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