later that mathematicians were able to prove that this type of<br />

construction is impossible. (Why)<br />

For more details visit<br />

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

In the unit three methods of solving the quadratic equations are discussed:<br />

A. Factorization method<br />

B. Completing the square<br />

C. Discriminant method<br />

Good practice is required to acquire the skill of finding the roots of quadratic equations.<br />

10


Geometrical interpretation of quadratic equations<br />

Some Geogebra activities are already described in the unit of polynomials.<br />

Students are familiar with the curves obtained for quadratic polynomials. They should be given<br />

clear understanding that all points on a curve satisfy the quadratic polynomial written as<br />

equation y= ax 2 +bx+c<br />

At the most two points satisfy the quadratic expression written as quadratic equation<br />

ax 2 +bx+c = 0. These points are values of variable x satisfying the quadratic equation ax 2 +bx+c =0<br />

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

activities examples of quadratic equations with repeated roots, with rational roots, with<br />

irrational conjugate roots and ask the students to verify the conditions on discriminant<br />

simultaneously to help them to internalize the concepts learnt rigorously.<br />

Some thought provoking extension activities as suggested below can be used to further<br />

enhance the interest in the topic. For example teacher can take two different irrational roots<br />

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

learnt<br />

x 2 ‐ (sum of roots) x+ product of roots =0<br />

For example if roots are √2 and √3, then equation is<br />

x 2 – (√2 +√3)x+ √6 =0<br />

which is a quadratic equation. Then why do we say that irrational roots of a quadratic equation<br />

occur in conjugate pairs.<br />

To answer this question following modification can be made in above statement:<br />

A quadratic equation with rational coefficients can have rational roots or irrational roots in<br />

conjugate pairs.<br />

<strong>Quadratic</strong> equations with irrational coefficients have different irrational roots (not in conjugate<br />

pairs).<br />

11


Common errors made by students<br />

1. While solving the equations like x 2 =16 students do not give the solution as x= ± 4.They<br />

write single solution i.e. x=4<br />

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

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

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

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

by x=3 in both factors. Similarly x=5 is one solution.<br />

These errors shall be discussed in the class with the students.<br />

Once the students have acquired the skill of solving the quadratic equations, word problems<br />

can be introduced to expose the students with daily life situations.<br />

12


Activity Skill Matrix<br />

Type of Activity Name of Activity Skill to be developed<br />

Warm UP(W1)<br />

Warm UP(W2)<br />

Pre‐Content (P1)<br />

Pre‐Content (P2)<br />

Content (CW 1)<br />

Content (CW 2)<br />

Content (CW 3)<br />

Content (CW 4)<br />

Content (CW 5)<br />

Content (CW 6)<br />

Content (CW 7)<br />

Recalling quadratic<br />

expressions<br />

Factors of a<br />

polynomial<br />

<strong>Quadratic</strong> Expression<br />

and quadratic<br />

equations<br />

Basics of quadratic<br />

equation<br />

Zeroes of a quadratic<br />

polynomial<br />

Roots of a quadratic<br />

equation<br />

Solving a quadratic<br />

equation<br />

Discriminant Method<br />

of Finding the roots<br />

Relation between sum<br />

and product of roots<br />

and coefficients of<br />

quadratic equations<br />

Forming a quadratic<br />

equation<br />

Application of<br />

quadratic equation in<br />

real life problems<br />

Observation, analytical skill<br />

Problem solving skill<br />

knowledge and creative skill<br />

Knowledge, thinking skill,<br />

Understanding, verification, application<br />

Application, Problem Solving Skill<br />

Observation, Analytical Skill<br />

Observation and analytical skills<br />

Analytical and synthesizing skills<br />

Observation, application<br />

Application of knowledge, analytical skills,<br />

Problem Solving Skill<br />

13


Post ‐ Content<br />

(PCW 1)<br />

Post ‐ Content<br />

(PCW 2)<br />

Post ‐ Content<br />

(PCW 3)<br />

Identifying <strong>Quadratic</strong><br />

Equation<br />

Assignment based<br />

Basics of <strong>Quadratic</strong><br />

equations and nature<br />

of roots<br />

Assignment based on<br />

method of solving<br />

<strong>Quadratic</strong> <strong>Equations</strong><br />

Knowledge, understanding analytical skill<br />

Analytical skills and computational skill<br />

Problem solving skills, Computational Skill<br />

Warm up Activity (W1)<br />

Identify the Polynomials of Degree 2<br />

Specific Objective<br />

To review and recall the concept of quadratic expressions<br />

Description<br />

Execution<br />

Parameters for<br />

assessment<br />

In the previous chapter, the students have learnt about quadratic<br />

expressions. They know how to differentiate between a<br />

polynomial and an algebraic expression. They know that a<br />

polynomial of degree 2 is called a quadratic expression.<br />

The teacher may give printouts of the worksheet and ask the<br />

students to filter down the quadratic expressions from the<br />

algebraic expressions in the funnel and write the quadratic<br />

expressions in the box provided.<br />

Understanding of the difference between a polynomial and an<br />

algebraic expression<br />

Finding out the degree of a given polynomial<br />

14


Warm up Activity (W2)<br />

Recognize Zeroes of a Polynomial<br />

Specific Objective: To recall the formation of a polynomial when the factors of the polynomial<br />

are given. To identify the zeroes of the polynomial from the given factors.<br />

Description:<br />

The task calls for forming a polynomial when the factors of the polynomial<br />

are given. The students shall form the polynomials by multiplying the<br />

factors given in the table. They shall also read the zeroes of the polynomial<br />

from the given factors.<br />

Execution: The teacher may write the factors on the board. The teacher may<br />

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

more than two factors.<br />

Parameters for Assessment:<br />

Students will be able to:<br />

• Understand that a polynomial is a product of its factors<br />

• Reading the zeroes of a polynomial from the given factors<br />

Pre Content Worksheet (P1)<br />

<strong>Quadratic</strong> <strong>Equations</strong> from <strong>Quadratic</strong> Polynomials<br />

Specific Objective<br />

Description<br />

To recall a polynomial equation and form a quadratic equation<br />

from the given quadratic expression. To review the factorization<br />

of the quadratic expression by splitting of the middle term<br />

method.<br />

In task 1, the students are expected to give answers to the given<br />

questions and form quadratic equations. In task 2, the students<br />

15


shall factorize the given quadratic expressions by splitting the<br />

middle term. They would not be able to factorize two of them viz.<br />

3x 2 +4x‐2 and √2x 2 ‐3x‐5. The teacher shall ask the students to list<br />

them down in the provided space and discuss them later while<br />

using discriminant of a quadratic equation.<br />

Execution<br />

Parameters for<br />

assessment<br />

Teacher may take printouts of the sheet or ask the questions of<br />

task 1 orally and by using the worksheet done in warm up<br />

worksheet for task 2.<br />

Understanding the concept of an equation<br />

Formation of an equation<br />

Factorization of a quadratic polynomials using the splitting of<br />

middle term method<br />

Specific Objective<br />

Description<br />

Execution<br />

Parameters for<br />

assessment<br />

Pre Content Worksheet (P2)<br />

Revisit Key Concepts<br />

To review, recall a polynomial equation, quadratic equation, zero<br />

of a polynomial and other concepts covered in the chapter on<br />

Polynomials. To review the factorization of the quadratic<br />

expression by splitting of the middle term method.<br />

The task describes some questions designed to test the basic<br />

understanding of the students on the Polynomials chapter.<br />

Teacher may take printouts of the sheet or ask the questions<br />

orally.<br />

Understanding the concept of a quadratic polynomial and<br />

equation<br />

Understanding the concept of number of zeroes of a quadratic<br />

polynomial<br />

General understanding of the concepts covered in the<br />

Polynomials chapter<br />

16


Activity 5: Content Worksheet (CW1)<br />

Zeroes of a <strong>Quadratic</strong> Polynomial<br />

Specific Objective<br />

Description<br />

Execution<br />

Parameters for<br />

assessment<br />

Recall the zeroes of a quadratic and match them with a given<br />

quadratic expression<br />

The students shall match the zeroes given in the form of flowers<br />

in the flower vase with the quadratic expressions given in the<br />

vase. Each quadratic expression corresponds to a pair of zeroes of<br />

a quadratic expression. They would then complete the table given<br />

in the task by writing the quadratic expression with the<br />

corresponding pair of zeroes.<br />

Teacher may take printouts of the sheet and ask the students to<br />

write in the given space<br />

Understanding of the concept of zeroes of a polynomial<br />

Finding out the zeroes of the polynomial<br />

17


Activity 6: Content Worksheet (CW2)<br />

Roots of a <strong>Quadratic</strong> <strong>Equations</strong><br />

Specific Objective<br />

Description of task<br />

Execution<br />

Parameters for assessment<br />

To understand the concept of roots and solution of a given<br />

quadratic equation and correlate it with the zeroes of a<br />

quadratic expression<br />

To check whether a given value of x is a solution of a given<br />

quadratic equation<br />

The students use their previous knowledge of factorization of<br />

a quadratic polynomial by splitting of the middle term method<br />

and find the zeroes of the quadratic. They shall then write that<br />

the found values of x are the roots and hence the solutions of<br />

the given quadratic equation. They shall complete the task on<br />

similar lines as the example.<br />

In task 2, the students shall verify whether a given value of x is<br />

a solution of the given quadratic equation by substituting the<br />

value of x in the quadratic equation.<br />

The teacher may give printouts of the given worksheet or write<br />

the questions on the board and discuss them with the class.<br />

Understanding of the concept quadratic equation<br />

Writing the zeroes of a polynomial and the roots and solutions<br />

of the quadratic equation<br />

Checking if the given value of x is a solution of the given<br />

quadratic equation<br />

Extra reading<br />

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

18


Activity7: Content Worksheet (CW3)<br />

Nature of Roots<br />

Specific Objective (s)<br />

Description of task<br />

To predict the nature of the roots by using the sign of the<br />

discriminant.<br />

The teacher may ask the students orally about the conditions<br />

of the roots to be real and distinct, real and equal and non real<br />

roots. For task 1, the teacher may discuss with the students.<br />

In task 2, the teacher may give printed worksheets and ask the<br />

students to find out the discriminants in each case and colour<br />

the grid following ther instructions.<br />

The teacher may draw the final answer figure on the board and<br />

may discuss the figure so obtained ‘swastik’ («) with the<br />

students and may, if feasible, ask them to explore more about<br />

it through net surfing.<br />

Execution Discussions in task 1 and printed worksheets in task 2<br />

Parameters of assessment<br />

Is able to state the conditions on the discriminant of a<br />

quadratic equation for the existing roots<br />

Is able to calculate the discriminant<br />

Is able to categorize the quadratic equations into 3 types<br />

according to the types of roots of the quadratic equation by<br />

finding the discriminant of the quadratic equation.<br />

Is able to complete the design correctly.<br />

Extra readings:<br />

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

19


Answer grid to the colouring worksheet<br />

20


Activity 8: Content Worksheet (CW4)<br />

Discriminant Method of Finding the Roots<br />

Specific Objective (s)<br />

Description of task<br />

To find the value of an unknown constant in a quadratic<br />

equation when the nature of the roots is given. To find the<br />

solution of a given quadratic equation by the discriminant<br />

method in particular and by any of the two methods in general.<br />

In task 1, the student shall complete the table by finding out the<br />

discriminat and categorizing the given equation to have real and<br />

distinct roots, real and equal roots or no real roots according as<br />

D>0, D=0 or D


Activity 9‐ Content Worksheet (CW5)<br />

Relations between Roots and Co‐efficients of a <strong>Quadratic</strong> <strong>Equations</strong><br />

Specific Objective:<br />

1. To know the relation between sum of the roots of quadratic equation and the coefficients<br />

of x 2 , x and constant.<br />

2. To know the relation between product of the roots of quadratic equation and the<br />

coefficients of x 2 , x and constant.<br />

Description: In this task students will first solve the equation and then recognize and write the<br />

coefficients of x 2 , x and constant term. They will find –b/a and c/a. Students will<br />

write sum of roots & product of roots. They will generalize the relationship<br />

between a, b & c with the sum of roots & product of roots for any quadratic<br />

equation ax 2 +bx+c=0.<br />

Execution: Teacher may distribute the photocopies of worksheets and students will solve the<br />

questions on their worksheets individually.<br />

Parameters for Assessment:<br />

Students will be able to:<br />

• Recognize coefficients of x 2 , x and constant terms.<br />

• Find roots of a quadratic equation.<br />

• Calculate sum of roots and products of roots correctly.<br />

• Relate sum & product of roots with the coefficients of x 2 , x and constant terms.<br />

Extra Reading:<br />

Sum of roots, product of roots and discriminant formula<br />

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

Solving quadratic equation<br />

http://library.thinkquest.org/20991/alg2/quad.html<br />

22


Activity 10‐ Content Worksheet (CW6)<br />

Formation of a <strong>Quadratic</strong> Equation<br />

Specific Objective:<br />

1. To form a quadratic equation if sum of roots and product of roots are known.<br />

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

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

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

equation of Reeta. Reeta lost his equations. In this task students are supposed to<br />

trace Reeta’s equations by making use of their roots.<br />

Execution:<br />

Teacher may show the dialogue script to the class on projector. Students will<br />

solve and write their equations in their notebooks.<br />

Parameters for Assessment:<br />

Students will be able to:<br />

• Find sum of roots of a quadratic equation.<br />

• Find product of roots of a quadratic equation.<br />

• Form a quadratic equation with given roots.<br />

23


Activity 11‐ Content Worksheet (CW7)<br />

Application of <strong>Quadratic</strong> Equation in Real Life Problems<br />

Specific Objective:<br />

1. To solve the problems from real life situations having application of quadratic equations.<br />

Description: The task is designed to make students apply quadratic equations to solve real life<br />

situations. Students will frame the quadratic equation after comprehending the<br />

word problem. They will then solve the equations to get the answers of the<br />

problems.<br />

Execution:<br />

Teacher may distribute the photocopies of worksheets and students will solve the<br />

questions on their worksheets individually.<br />

Parameters for Assessment:<br />

Students will be able to:<br />

• Comprehend the word problem.<br />

• Express the word problem in the form of quadratic equation.<br />

• Solve the quadratic equation.<br />

• Write the answer of the given word problem.<br />

Extra Reading:<br />

http://www.youtube.com/watchv=EhPPci8shA8<br />

http://www.youtube.com/watchv=Vu3px08WX_8<br />

http://www.youtube.com/watchv=lS9S1iEjlPI<br />

24


Activity 12‐ Post Content Worksheet (PCW1)<br />

Students will be assessed on the worksheet containing questions based of recognizing<br />

quadratic polynomials & quadratic equations.<br />

Activity 13‐ Post Content Worksheet (PCW2)<br />

Students will be assessed on the worksheet containing questions based on nature of roots of a<br />

quadratic equation.<br />

Activity 14‐ Post Content Worksheet (PCW3)<br />

Students will be assessed on the worksheet containing questions based on finding the roots of<br />

a quadratic equation.<br />

Activity 15‐ Post Content Worksheet (PCW4)<br />

Students will be assessed on the worksheet containing questions based on the complete<br />

chapter including application of quadratic equations in solving real life problems.<br />

• Read about the applications of <strong>Quadratic</strong> equations in real life at<br />

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

25


Assessment guidance plan for teachers<br />

Assessment Plan<br />

2.22 Assessment Plan<br />

Assessment guidance plan for teachers<br />

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

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

These rubrics are meant to develop the learner as the self motivated learner.<br />

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

formative assessment and one is for summative assessment.<br />

Suggestive Rubric for Formative Assessment (exemplary)<br />

Parameter Mastered Developing Needs motivation Needs personal<br />

attention<br />

Method of<br />

Able to apply<br />

Able to apply<br />

Able to apply<br />

Not Able to apply<br />

solving<br />

factorization<br />

factorization<br />

factorization<br />

factorization<br />

quadratic<br />

method to solve<br />

method to<br />

method to solve<br />

method to solve<br />

equations .<br />

the quadratic<br />

solve the pair of<br />

the quadratic<br />

the quadratic<br />

equations , get<br />

linear<br />

equations ,not able<br />

equations.<br />

correct values of x,<br />

equations ,get<br />

to get the correct<br />

,can verify the<br />

correct value of<br />

value of x<br />

correctness of<br />

x, cannot verify<br />

solution<br />

the correctness<br />

of solution<br />

Able to apply<br />

Able to apply<br />

Able to find<br />

Not Able to apply<br />

discriminant<br />

discriminant<br />

discriminant to<br />

discriminant<br />

method to solve<br />

method to<br />

solve the quadratic<br />

method to solve<br />

26


the quadratic<br />

solve the<br />

equations , but<br />

the quadratic<br />

equations ,get<br />

quadratic<br />

cannot get correct<br />

equations .<br />

correct values of x,<br />

equations ,get<br />

values of x, cannot<br />

can verify the<br />

one correct<br />

verify the<br />

correctness of<br />

value of x,<br />

correctness of<br />

solution<br />

cannot verify<br />

solution<br />

the correctness<br />

of solution<br />

From above rubric it is very clear that<br />

• Learner requiring personal attention is poor in concepts and requires the training of basic<br />

concepts before moving further.<br />

• Learner requiring motivation has basic concepts but face problem in calculations or in<br />

making decision about suitable substitution etc. He can be provided with remedial<br />

worksheets containing methods of solving the given problems in the form of fill‐ups.<br />

• Learner who is developing is able to choose suitable method of solving the problem and is<br />

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

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

improve the results. Learner at this stage may not have any mathematical problem but may<br />

have attitudinal problem. Mathematics teachers can avail the occasion to bring positive<br />

attitudinal changes in students’ personality.<br />

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

equations in two variables.<br />

27


Teachers’ Rubric for Summative Assessment of the <strong>Unit</strong><br />

Parameter 5 4 3 2 1<br />

Solving the<br />

quadratic<br />

equation<br />

Nature of<br />

roots<br />

Forming<br />

quadratic<br />

equation<br />

• Able to solve quadratic<br />

equation by factorization<br />

method<br />

• Able to identify which<br />

quadratic equation cannot be<br />

solved by factorization.<br />

• Able to apply discrminant<br />

method to solve the<br />

quadratic equation<br />

• Can find the discriminant<br />

accurately<br />

• Can predict the nature of<br />

roots on the basis of<br />

discriminant<br />

• Can state the conditions for<br />

existence of real roots<br />

• Can find the unknown<br />

coefficient for prescribed<br />

nature of roots<br />

• Can state the formula for<br />

forming quadratic equation<br />

• Can form correct quadratic<br />

equation when both roots are<br />

known<br />

• Not able to solve quadratic<br />

equation by factorization<br />

method<br />

• Not able to identify which<br />

quadratic equation cannot be<br />

solved by factorization.<br />

• Not able to apply discrminant<br />

method to solve the quadratic<br />

equation<br />

• Cannot find the discriminant<br />

accurately<br />

• Cannot predict the nature of<br />

roots on the basis of<br />

discriminant<br />

• Cannot state the conditions for<br />

existence of real roots<br />

• Cannot find the unknown<br />

coefficient for prescribed<br />

nature of roots<br />

• Cannot state the formula for<br />

forming quadratic equation<br />

• Cannot form correct quadratic<br />

equation when both roots are<br />

known<br />

28


Application<br />

in word<br />

problems<br />

• Can form quadratic equation<br />

with rational real<br />

co‐effecients when one<br />

irrational root is given<br />

• Able to identify the variables<br />

from given statement<br />

• Able to form quadratic<br />

equations correctly from<br />

given statement<br />

• Able to solve the equations by<br />

any of the above methods<br />

• Able to verify the solution.<br />

• Cannot form quadratic<br />

equation with rational<br />

co‐effecients when one<br />

irrational root is given<br />

• Not able to identify the<br />

variables in given statement<br />

• Not able to form quadratic<br />

equation correctly from given<br />

statement<br />

29


STUDY<br />

MATERIAL<br />

30


<strong>Quadratic</strong> <strong>Equations</strong><br />

Introduction<br />

You have already studied about linear equations in one and two variables. You have also<br />

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

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

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

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

water.”<br />

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

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

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

Now, time taken to cover 30 Km downstream =<br />

And time taken to cover 30km upstream = <br />

hours<br />

So, total time= <br />

+ <br />

hours<br />

Therefore, according to the given condition,<br />

<br />

hours<br />

<br />

+ <br />

= 4<br />

<br />

or<br />

<br />

+ <br />

= <br />

<br />

or<br />

<br />

= <br />

or<br />

<br />

= <br />

<br />

31


Or<br />

120x = 9x 2 −225<br />

Or 9x 2 −120x −225=0<br />

Or 3x 2 −40x −75=0<br />

(I)<br />

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

such problems Definitely not, because this equation is different from those equations which<br />

we have solved so far. That is, it is not a linear equation.<br />

However, this equation appears to have some resemblance with the polynomials of the type<br />

ax 2 +bx + c. Can you recall that polynomials of the type ax 2 +bx + c are called quadratic<br />

polynomials<br />

Keeping in view this rememblance, equations of the type 3x 2 −40x −75=0 are called quadratic<br />

equations in one variable . In general, an equation of the type<br />

ax 2 + bx +c =0 is called a quadratic equation in one variable, where a, b, and c are real numbers<br />

and a ≠ o.<br />

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

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

the roots and coefficients of given quadratic equation and solve some daily life problems,<br />

with the help of quadratic equations.<br />

(1) Introductions to <strong>Quadratic</strong> <strong>Equations</strong><br />

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

equations of the type 3x 2 – 40x −75=0 which are of the form ax 2 +bx+c=0, where a, b and c<br />

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

variable. ax 2 +bx+c=0 is sometimes also referred to as the general form or the standard<br />

form of a quadratic equation; a is called the coefficient of x 2 , b the coefficient of x and c is<br />

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

32


Some more examples of quadratic equations are 3x 2 − 5x+7=0 , −7x 2 +2x + 8 = 0,<br />

2y 2 + 3y −8 = 0, x 2 −2x +3=0, x 2 − 4 = 0, 2y 2 −3y = 0, 8z 2 + 4z + 9 = 0.<br />

Let us now consider an example to identify the quadratic equations in one variable.<br />

Example 1:‐<br />

Which of the following are quadratic equations in one variable and which are not<br />

(i)<br />

3x 2 +7x−2=0<br />

(ii) (x+2) 2 =x (x+1) +2<br />

(ii) 2x 3 +5x 2 −6=2x 2 (x−2) +4x<br />

(IV) 3x 2 −5=0<br />

(v)<br />

(vi)<br />

5y 2 −12y−9=0<br />

(2z+5) 2 ‐5z = z (4z+9)<br />

(vii) 2p −7 =9p 2<br />

(viii) (2p −7) 2 = 4p (p 2 – 7)<br />

Solutions:‐<br />

(i) 3x 2 + 7x − 2 = 0 is <strong>Quadratic</strong> equation as it is of the form ax 2 +bx +c=0<br />

(ii) (x+2)² = x (x+1) +2 gives<br />

x 2 + 4x + 4 = x 2 + x + 2<br />

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

(iii) From 2 + 5x −6 = 2x 2 (x−2) + 4x, we have :<br />

2x 3 + 5x −6 = 2x 3 − 4x 2 + 4x<br />

Or 2x 3 −2x 3 + 4x 2 + 5x − 4x − 6 = 0<br />

or 4x 2 + x−6=0, so it is a quadratic equation.<br />

(iv) 3x 2 − 5 = 0. It is of the form ax 2 + bx + c = 0,<br />

Where a = 3, b = 0and c = −5.<br />

33


So, it is a <strong>Quadratic</strong> equation.<br />

(v) 5y 2 −12y − 9 = 0. It is of the form ax 2 + bx + c = 0.<br />

So, it is a quadratic equation.<br />

(vi) From (2z+5) 2 −5z = z (4z+9), we have<br />

4z 2 +20z + 25−5z = 4z 2 + 9z<br />

or 4z 2 +15z +25 −4z 2 − 9z = 0<br />

or 6z + 25 = 0. So it is not a quadratic equation.<br />

(vii) From 2p −7 = 9p 2 , We have<br />

−9p 2 + 2p −7 = 0. So, it is a quadratic equation.<br />

(viii) From (2p −7) 2 = 4p (p 2 −7), we have:<br />

4p 2 − 28p + 49 = 4p 3 −28p<br />

or 4p 3 − 4p 2 − 49 = 0.<br />

It is not of the type ax 2 + bx + c=0 (though it has three terms in L H S). So, it is not a<br />

quadratic equation.<br />

(2) Solving a quadratic Equation<br />

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

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

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

substituted in the equation we get L H S = R H S.<br />

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

consider the equation 3x 2 − 40x −75=0<br />

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

L H S = 3 x15 2 − 40 x 15 − 75<br />

=675 − 600 −75 = 0 = R H S.<br />

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

34


Again, let us substitute x = <br />

<br />

in the L H S of above equation. We have:<br />

L H S = 3 x <br />

2 −40 x 5<br />

3 −75<br />

= <br />

+ <br />

− 75<br />

= <br />

<br />

– 75 = 75 −75 = 0 = R H S.<br />

Thus, we can also say that x = <br />

is also a solution of the equation 3x 2 − 40x −75=0<br />

Let us check whether x = 2 is a solution of this equation or not<br />

L H S = 3x 2 – 40x −75<br />

= 3(4) – 40(2) – 75<br />

= 12 − 80 −75<br />

= −143<br />

R H S = 0<br />

So, L H S ≠ R H S<br />

Hence x = 2, is not a solution of the given equation.<br />

<br />

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

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

3x 2 − 40x −75 =0. But the problem before us is how to find such solutions. Let us discuss<br />

method of factorization of solving a quadratic equation.<br />

Factorization Method<br />

You are already familiar with the factorization of the trinomials of the type ax 2 + bx + c, by<br />

splitting the middle term. In the factorization method of solving a quadratic equation, we<br />

first write the equation in the standard form ax 2 + bx + c = 0 and factorize the L H S of the<br />

equation by splitting the middle term. We explain the process by taking again quadratic<br />

equation 3x 2 − 40x −75=0,<br />

We have 3x 2 − 40x −75 = 0 splitting − 40x as − 45x + 5x<br />

35


Or 3x 2 − 45x + 5x −75=0<br />

Why Because −45 x 5 = 3 x (‐75)<br />

Or 3x (x−15) + 5 (x−15) = 0<br />

Or (x−15) (3x+5) =0<br />

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

zero.)<br />

So , (x−15)=0 or (3x+5)=0<br />

i.e. x=15 or x = <br />

<br />

So, the required solution of two quadratic equations3 — 40 75 0 <br />

15 <br />

.<br />

Note: .<br />

How many solutions are possible for<br />

a quadratic equation<br />

Are there always two solutions<br />

: Solve the following quadratic equation by factorization method:<br />

i x 6x50<br />

ii 5x 3x20<br />

iii 8x 22x 21 0<br />

36


iv 6x x20<br />

v z 60<br />

vi y ‐ 2√3x +3=0<br />

vii abx +(b acxbc0<br />

viii 2y aya 0<br />

ix 5z <br />

<br />

z<br />

0<br />

x 3√2 x 32√3x30<br />

: i we have:<br />

x 6x50<br />

Or x 5xx50 splitting the middle term 6 x<br />

Or xx 5 1 x 5 0<br />

Or x 5 x 1 0<br />

so, x 5 0 or x1 0<br />

x5 or x1<br />

Thus x 5 and x 1 are solution of the given equations<br />

Check; LHS 5 65 50RHS,when x 5<br />

LHS 1 61 50RHS, when x = −1<br />

ii 5x 3x20<br />

Or 5x 5x2x20 (splitting the middle term)<br />

Or 5xx1 2 x 1 0<br />

Or x 15x 2 0<br />

so, x 1 0 or 5x 2 0<br />

i. e., x 1 or x <br />

<br />

so, 1 and x <br />

<br />

are solutions of the equation<br />

Check: LHS 5 1 31 20RHS<br />

37


and LHS 5 2<br />

5 3 2<br />

5 2<br />

= <br />

2<br />

= 0RHS<br />

iii 8x 22x 21 0<br />

Or 8x 28x 6x 21 0<br />

Or 4x2x 7 32x 7 0<br />

Or 2x 7 4x 3 0<br />

so,2x70 or 4x30<br />

i. e, x <br />

so, x <br />

or x <br />

<br />

<br />

and x <br />

are solution of the equation.<br />

iv<br />

6x x20<br />

Or 6x 4x3x20<br />

Or 2x3x 2 13x 2 0<br />

Or 3x 22x 1 0<br />

so, 3x 2 0 or 2x 1 0<br />

i. e. x 2 3 or x 1 2<br />

so, x 2 3<br />

and x 1 2<br />

are solution of the equation.<br />

v<br />

z 60<br />

Or z 6 z 6 0<br />

so, z 6 0 or z 6 0<br />

i. e z √6 or z √6<br />

38


so, z √6 and z √6 are solution of the equation<br />

vi y 2√3 y 3 0<br />

Or y √3y √3y 30<br />

Or yy 3 − √3 (y − 3 =0<br />

Or y 3 (y −3 = 0<br />

so, y √3 0 or y√3 0<br />

i. e. y √3 or y √3<br />

Note In this case the two solutions are the same.<br />

vii<br />

abx b acx bc0<br />

Or abx b xacx bc0<br />

Or bxax b c ax b 0<br />

so, ax b 0 or bx c 0<br />

i. e. x or x <br />

so, x and x <br />

are solution of the equation.<br />

Check for x ,<br />

LHS ab ² b ac – bc<br />

<br />

³<br />

³<br />

<br />

bc bc<br />

0 RHS<br />

For x= ,<br />

39


LHS ab ( )2 + (b 2 –ac) ( – bc<br />

ac<br />

b bcac b<br />

bc0<br />

RHS<br />

x = and x = both are solutions of the equation.<br />

<br />

viii<br />

2y ay a 0<br />

Or 2y 2ayay a 0<br />

Or 2yy a ay a 0<br />

Or y a 2y a 0<br />

so, y a and y are solution of the equation<br />

<br />

ix<br />

5z z 0<br />

Or 5z <br />

z <br />

0<br />

Or 5z z 2z 3 0<br />

<br />

Or 5z z3 2 0<br />

so, 5z 0 or z 0<br />

i. e, z 1 10 or z <br />

so, z 1 10 and z 3 2<br />

are solutions of the equation<br />

x 3√2 x 3 √2 √3 x −3 =0<br />

Or 3√2 x 3√3x √6x 30<br />

Or 3x(√2x √3 √3 √2x √3 0<br />

40


Or (√2x √3 (3x −√3)=0<br />

so, √2x √3 0 or 3x√3 =0<br />

i. e., x or x 1 √3<br />

so, x <br />

and x 1 √3<br />

are solutions of the eqution.<br />

You are advised to check the solutions you obtain, with the given original equation.<br />

: <br />

You are already familiar with the identities<br />

² 2 2 <br />

We use these identities in solving a quadratic equation by the method of completing the<br />

square. Let us explain it through an example:<br />

Consider the equation<br />

3 40 75 0<br />

or 40 3<br />

25 0 Divide the equation by 3<br />

40 3 40 40<br />

² ² 250 (Adding and Subtracting (½ of the<br />

2 x 3 2 x 3<br />

coefficient of x)² in L H S<br />

or 20<br />

3<br />

or 20<br />

3 25 3<br />

² <br />

400 225<br />

9<br />

625<br />

9<br />

<br />

25<br />

3 <br />

, 20 3 25 3 20 3 25 3<br />

15 <br />

<br />

We have again, got two solutions as obtained earlier.<br />

41


Let us now consider the general or standard form of equation 0<br />

We have:<br />

0<br />

Or 0 <br />

Or x 2 + 2<br />

⎛ b ⎞<br />

<br />

⎜<br />

⎝<br />

⎟<br />

2a ⎠<br />

<br />

⎛ b ⎞<br />

⎜<br />

⎝<br />

⎟<br />

2a ⎠<br />

2<br />

= 0 (Adding and subtracting the square of <br />

of the<br />

coefficient of 2x) in LHS)<br />

Or<br />

2 2<br />

⎛ b ⎞ ⎛ b ⎞<br />

⎜x<br />

+ ⎟ − ⎜ ⎟ + ⎝ 2a ⎠ ⎝ 2a ⎠ <br />

= 0 (we obtain a complete & same a ( <br />

<br />

)2<br />

2<br />

⎛ b ⎞ b² c<br />

or ⎜x<br />

+ ⎟ = −<br />

2<br />

⎝ 2a ⎠ 4a a<br />

Or<br />

2 2<br />

⎛ b ⎞ b − 4ac<br />

⎜x<br />

+ ⎟ =<br />

2<br />

⎝ 2a ⎠ 4a<br />

2 2<br />

⎛ b ⎞ ⎛± b − 4ac⎞<br />

or ⎜x<br />

+ ⎟ =<br />

⎝ 2a ⎠ ⎜ 2a ⎟<br />

⎝ ⎠<br />

2<br />

Or<br />

2 2<br />

b b −4ac b b − 4ac<br />

x + = −<br />

or x + =<br />

2a 2a 2a 2a<br />

Or<br />

− − − − −<br />

x = or x =<br />

2a 2a<br />

2 2<br />

b b 4ac b + b 4ac<br />

Note that<br />

√ 4<br />

2<br />

√ 4<br />

2<br />

.<br />

<br />

42


x = √ <br />

<br />

√ <br />

<br />

are the two roots of the equation.<br />

, 5 5<br />

3 3 40 75 0<br />

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

Sridhar (around 1029AD)<br />

Example 3:<br />

Solve the equation<br />

6 5 0 .<br />

Solution: 6 2 − 2 + 5=0<br />

Or 69950<br />

Or 3 2 = 4<br />

Or<br />

:<br />

Solution:<br />

or<br />

3 √4 2<br />

, 3 2 or 32<br />

1 5 [See example 2 (i)]<br />

1, 5 <br />

1, 5 .<br />

5 −3x−2=0 by the method completing the square.<br />

5 320<br />

+ − = (why)<br />

or<br />

or<br />

or<br />

or<br />

<br />

<br />

0<br />

<br />

<br />

100 0<br />

= <br />

<br />

<br />

10 =1<br />

43


or<br />

<br />

<br />

<br />

1 <br />

<br />

1 , 2 <br />

5<br />

Example 5:<br />

Solve the equation 8 22 21 0 .<br />

Solution: Comparing the given equation with , 8, 22, 21.<br />

We have,<br />

√ <br />

<br />

(<strong>Quadratic</strong> Formula )<br />

²<br />

<br />

22 484672<br />

16<br />

<br />

<br />

√<br />

<br />

<br />

<br />

So, = <br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

, 7 2 3 4<br />

<br />

7 3<br />

<br />

2 4<br />

Example 6: Solve the equation 3√2 ² 3 √2) √3 3 0 By using the quadratic<br />

formula<br />

Solution :<br />

Here a = 3√2 , √3 3 2 =3 √3 √6 , 3<br />

44


√ <br />

<br />

(<strong>Quadratic</strong> formula)<br />

=<br />

√ √√² <br />

√<br />

=<br />

√ 27√√<br />

√<br />

=<br />

√ √<br />

√<br />

=<br />

√ √²<br />

√<br />

=<br />

√ √<br />

√<br />

, √<br />

√<br />

√<br />

<br />

<br />

√<br />

<br />

√<br />

√<br />

<br />

thus, <br />

√ and <br />

<br />

are the roots of the given equation.<br />

Example 7 : Solve the equation 2√3 3 0 .<br />

Solution: here 1,2√3, c=3<br />

So roots are<br />

<br />

√ 4<br />

2<br />

√ 4<br />

2<br />

45<br />

, √ 4<br />

2


. <br />

23 23² 413<br />

21<br />

,<br />

23 23² 413<br />

21<br />

. <br />

2√3 0<br />

2<br />

,<br />

2√3 0<br />

2<br />

. . √3 , √3 2 <br />

(4) Nature of Roots<br />

You have seen that the roots of the quadratic equation<br />

,:<br />

0 , <br />

√ 4<br />

2<br />

, √ 4<br />

2<br />

Observe that:<br />

In Example 5, the value of b 4ac 1156 0 <br />

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

<br />

√<br />

<br />

In Example 7, the value of<br />

4 0 √3 , √3<br />

(i)<br />

(ii)<br />

We can say that roots of a quadratic equation are real and unequal or distinct, if<br />

4 0<br />

roots of a quadratic equation are real and equal (coincident) if 40<br />

What will happen, if 4 0, , 36 Can you find √ 4 i.e. √36 .<br />

Can you find a real number whose square is −36<br />

There is no real number whose square is −36.<br />

So,<br />

(iii) Roots of a quadratic equation are not real when 40.<br />

46


Note that the nature of roots of a quadratic equation are depending on the value of the<br />

expression 4. That is why we call as the discriminant of the quadratic<br />

equation , where a, b,c are real numbers with a 0.<br />

We denote the expression 4 .<br />

So, D is the discriminant of the equation.<br />

Let 0. a 0<br />

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

if D=0, then the equation has two coincident real roots<br />

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

A quadratic equation can have at the most two roots<br />

Example 8: Find the discriminant of the following quadratic equations:<br />

(i)<br />

(ii)<br />

3 520<br />

2 630<br />

(iii) 5 270<br />

(iv) 20<br />

Solution:<br />

(i) Here, 3 , 5 , 2.<br />

iscriminantb 4 5² 432 25241<br />

(ii) Here , a 2 , 6 , 3<br />

b 4 6 2 423 362412<br />

(iii)<br />

5 270<br />

Here 5 , 0 . 27<br />

47


404527 540<br />

(iv)<br />

20<br />

Here, 1 , 2 , 0 <br />

b 44410 4<br />

Example 9: State the nature of roots of the following equations:<br />

(i) 2 10<br />

(ii) 10<br />

(iii) 9 12 4 0<br />

(iv) 2 550<br />

(v) 410<br />

(vi) 16 40 25 0<br />

Solution: (i) 2 10<br />

Here 2 , 1 , 1<br />

b 41 421 1890<br />

So, the equation has two distinct real roots<br />

(ii) 10<br />

Here , 1 , 1 , 1<br />

b 41411 30,<br />

So, the equation does not have real roots<br />

(iii) 9 12 4 0<br />

Here, 9 , 12 , 4<br />

b 4 12 2 4(9)(4) = 144−144=0<br />

So, the equation has two coincident real roots<br />

(iv) 2x 2 +5x + 5=0<br />

Here a=2, b=5, c=5<br />

48


D=b 2 −4ac = 2574 (2) (5) = 15


As, the roots are distinct and real, or 0<br />

. . 16 4 0 16 4 4 16 4<br />

(iii) 9 12 0 .<br />

9,12,<br />

144 36<br />

Since the roots are not real, so<br />

0<br />

or 144 36 0<br />

or 144144 or k>4<br />

(v)<br />

√kz 80<br />

, √ 2 418<br />

= k32<br />

Since, roots are real, D 0<br />

Therefore 320 320<br />

i.e. k>32 or k=32<br />

, 32<br />

(5) Relation between the Roots and Coefficients of a <strong>Quadratic</strong> Equation<br />

You know that roots of the equation<br />

0<br />

0<br />

are √ <br />

<br />

and<br />

√ <br />

<br />

We see that<br />

Sum of the roots<br />

= √ <br />

<br />

√ <br />

<br />

50


= √ √ <br />

<br />

<br />

<br />

<br />

<br />

or Cf<br />

Coefficients <br />

Product of the roots<br />

= √ <br />

<br />

√ <br />

<br />

<br />

= <br />

√ <br />

<br />

√ <br />

<br />

<br />

=<br />

2<br />

2 2<br />

⎛ b ⎞<br />

⎜ − ⎟<br />

⎝ 2a ⎠ – ⎛ b − 4ac ⎞<br />

⎜ 2a ⎟<br />

⎝<br />

⎠<br />

<br />

=<br />

<br />

<br />

<br />

= <br />

<br />

. .<br />

<br />

Cf <br />

Thus, if pand q are roots of 0, 0<br />

Then <br />

<br />

i.e.<br />

Cf <br />

<br />

Cf <br />

And pq <br />

. . <br />

<br />

Cf <br />

• In Example 5, the roots are <br />

and<br />

<br />

<br />

Sum of the roots = <br />

Also<br />

Cf <br />

Cf <br />

= <br />

<br />

<br />

<br />

51


Thus, we have verified that<br />

Cf <br />

Sum of roots = <br />

Cf <br />

Similarly, Product of roots = <br />

= C <br />

Cf <br />

Example 11: Find the roots of the equation<br />

9 320<br />

And verify the relations between the roots and coefficients.<br />

Solution:<br />

Roots are given by<br />

x √ <br />

<br />

=<br />

=<br />

<br />

<br />

√<br />

<br />

=<br />

<br />

<br />

, <br />

<br />

Sum of roots =<br />

<br />

<br />

Also, sum of roots =<br />

Cf <br />

Coefficients <br />

= <br />

<br />

<br />

Coefficients <br />

Hence, Sum of roots =<br />

Cf <br />

<br />

52


Similarly,<br />

Product of roots = <br />

<br />

<br />

<br />

Example 12:<br />

<br />

Also<br />

<br />

Cf <br />

<br />

So, product of roots =<br />

Cf <br />

<br />

Find the roots of the equation<br />

410<br />

And hence verify the relations between the roots and coefficients.<br />

Solution:<br />

Roots are given by<br />

√ <br />

<br />

<br />

√<br />

<br />

√<br />

<br />

2 √3<br />

So, the roots are 2√3 and 2 √3<br />

Sum of roots 2√3 2√3=4<br />

Also<br />

Coefficients <br />

Cf = <br />

4<br />

<br />

Cf <br />

Hence, sum of roots =<br />

Cf <br />

Similarly the product of roots<br />

2 √32 √3 431<br />

• Note that roots of equation are<br />

2√3 2√3<br />

Such roots are called conjugate roots.<br />

Constant term<br />

Coefficients of x 2<br />

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

53


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

What will be the other root<br />

Solution:<br />

Since, irrational roots occur in conjugate pairs when coefficients are rational<br />

number, so the other root will be<br />

3√15<br />

(5) Forming a <strong>Quadratic</strong> Equation with given roots<br />

We know that for the quadratic equation<br />

0 ,0,<br />

<br />

<br />

Now 0 , written as<br />

0 , 0<br />

Or 0<br />

Or<br />

0<br />

Example 14: Form the quadratic equation whose roots are 6 and ‐3<br />

Solution:<br />

6 3 3<br />

6 3 18<br />

So, required quadratic equation will be<br />

3180<br />

54


Alternatively, the equation with roots 6 and ‐3 can also be found as<br />

6 3 =0<br />

or<br />

3180<br />

Note that the equations<br />

2 318 0 ,<br />

3 318 0<br />

<br />

318 0<br />

<br />

2 318 0 etc. all have the same roots 6 and −3 (check!!)<br />

In fact, k 318 0 6 3 .<br />

Example 15: Find the quadratic equation whose roots are 1 √5 1 √5<br />

Solution:<br />

Sum of roots =1 √5 1√5 2<br />

1 √5 1√5 154<br />

So, the required equation is<br />

2 40<br />

Alternatively, the quadratic equation with<br />

1 √5 1 √5 <br />

1 √5 1 √5 0<br />

. 240<br />

Note that the roots 1√5 1 √5 are called conjugate roots and they always occur in<br />

pairs if the coefficients of the quadratic equation are rational numbers. In other words, if the<br />

55


coefficients of the quadratic equation are rationals and one of the roots is a√ , then the<br />

other roots must be √ .<br />

Example 16: Find the quadratic equation the sum of whose roots is <br />

and the Product of the<br />

roots is 5 9<br />

Solution:<br />

the required equation is 6 7 5 9 0<br />

or 63 54 35 0<br />

(6) Applications of <strong>Quadratic</strong> <strong>Equations</strong><br />

Now, we will discuss applications of quadratic equations in solving problems related to daily<br />

life.<br />

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

numbers.<br />

Solution:<br />

Let the three consecution natural numbers be , 1,2.<br />

According to given condition,<br />

1 2 2 2 =110<br />

Or 2350<br />

Or 75350<br />

Or 7 5 7 0<br />

Or 7 5 0<br />

So, 7 5<br />

It cannot be ‐7, (why) .<br />

Thus x=5,<br />

And the numbers are 5, 5 1, 5 2, . 5,6,7,<br />

56


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

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

motorboat in still water.<br />

Solution:<br />

Let r be the speed of the boat in still water.<br />

Down the river (with<br />

the current)<br />

Upriver (against the<br />

current)<br />

Distance Speed Time<br />

12 km r+2 12<br />

2<br />

12 km r−2 12<br />

2<br />

Thus<br />

12<br />

2 12<br />

2 5<br />

Or <br />

<br />

=5<br />

Or 24 5 2 2<br />

Or 24 5 4<br />

Or 5 24 20 0<br />

Therefore ²<br />

<br />

√<br />

<br />

5.5 0.7 <br />

Since speed cannot be negative, so<br />

5.5 km/hr<br />

57


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

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

other could do it alone. How long would it take each pump to complete the job alone<br />

Solution:<br />

Let t be the number of hours for slower pump to complete the work itself.<br />

Then 2 would be the number of hours for the faster pump to complete the job<br />

by itself.<br />

So,<br />

<br />

<br />

1<br />

<br />

or<br />

<br />

<br />

1<br />

or<br />

or<br />

<br />

2 1<br />

12 12 2<br />

or ² 14 12 0<br />

t ²<br />

<br />

√<br />

<br />

13.1 0.9 .<br />

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<br />

reject the value of t=0.9.<br />

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

the job.<br />

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

added to the number the digits interchange their places. Find the numbers.<br />

Solution: Let digit at ten’s place be x and at unit’s place be y.<br />

So, the number = 10 <br />

When digits are interchanged, the new number 10 <br />

According to the problem,<br />

xy 12<br />

1<br />

58


and 10 36 10 <br />

Or 9 9 36<br />

Or 4 2<br />

Putting the value of from (2) in (1).<br />

4 12<br />

Or 4 12 0 )<br />

∴ √<br />

<br />

= √64<br />

<br />

= <br />

<br />

6,2<br />

Rejecting 2 . , we get<br />

6<br />

So, <br />

<br />

Thus, the number is 26.<br />

2 [(using (1)]<br />

Check : Product of digits =2 612<br />

Also<br />

26+36=62<br />

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

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

Solution:<br />

Let father’s age be years, then daughter’s age = 45 years<br />

5 Years ago<br />

Father’s age = 5 .<br />

Daughter’s age =45 5.<br />

59


= (40x) years<br />

According to the problem,<br />

Product of<br />

their ages (in<br />

years)<br />

540 = 4 5<br />

4 times the<br />

father’s age (in<br />

years)<br />

40 200 5420<br />

45 200 4 20<br />

41 180 0<br />

41 180 0<br />

So,<br />

41 <br />

<br />

= <br />

<br />

36,5<br />

x can not be 5 (why )<br />

Thus, 36<br />

i.e father’s present age =36 years and daughter’s presents age= 9 years.<br />

Check :<br />

(i) Sum of Ages = 36 + 9 = 45<br />

(ii )<br />

5 years ago, father’s age = 36‐5=31 years<br />

Daughter’s age = 9‐5 =4 years.<br />

So, 31 4 =4x <br />

= 4 31<br />

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

level, after seconds is given by<br />

5 200<br />

Determine, in how many seconds will the rocket be 1875 metres above the ground.<br />

60


Solution: Here y=1875<br />

So, we have<br />

1875 5 200<br />

Or 40 375 0<br />

Or 15 25 0<br />

i.e. 15 25.<br />

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

Check: 515 20015 1875<br />

525² 20025 1875<br />

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

and other while coming down.<br />

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

sides AB and EF.<br />

Solution:<br />

As triangles ABC and CEF are similar<br />

∴ we have<br />

<br />

<br />

<br />

61


= 8 <br />

Or x (x −3) = 40<br />

Or x² − 3x − 40 =0<br />

Or (x−8) (x+5) = 0<br />

∴ x −8 =0 or x + 5 = 0<br />

∴ x = 8 or x = − 5<br />

x cannot be equal to −5 (why).<br />

So, x = 8.<br />

Hence AB = 8 − 3 = 5 units<br />

EF = 8 units.<br />

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

62


STUDENT’S<br />

SUPPORT<br />

MATERIAL<br />

63


Student’s Worksheet 1 (SW1)<br />

Warm up Worksheet (W1)<br />

Identify the Polynomials of Degree 2<br />

Name of Student___________<br />

Date________<br />

Send the balloons to their respective homes<br />

0x 2 ‐3x+4<br />

2x ‐7/4 ‐5x+7<br />

x 2 ‐3x‐4<br />

x 2 ‐x‐72<br />

5x 2 ‐<br />

7x 3 +2x+2<br />

2√2x 2 ‐3x‐5<br />

3x 2 +4x‐2<br />

x 2 ‐2x‐3<br />

x 2 ‐4x+4<br />

5x 1/2 ‐3x+4<br />

x 2 ‐5x +6<br />

64


NON<br />

POLYNOMIALS<br />

POLYNOMIALS<br />

OF DEGREE2<br />

POLYNOMIALS OF<br />

DEGREE OTHER<br />

THAN2<br />

Self Assessment Rubric<br />

Parameters of assessment<br />

Understanding of the difference<br />

between a polynomial and an<br />

algebraic expression<br />

Finding out the degree of a given<br />

polynomial<br />

65


Student’s Worksheet 2 (SW2)<br />

Warm up Worksheet (W2)<br />

Recognize Zeroes of Polynomials<br />

Name of Student___________<br />

Date________<br />

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

the zeroes of the polynomials in the table below. The first one is done for you<br />

Factors Polynomials Zeroes of polynomials<br />

x‐2, x+3<br />

(x‐2)(x+3)=x 2 +x‐6<br />

2,‐3<br />

2x‐3, x+2<br />

5x+4, 4x+1<br />

x‐3, 7x‐1/2<br />

x+5/2, 3x+2<br />

7x‐1, 3x‐2<br />

3x‐2,3x+2,2x+1<br />

66


Self Assessment Rubric<br />

Parameters of assessment<br />

Understanding that a polynomial<br />

is formed as a product of the given<br />

factors<br />

Reading the zeroes of a<br />

polynomial from the given factors<br />

67


Student’s Worksheet 3 (SW3)<br />

Pre Content Worksheet (P1)<br />

<strong>Quadratic</strong> <strong>Equations</strong> from <strong>Quadratic</strong> Polynomials<br />

Name of Student___________<br />

Date________<br />

Task 1<br />

Referring to the activity done in the warm up worksheet 1, answer the following questions:<br />

1. What are the expressions in the balloons called<br />

___________________________________________________________________<br />

2. What are the polynomials with degree 2 called<br />

___________________________________________________________________<br />

Write the quadratic expression of your choice in the table below.<br />

Now form the quadratic equations from the quadratic expressions.<br />

<strong>Quadratic</strong> Expression, Q(x)<br />

<strong>Quadratic</strong> Equation, Q(x)=0<br />

68


Task 2<br />

Identify <strong>Quadratic</strong> Polynomials which cannot be factorized<br />

Referring to the warm up worksheet, factorize the quadratic expressions by using the splitting<br />

of middle term method. Is there any quadratic polynomial which cannot be factorized List<br />

them down here. We shall talk about them later.<br />

List the quadratics which could not be factorized:<br />

______________________________________________________________________________<br />

______________________________________________________________________________<br />

______________________________________________________________________________<br />

69


Self Assessment Rubric<br />

Parameters of assessment<br />

Understanding of the concept of<br />

a polynomial and an equation<br />

Formation of equation<br />

Factorization of quadratic<br />

polynomials using the splitting of<br />

middle term method<br />

70


Student’s Worksheet 4 (SW4)<br />

Pre Content Worksheet (P2)<br />

Revisit Key Concepts<br />

Name of Student___________<br />

Date________<br />

Answer the following questions<br />

1. The general form of a quadratic equation is ________________<br />

2. The conditions on the coefficient of x 2 is__________<br />

3. The coefficients and the constant terms are ___________ numbers.<br />

4. The number of zeroes of a polynomial of degree 2 is ____________<br />

5. What is the degree of the expression√2 9 13<br />

_________________. Does it represent a <strong>Quadratic</strong> equation (Y/N)<br />

Why__________________<br />

6. Is (x‐3)(2x+1) = x(x+5) a quadratic equation Why<br />

_________________________________________________________<br />

7. The product of two linear equations is always a _______________ equation<br />

8. The equation <br />

=3 is a _____________ equation<br />

<br />

9. One of the zeroes of the quadratic polynomial ax 2 +bx is always ________ irrespective of the<br />

values of a and b .<br />

10. While factorizing a quadratic polynomial by the splitting of the middle term method, the<br />

product of the numbers chosen should be equal to the product of coefficients of<br />

____________ and ____________________________________the sum of the numbers<br />

chosen should be equal to the coefficient of _________.<br />

71


Self Assessment Rubric<br />

Parameters of assessment<br />

Understanding the concept of a<br />

quadratic polynomial and a<br />

quadratic equation<br />

Understanding the concept of<br />

number of zeroes of a quadratic<br />

polynomial<br />

General understanding of the<br />

concepts covered in the chapter on<br />

Polynomials<br />

72


Student’s Worksheet 5 (SW5)<br />

Content Worksheet (CW1)<br />

Zeroes of a <strong>Quadratic</strong> Polynomial<br />

Name of Student___________<br />

Date________<br />

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

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

their flowers.<br />

2, 4<br />

1,3<br />

½,2<br />

5,7<br />

‐2,1<br />

‐2,‐3<br />

‐4,‐1<br />

‐½,1<br />

3/2,1<br />

2x 2 ‐5x+2, x 2 ‐6x+8,<br />

x 2 +x‐2, x 2 +5x+4,<br />

x 2 ‐5x+6,2x 2 +5x+3,<br />

2x 2 ‐x‐1, x 2 +5x+6,<br />

x 2 ‐12x+35, x 2 ‐x+3<br />

2,3<br />

73


<strong>Quadratic</strong> polynomial<br />

Zeroes<br />

74


Self Assessment Rubric<br />

Parameters of assessment<br />

Understanding of the concept of<br />

zeroes of a polynomial<br />

Identifying the zeroes of the<br />

polynomial<br />

75


Student’s Worksheet 6 (SW6)<br />

Content Worksheet (CW2)<br />

Roots of a <strong>Quadratic</strong> Equation<br />

Name of Student___________<br />

Task 1<br />

Date________<br />

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

x = a is a root of the equation p(x) = 0.<br />

Now choose any 5 of the quadratics from CW1 and write the above statement in the following<br />

manner e.g.<br />

<strong>Quadratic</strong> polynomial: p(x) = x 2 − 6x + 8<br />

<strong>Quadratic</strong> equation: x 2 − 6x + 8 = 0<br />

(x − 2)(x − 4) = 0<br />

x = 2 and x = 4 are the zeroes of the polynomial p(x) = x 2 − 6x + 8<br />

1.<br />

x = 2 and x = 4 are the roots of the given equation x 2 −6x + 8 = 0<br />

2.<br />

76


3.<br />

4.<br />

5.<br />

Roots of a quadratic equation are also called the solutions of a quadratic equation as they<br />

always satisfy the equation.<br />

77


Task 2<br />

Check if the following values of x are the solutions of the given quadratic equations. Complete<br />

the table below. The first one is done for you<br />

<strong>Quadratic</strong> equations Value of x Your working Conclusion<br />

2x 2 ‐x‐2=0 2 2.2 2 ‐2‐20 x=2 is not a solution<br />

of the given equation<br />

x 2 ‐x‐2=0 ‐1<br />

x 2 +3x‐20=0 4<br />

x 2 ‐8x‐12=0 ‐3<br />

x 2 ‐3x‐10=0 5<br />

x 2 ‐4x‐12=0 ‐2<br />

78


Self Assessment Rubric<br />

Parameters of assessment<br />

Is able to find the roots of the given<br />

quadratic equation by factorization<br />

method.<br />

Writing the zeroes of a polynomial<br />

and the roots or solutions of the<br />

quadratic equation<br />

Checking if the given value of x is a<br />

solution of the given quadratic<br />

equation<br />

79


Student’s Worksheet<br />

7 (SW7)<br />

Content Worksheet (CW3)<br />

Nature of Roots<br />

Name of<br />

Student___ _________<br />

Date_____ ____<br />

For a given quadratic<br />

equation ax 2 +bx+c=0 where a0 and a, b, c are real numbers, we define<br />

discriminant of the equation as D = b 2 ‐4ac<br />

Remember: For any quadratic equation to have real roots, the discriminant<br />

has to be a non negative real number.<br />

Task 1<br />

Now complete the table for a given quadratic equation ax 2 +bx+c=0<br />

D=b<br />

2 ‐4ac<br />

D=0 then the roots of<br />

the equation<br />

are_______ __<br />

D>0 then the roots of<br />

the equation<br />

are___ ______<br />

D


Task 2<br />

Given below is a grid<br />

of 25 squares with polynomials. Colour the grid as indicated<br />

• Red if the quadratic equation<br />

has real and distinct roots<br />

• Mauve if the quadratic equation has real and equal roots<br />

• Leavee them white if the quadratic equation has no real roots.<br />

81


Self Assessment Rubric<br />

Parameters of assessment<br />

Is able to state the conditions on<br />

the discriminant of a quadratic<br />

equation for the existence of real<br />

roots<br />

Is able to calculate the discriminant<br />

Is able to predict the nature of<br />

roots using discriminant<br />

82


Student’s Worksheet 8 (SW8)<br />

Content Worksheet (CW4)<br />

Discriminant Method<br />

Name of Student___________<br />

Date________<br />

Task 1<br />

To find the value of an unknown constant when the nature of the roots of a quadratic equation<br />

and the equation are given.<br />

Complete the following table. The first one is done for you<br />

Polynomial Nature of roots Your working Value of unknown<br />

constant<br />

kx 2 −2√5x+4=0 Real and equal roots D=(‐2√5) 2 −4.k.4<br />

k=5/4<br />

= 20−16k<br />

D=0 gives k=5/4<br />

x 2 −4x+k=0<br />

Real and equal roots<br />

kx 2 −2x−1=0<br />

Real and unequal roots<br />

(k+1)x 2 +2x+1=0<br />

Real and equal roots<br />

x 2 −2(k−1)x+1=0<br />

Real and equal roots<br />

3x 2 −5x−k=0<br />

Real roots<br />

83


Task 2<br />

Solving a quadratic equation by the discriminant method (or method of completing the<br />

squares):<br />

Recall that D=b 2 −4ac<br />

The solution of a quadratic equation ax 2 +bx+c=0, a0 and a,b,c are real numbers,<br />

The solution of the quadratic equation is given by<br />

x= √<br />

<br />

i.e. x= √<br />

√<br />

<br />

<br />

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

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

solution of those equations.<br />

Now to solve the following equations by the method of completing the squares (or discriminant<br />

method); fill up the blank spaces.<br />

1. Solve the equation √5x 2 ‐3x‐√5=0<br />

Soln.: a=________, b=________, c=__________<br />

D=_______________<br />

x= √<br />

<br />

i.e. x= √<br />

√<br />

<br />

<br />

x<br />

or x<br />

84


2. Solve the equation x 2 +3√5x+6=0<br />

Soln.: a=________, b=________, c=__________<br />

D=_______________<br />

x= √<br />

<br />

i.e. x= √<br />

√<br />

<br />

<br />

x<br />

or x<br />

3. Solve the equation x 2 +2√7x−5=0<br />

Soln.: a=________, b=________, c=__________<br />

D=_______________<br />

x= √<br />

<br />

ie. x= √<br />

√<br />

<br />

<br />

x<br />

or x<br />

4. Solve the equation x 2 +5x+5=0<br />

Soln.: a=________, b=________, c=__________<br />

D=_______________<br />

x= √<br />

<br />

ie. x= √<br />

√<br />

<br />

<br />

85


x<br />

or x<br />

5. Solve the equation x 2 −5x+2=0<br />

Soln.: a=________, b=________, c=__________<br />

D=_______________<br />

x= √<br />

<br />

ie. x= √<br />

√<br />

<br />

<br />

x<br />

or x<br />

Task 3<br />

Solve the following quadratic equations using discriminant method and complete the table<br />

given below. If no real roots exist, please mention that.<br />

1. 100x 2 −20x+1=0<br />

2. 2x 2 +14x+9=0<br />

3. 9x 2 −30x+25=0<br />

4. 4x 2 −4x+1=0<br />

5. X 2 +2x+4=0<br />

6. 4x 2 +4√3x+3=0<br />

7. 3x 2 +2√5x+5=0<br />

8. 25x 2 +20x+7=0<br />

9. 6x 2 +23x+20=0<br />

10. 2x 2 +5x+5=0<br />

Can you solve all equations by factorization method<br />

86


Now complete the table below<br />

<strong>Quadratic</strong> equation Discriminant Nature of roots Roots (if they exist)<br />

87


Self Assessment Rubric ‐ Content Worksheet (CW4)<br />

Parameters of assessment<br />

Is able to find the value of an<br />

unknown constant when the<br />

nature of the roots is given<br />

Is able to find the roots of a given<br />

quadratic equation<br />

Is able to identify the situation<br />

when the equation has no real<br />

roots<br />

Is able to understand that while the<br />

method of factorization is not<br />

universally applicable for all<br />

quadratic equations, the method of<br />

discriminant is universally<br />

applicable.<br />

88


Student’s Worksheet 9 (SW9)<br />

Content Worksheet (CW5)<br />

Relation Between Sum & Product of Roots and Coefficients of <strong>Quadratic</strong> Equation<br />

Name of Student___________<br />

Date________<br />

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

columns considering the general form of quadratic equation as ax 2 +bx +c =0.<br />

I II III IV V VI VII VIII IX<br />

Equation<br />

&<br />

Solution<br />

Coefficients of x 2 , x and<br />

a =<br />

constant terms (a, b & c)<br />

<br />

<br />

<br />

<br />

Roots<br />

(α, β)<br />

α = ………<br />

Sum of Roots<br />

(α + β)<br />

Product of Roots<br />

(αβ)<br />

Relation between Column III<br />

& VI<br />

Relation between Column IV<br />

& VII<br />

<br />

……….<br />

β = ………<br />

b =<br />

……….<br />

c =<br />

……….<br />

89


a =<br />

……….<br />

b =<br />

……….<br />

α = ………<br />

β = ………<br />

<br />

<br />

c =<br />

……….<br />

a =<br />

……….<br />

b =<br />

……….<br />

α = ………<br />

β = ………<br />

<br />

<br />

c =<br />

……….<br />

a =<br />

……….<br />

b =<br />

……….<br />

α = ………<br />

β = ………<br />

c =<br />

……….<br />

90


a =<br />

……….<br />

b =<br />

……….<br />

α = ………<br />

β = ………<br />

c =<br />

……….<br />

Is there any relationship between the coefficients of x 2 , x and the constant terms with the<br />

roots of the equations Reflect.<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

_________________________________________________________________________________________________________<br />

91


Self Assessment Rubric 1 – Content Worksheet (CW5)<br />

Parameter<br />

Able to recognize coefficients of x 2 ,<br />

x and constant terms.<br />

Able to find roots of a quadratic<br />

equation.<br />

Able to calculate sum of roots and<br />

products of roots correctly.<br />

Able to relate sum & product of<br />

roots with the coefficients of x 2 , x<br />

and constant terms.<br />

92


Student’s Worksheet<br />

10 (SW10)<br />

Content Worksheet (CW6)<br />

Forming a <strong>Quadratic</strong> Equation<br />

Name of<br />

Student___ _________<br />

Date_____ ____<br />

Reeta lost her bag containing 5 equations. Fortunately, Ranjeeta and Saleem had one root each<br />

for Reeta’s equation in their bags. Work<br />

with Ranjeeta and Saleem to get back Reeta’s<br />

<strong>Equations</strong>.<br />

93


The following table may help you trace Reeta’s <strong>Equations</strong>.<br />

First Root<br />

Second Root<br />

Sum of Roots<br />

Product of<br />

Reeta’s Equation<br />

(α)<br />

(β)<br />

(S = α + β)<br />

Roots<br />

x 2 ‐ (S)x + P = 0<br />

(P = αβ)<br />

2 4 2 + 4=6 2 X 4 = 8 x 2 ‐ 6x + 8 = 0<br />

‐7 2<br />

3 ‐5<br />

1 6<br />

‐0.5 9<br />

Self Assessment Rubric – Content Worksheet (CW6)<br />

Parameter<br />

Able to find sum of roots.<br />

Able to find product of roots.<br />

Able to form an equation with<br />

given roots.<br />

94


Student’s Worksheet 11 (SW11)<br />

Content Worksheet (CW7)<br />

Application of <strong>Quadratic</strong> Equation in Real Life Problems<br />

Name of Student___________<br />

Date________<br />

Represent the following in the language of mathematics:<br />

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

product of the number of marbles they now have is 124.<br />

2. The area of a rectangular plot is 528 m 2 . The length of the plot (in metres) is one more than<br />

twice its breadth.<br />

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

from now will be 360.<br />

Solve the following:<br />

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

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

original fraction<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

95


___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

2. Find two consecutive positive integers whose product is 306.<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

96


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

now will be 360. Find Sazi’s present age.<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

___________________________________________________________________________<br />

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4. A rectangle has a perimeter of 23 cm and an area of 33 cm 2 . Find the dimensions.<br />

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5. The sum of the reciprocals of two consecutive even integers is . What are the integers<br />

<br />

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

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

has a volume of 128 cu units.<br />

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

other two sides.<br />

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

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

side is 30 metres more than the shorter side, find the sides of the field.<br />

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

years) would have been 11 more than five times her actual age. What is her age now<br />

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Self Assessment Rubric – Content Worksheet (CW7)<br />

Parameter<br />

Able to comprehend word<br />

problems.<br />

Able to express a word problem in<br />

the form of quadratic equation.<br />

Able to solve quadratic equations.<br />

Able to write the answer of the<br />

given word problem.<br />

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Student’s Worksheet<br />

12 (SW12)<br />

Post Contentt Worksheet (PCW1)<br />

Name of Student_ ____________<br />

1. Understand the differencee between quadratic polynomial<br />

Separate quadratic equations and quadratic polynomials from<br />

boxes<br />

given below:<br />

Date_____ ____<br />

and quadratic equations.<br />

the given box into the<br />

two<br />

e.g.<br />

<strong>Quadratic</strong> Polynomial<br />

2x 2 + 3x – 4<br />

x 2 + 3/4 x – 4<br />

<strong>Quadratic</strong> <strong>Equations</strong><br />

2x 2 + 3x – 4 = 0<br />

x 2 + 3/4<br />

x – 4 = 0<br />

‐4x 2 +3x+2<br />

6x 2 ‐2x‐1= =6<br />

4x 2 ‐3x‐33<br />

3x<br />

2 +4x‐7=0<br />

x 2<br />

+3x‐7=0<br />

√7x 2 +4x+7<br />

2x 2 ‐10x=7<br />

6x 2 ‐14x‐7<br />

4x 2 ‐12x+3<br />

Funnel<br />

103


<strong>Quadratic</strong><br />

Polynomial<br />

<strong>Quadratic</strong><br />

<strong>Equations</strong><br />

2. Which of the following are <strong>Quadratic</strong> <strong>Equations</strong><br />

Student’s Worksheet 13 (SW13)<br />

Post Content (PCW2)<br />

Name of Student___________<br />

Date________<br />

1. What is a quadratic equation What is the degree of quadratic equation<br />

2. What do you understand by root of an equation How many roots will a quadratic equation<br />

have<br />

3. What is discriminant Does it help to predict upon the nature of roots of a quadratic<br />

equation Explain.<br />

4. Write the nature of roots for each quadratic equation (ax 2 +bx+c=0) given below:<br />

104


Equation<br />

x 2 +4x+5=0<br />

a, b<br />

& c<br />

a=…<br />

D=b 2 ‐4ac D>0 D=0 D


c=…<br />

5x 2 ‐6x+2=0<br />

a=…<br />

b=…<br />

c=…<br />

5x 2 ‐6x‐2=0<br />

a=…<br />

b=…<br />

c=…<br />

3x 2 ‐5x+2=0<br />

a=…<br />

b=…<br />

c=…<br />

106


Student’s Worksheet 14 (SW14)<br />

Post Content (PCW3)<br />

Name of Student___________<br />

Date________<br />

Find the roots of the following quadratic equations using:<br />

• Factorization.<br />

• Completion of Squares.<br />

• <strong>Quadratic</strong> Formula.<br />

Student’s Worksheet 15 (SW15)<br />

Post Content (PCW4)<br />

Name of Student___________<br />

Do as directed:<br />

Date________<br />

1. If x 2 +5x+1=0, find the value of .<br />

2. Solve for x : √3x 2 − 2√2x − 2√3=0<br />

3. Solve for x :<br />

<br />

<br />

3<br />

4. Find the roots of the following equations:<br />

a) x 3 x180<br />

b) 4 x 4 x240<br />

c) 5 x 25 x 30 0<br />

(x 1,2)<br />

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d) 2 x 4 x160<br />

e) 8 x 24 x 32 0<br />

f) 4 x 12 x 40 0<br />

g) 2 x 2 x40<br />

h) 2 x 2 x400<br />

5. Verify the relationship between the sum of roots, products of roots and coefficient of x 2 , x<br />

and constant term for the equations given in question 4.<br />

6. Solve for x:<br />

2 <br />

<br />

3 <br />

; x −3, x <br />

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

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

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

9. Is 0.3 a root of the equation x 2 – 0.9 = 0 Justify.<br />

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

would have been the square of his actual marks. How many marks did he get in the test<br />

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

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

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

area is 800m² If so, find its length and breadth.<br />

108


Useful Online Links<br />

http://www.purplemath.com/modules/quadform.htm<br />

http://mathworld.wolfram.com/<strong>Quadratic</strong>Equation.html<br />

History of <strong>Quadratic</strong> Equation<br />

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

Sum of roots, product of roots and discriminate formula<br />

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

Solving quadratic equation<br />

http://library.thinkquest.org/20991/alg2/quad.html<br />

101 uses of quadratic equation<br />

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

Introduction to <strong>Quadratic</strong> equation<br />

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

Derivation of quadratic formula<br />

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

Online <strong>Quadratic</strong> Equation solver<br />

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

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

http://kselva.tripod.com/quad.html<br />

Test on quadratic equation<br />

http://www.alexmaths.com/cbse10/quadratic/quadratic.html<br />

109


Videos<br />

5 ways to solve quadratic equation:<br />

http://www.youtube.com/watchv=zAjeVyUFaSc&feature=fvwrel<br />

Solving quadratic equation by Square root method<br />

http://www.youtube.com/watchv=zAjeVyUFaSc&feature=fvwrel<br />

Solving quadratic equation by factoring<br />

http://www.youtube.com/watchv=lMU5wMDcJNg&feature=related<br />

Introduction to quadratic equation<br />

http://www.youtube.com/watchv=IWigvJcCAJ0&feature=related<br />

<strong>Quadratic</strong> Formula<br />

http://www.youtube.com/watchv=IvXgFLV2gOk&feature=related<br />

Using UnFOIL to Factor <strong>Quadratic</strong> <strong>Equations</strong><br />

http://www.youtube.com/watchv=z57PKs3Bm4U&feature=related<br />

Solving Word Problems<br />

http://www.youtube.com/watchv=EhPPci8shA8<br />

http://www.youtube.com/watchv=Vu3px08WX_8<br />

http://www.youtube.com/watchv=lS9S1iEjlPI<br />

110


CENTRAL BOARD OF SECONDARY EDUCATION<br />

Shiksha Kendra, 2, Community Centre, Preet Vihar,<br />

Delhi-110 092 India

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