Find the best tutors and institutes for Class 10 Tuition
Q1(iii):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(iii) $6x^2 - 3 - 7x$
Solution :
Given: A quadratic polynomial $p(x) = 6x^2 - 3 - 7x$.
To Find: The zeroes of the polynomial and verify the relationship between the zeroes and the coefficients.
Step 1: Standardizing the Polynomial
The standard form of a quadratic polynomial is $ax^2 + bx + c$. We rearrange the given polynomial in descending order of the powers of $x$:
$p(x) = 6x^2 - 7x - 3$
Step 2: Finding the Zeroes by Factorization
To find the zeroes, we set $p(x) = 0$.
$6x^2 - 7x - 3 = 0$
We use the splitting the middle term method. We look for two numbers whose product is $a \times c = 6 \times (-3) = -18$ and whose sum is $b = -7$.
The factors of $-18$ that sum to $-7$ are $-9$ and $2$.
$6x^2 - 9x + 2x - 3 = 0$
Group the terms:
$(6x^2 - 9x) + (2x - 3) = 0$
Factor out the common terms:
$3x(2x - 3) + 1(2x - 3) = 0$
$(3x + 1)(2x - 3) = 0$
Step 3: Solving for $x$
Setting each factor to zero:
1) $3x + 1 = 0 \implies 3x = -1 \implies x = -\frac{1}{3}$
2) $2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$
Thus, the zeroes are $\alpha = \frac{3}{2}$ and $\beta = -\frac{1}{3}$.
Step 4: Verifying the Relationship between Zeroes and Coefficients
For the polynomial $6x^2 - 7x - 3$, the coefficients are $a = 6$, $b = -7$, and $c = -3$.
Verification of Sum of Zeroes:
Formula: $\alpha + \beta = -\frac{b}{a}$
LHS: $\alpha + \beta = \frac{3}{2} + (-\frac{1}{3}) = \frac{9 - 2}{6} = \frac{7}{6}$
RHS: $-\frac{b}{a} = -\frac{-7}{6} = \frac{7}{6}$
Since LHS = RHS, the relationship is verified.
Verification of Product of Zeroes:
Formula: $\alpha \cdot \beta = \frac{c}{a}$
LHS: $\alpha \cdot \beta = (\frac{3}{2}) \cdot (-\frac{1}{3}) = -\frac{3}{6} = -\frac{1}{2}$
RHS: $\frac{c}{a} = \frac{-3}{6} = -\frac{1}{2}$
Since LHS = RHS, the relationship is verified.
Final Answer: The zeroes of the polynomial are $\frac{3}{2}$ and $-\frac{1}{3}$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $\frac{7}{6}$ and the product of zeroes is $-\frac{1}{2}$.
More Questions from Class 10 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) $x^2 - 2x - 8$
- Q1(ii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (ii) $4s^2 - 4s + 1$
- Q1(iv): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iv) $4u^2 + 8u$
- Q1(v): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (v) $t^2 - 15$
- Q1(vi): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (vi) $3x^2 - x - 4$
- Q2(i): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) $\frac{1}{4}, -1$
- Q2(ii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (ii) $\sqrt{2}, \frac{1}{3}$
- Q2(iii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iii) $0, \sqrt{5}$
- Q2(iv): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iv) $1, 1$
- Q2(v): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (v) $-\frac{1}{4}, \frac{1}{4}$
- Q2(vi): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (vi) $4, 1$
CBSE Solutions for Class 10 Mathematics Polynomials
Chapters in CBSE - Class 10 Mathematics
Top Tutors who teach Polynomials
Top Tutor
Nehru Nagar, Bangalore
10 yrs of
Exp
61 students
1163 hrs of classes completed
167
per hour
Online Classes
I am Mopidevi Padmaja, an experienced Math Teacher, Quantitative Aptitude Coach, and Certified Vedic math trainer. I teach CBSE 9th and 10th Math, Quantitative Aptitude for Competitive exams (CAT, SSC, RRB, etc.) & Vedic Maths starting from grade 6 to graduation level students. -------------------------------------------------------------------------------------------------- I hold an Engineering degree & am an Associate Member of the Institution of Engineers (India) I am also a certified Vedic Math trainer -------------------------------------------------------------------------------------------------- I have over 10+ years of teaching experience both online & offline. I have taught over 1000+ students to date. I have been teaching online for the last 7 years for Indian and overseas students. I am familiar with various teaching platforms available online and comfortable using them. -------------------------------------------------------------------------------------------------- I have a passion for making Math the easiest, simple, and most fun subject for students. I work on various learning issues in Math, like Math phobia, focus problems, word problem understanding, concept understanding, confusion, time management in math exams, and silly mistakes which is a major problem for most students. I could help my students to score more than 90% marks in Maths subjects who were losing around 20 to 25 marks by confusion in concepts, silly mistakes, lack of time management in math exams, etc. I am successful in making Math an interesting and easy subject for my students who have hated Math. -------------------------------------------------------------------------------------------------- I thoroughly teach fundamentals, concepts, formulae, and guide with smart ways of doing calculations, smart ways of doing word problems, strategies for solving questions, etc. I use visual aids in teaching like 2D animated PowerPoint presentations. I make students comfortable and at ease and encourage them to ask their doubts during a session and even after sessions through messages. I respond to messages within a short time. If more students have doubts about a specific topic I even take a special session for them. I always try to improve my techniques of assessment by using a quiz, worksheets, test papers, Sample papers, etc., so that I can do 100 % assessment and take the necessary steps to improve my students. Every student is important to me. In each session, I teach according to the needs and understanding levels of my students. I work with students having various intellectual and behavioral capabilities. I learn to teach and teach to learn. My ultimate goal is to help the student to become an independent learner. I believe marks are not about grades and ranks but about building confidence levels in students. ---------------------------------------------------------------------------------------------- I continuously upgrade myself by attending workshops, and webinars, browsing the internet and reading lots of books. I use Digital Media for a lot of learning new developments in the field of my teaching. I attend webinars & seminars and interact with other teachers & professionals from across the world.
I would like to express my sincere gratitude to the teacher Padmaja Mopidevi for the exceptional guidance provided to my daughter throughout the CBSE 10th grade. Her dedication and teaching methods truly helped simplify complex mathematical concepts, making the subject much more approachable. Thanks to her constant support and encouragement, my daughter was able to secure a distinction in her board exams. We are incredibly grateful for the positive impact she has had on my daughter's academic journey and confidence. I highly recommend her tuition classes to any student looking to excel in mathematics.
Top Tutor
Karampura, Delhi
10 yrs of
Exp
2500
per hour
Online Classes
Student's Home
I am an IIM Kashipur alumnus, passionate about guiding students to achieve their immediate and long-term goals. As a personal tutor and founder of the online educational startup ‘MinDely,’ I have gained deep insights into child psychology and student needs, allowing me to tailor my teaching methods accordingly. I design lesson plans based on each student's grasp of concepts, ensuring a student-friendly and easy-to-understand approach. My weekly tests provide valuable insights for parents to track their child’s progress. I take pride in the success of my students, many of whom have graduated from IITs, DTU, and NSIT.
Top Tutor
Sector 83, Gurgaon
20 yrs of
Exp
4 students
3678 hrs of classes completed
650
per hour
Online Classes
Tutor's home
For me, teaching is part of learning. As I am teaching I am also learning from my learners as they are diverse. Each learner in the classroom brings his or her unique background and ability. These are the elements that I invest in my teaching. Moreover, I allow learners to be themselves. This is a way of creating a conducive environment for learning.
Her method of teaching is very clear and student friendly. It’s really very easy for me to understand and also she has a lot of patience.
Top Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
1 student
688 hrs of classes completed
1850
per hour
Online Classes
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
Super Tutor
Nerul Sector 20, Thane
10 yrs of
Exp
62 students
2871 hrs of classes completed
1200
per hour
Online Classes
I have done b.Tech in electrical engineering from IIT KANPUR. I taught one year in akash institute patna, 2 years in fiitjee patna. Currently I am teaching in unacademy. I have taught more than 6000 jee and neet aspirants. In which more than 500 students are selected in iitand aiims. If some one is interested to improve their physics upto jee advanced level take help with me .And see the magic. I can help students in their all type of assignment from any institute like allen, fiitjee, narayna, akash, resonance. I can help students to solve both volume of h.C verma within two month if some one is intrested in solving I.E.Irodov join me and see the next level of physics. My way of teaching is so easy and so advanced that student will realyy enjoy it. Don't waste your time in making decision just join and experience my level
He is a very dedicated teacher with excellent knowledge on topic. He knows Subject very well and teach accordingly to kids.
Find more Tutor for Polynomials in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
