Find the best tutors and institutes for Class 10 Tuition
Q15(ii):
Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(ii) Area : $35y^2 + 13y –12$
Solution :
Initial Setup & Theoretical Foundation
The geometric area of a rectangle is defined by the product of its two adjacent spatial dimensions, length and breadth. Mathematically, this is expressed as:
[$Area = Length \times Breadth$]
We are given the area of a rectangle as a quadratic polynomial in terms of the variable $y$:
$Area = 35y^2 + 13y - 12$
To find the possible expressions for the length and breadth, we must factorize this quadratic polynomial into the product of two linear binomials. Each resulting binomial will represent one of the dimensions of the rectangle.
Step 1: Coefficient Analysis and the Product-Sum Method
We will factorize the quadratic polynomial $ay^2 + by + c$ using the method of splitting the middle term. First, we identify the coefficients:
- Leading coefficient ($a$) = $35$
- Middle coefficient ($b$) = $13$
- Constant term ($c$) = $-12$
According to the product-sum factorization theorem, we must find two real numbers, let's call them $p$ and $q$, such that:
1. Their product equals $a \times c$:
$p \times q = 35 \times (-12) = -420$
2. Their sum equals $b$:
$p + q = 13$
Step 2: Prime Factorization to Determine Split Values
To systematically find the values of $p$ and $q$, we analyze the prime factorization of the absolute value of the product ($420$):
$420 = 2 \times 210 = 2^2 \times 105 = 2^2 \times 3 \times 35 = 2^2 \times 3 \times 5 \times 7$
We need to group these prime factors into two numbers whose difference is $13$ (since the product is negative, one number must be positive and the other negative). Let us test combinations:
- Combination 1: $(2^2 \times 5) = 20$ and $(3 \times 7) = 21$. Difference is $1$. (Incorrect)
- Combination 2: $(2^2 \times 7) = 28$ and $(3 \times 5) = 15$. Difference is $13$. (Correct)
Since the sum must be positive ($+13$), the larger number must be positive. Therefore, our two numbers are $28$ and $-15$.
[$28 \times (-15) = -420$ and $28 + (-15) = 13$]
Step 3: Splitting the Middle Term and Grouping
We substitute the middle term $13y$ with $28y - 15y$ in the original polynomial:
$35y^2 + 28y - 15y - 12$
Next, we apply factorization by grouping. We group the first two terms and the last two terms:
$(35y^2 + 28y) - (15y + 12)$
Extract the greatest common divisor (GCD) from each group:
For $(35y^2 + 28y)$, the GCD is $7y$:
$7y(5y + 4)$
For $-(15y + 12)$, the GCD is $-3$:
$-3(5y + 4)$
Now, substitute these back into the expression:
$7y(5y + 4) - 3(5y + 4)$
Notice that the binomial $(5y + 4)$ is a common factor. Factoring it out yields the final product of two linear expressions:
$(7y - 3)(5y + 4)$
Step 4: Geometric Representation and Dimensional Assignment
Because multiplication is commutative [by the Commutative Property of Multiplication, $A \times B = B \times A$], either of these binomial factors can represent the length, and the other will represent the breadth.
Final Solution: The possible expressions for the dimensions of the rectangle are Length = $(7y - 3)$ and Breadth = $(5y + 4)$, or vice versa.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.4
- Q1(i): Use suitable identities to find the following products: (i) $(x + 4) (x + 10)$
- Q1(ii): Use suitable identities to find the following products: (ii) $(x + 8) (x – 10)$
- Q1(iii): Use suitable identities to find the following products: (iii) $(3x + 4) (3x – 5)$
- Q1(iv): Use suitable identities to find the following products: (iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
- Q1(v): Use suitable identities to find the following products: (v) $(3 – 2x) (3 + 2x)$
- Q10(i): Factorise each of the following: (i) $27y^3 + 125z^3$ [Hint : See Question 9.]
- Q10(ii): Factorise each of the following: (ii) $64m^3 – 343n^3$ [Hint : See Question 9.]
- Q11: Factorise : $27x^3 + y^3 + z^3 – 9xyz$
- Q12: Verify that $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q13: If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
- Q14(i): Without actually calculating the cubes, find the value of each of the following: (i) $(–12)^3 + (7)^3 + (5)^3$
- Q14(ii): Without actually calculating the cubes, find the value of each of the following: (ii) $(28)^3 + (–15)^3 + (–13)^3$
- Q15(i): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : $25a^2 – 35a + 12$
- Q16(i): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : $3x^2 – 12x$
- Q16(ii): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (ii) Volume : $12ky^2 + 8ky – 20k$
- Q2(i): Evaluate the following products without multiplying directly: (i) $103 \times 107$
- Q2(ii): Evaluate the following products without multiplying directly: (ii) $95 \times 96$
- Q2(iii): Evaluate the following products without multiplying directly: (iii) $104 \times 96$
- Q3(i): Factorise the following using appropriate identities: (i) $9x^2 + 6xy + y^2$
- Q3(ii): Factorise the following using appropriate identities: (ii) $4y^2 – 4y + 1$
- Q3(iii): Factorise the following using appropriate identities: (iii) $x^2 – \frac{y^2}{100}$
- Q4(i): Expand each of the following, using suitable identities: (i) $(x + 2y + 4z)^2$
- Q4(ii): Expand each of the following, using suitable identities: (ii) $(2x – y + z)^2$
- Q4(iii): Expand each of the following, using suitable identities: (iii) $(–2x + 3y + 2z)^2$
- Q4(iv): Expand each of the following, using suitable identities: (iv) $(3a – 7b – c)^2$
- Q4(v): Expand each of the following, using suitable identities: (v) $(–2x + 5y – 3z)^2$
- Q4(vi): Expand each of the following, using suitable identities: (vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
- Q5(i): Factorise: (i) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- Q5(ii): Factorise: (ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
- Q6(i): Write the following cubes in expanded form: (i) $(2x + 1)^3$
- Q6(ii): Write the following cubes in expanded form: (ii) $(2a – 3b)^3$
- Q6(iii): Write the following cubes in expanded form: (iii) $(\frac{3}{2}x + 1)^3$
- Q6(iv): Write the following cubes in expanded form: (iv) $(x - \frac{2}{3}y)^3$
- Q7(i): Evaluate the following using suitable identities: (i) $(99)^3$
- Q7(ii): Evaluate the following using suitable identities: (ii) $(102)^3$
- Q7(iii): Evaluate the following using suitable identities: (iii) $(998)^3$
- Q8(i): Factorise each of the following: (i) $8a^3 + b^3 + 12a^2b + 6ab^2$
- Q8(ii): Factorise each of the following: (ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
- Q8(iii): Factorise each of the following: (iii) $27 – 125a^3 – 135a + 225a^2$
- Q8(iv): Factorise each of the following: (iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
- Q8(v): Factorise each of the following: (v) $27p^3 – \frac{1}{216} – \frac{9}{2}p^2 + \frac{1}{4}p$
- Q9(i): Verify : (i) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- Q9(ii): Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
7 yrs of
Exp
9 students
217 hrs of classes completed
300
per hour
Online Classes
I am an experienced , qualified with experience of 5 years , across different boards including CBSE, ICSE, IGCSE and State Board. Passionate about teaching, over the years I have helped thousands of students .
Top Tutor
Nerul, Mumbai
6 yrs of
Exp
3 students
410 hrs of classes completed
450
per hour
Online Classes
Student's Home
Tutor's home
I have specifically guided Class 10 students for board exams, with a strong focus on NCERT syllabus, and topics like Real Numbers, Polynomials, Quadratic Equations, and Statistics. I provide regular tests, personalized feedback, and exam-oriented strategies to improve accuracy and speed. Many of my students have shown remarkable improvement and scored above 90% in their board exams.
It was an amazing experience learning and understanding maths with Saurav sir. He really helped me overcome my fear of maths and helped me score good marks in a subject that previously scared me. It really shows how dedicated he is as a teacher. Really good experience!
Top Tutor
Kaloor, Kochi
12 yrs of
Exp
2 students
2 hrs of classes completed
500
per hour
Online Classes
Student's Home
Tutor's home
With 12 years of teaching experience, I specialize in concept-based instruction, catering to both weaker students and advanced learners. I help struggling students grasp core concepts while providing advanced-level teaching to high achievers, enabling them to consistently reach 95% or higher in their academic performance.
Super Tutor
Badarpur, Delhi
12 yrs of
Exp
153 students
3900 hrs of classes completed
200
per hour
Online Classes
Tutor's home
Akshay is an excellent teacher. He has taught my son in 10 th standard science , with which he understood the concepts clearly and he also gave so many past question papers for my son to work out, all these made my son score 95 in board exams . I would definitely recommend Akshay for students who want to improve and get good scores in science .
Top Tutor
Ashok Vihar, Gurgaon
10 yrs of
Exp
1 student
214 hrs of classes completed
350
per hour
Online Classes
With a strong academic foundation in Mathematics, including an M.Sc. in Mathematics, B.Sc., BTC, and CTET qualification, I am a dedicated Mathematics educator committed to helping students build conceptual clarity, logical reasoning, and analytical problem-solving skills. I have been successfully teaching Mathematics for over 12 years through my coaching institute in Gurgaon, while also conducting engaging online classes for students across India and internationally. I teach students across CBSE, ICSE, NIOS, and State Boards, adapting my teaching methodology to suit each curriculum and the individual learning needs of every student. My teaching approach focuses on simplifying complex mathematical concepts through structured lesson planning, personalised instruction, continuous assessment, and interactive learning techniques. I believe every student can excel in Mathematics with the right guidance, consistent practice, and a strong conceptual foundation. I also specialise in Vedic Mathematics, helping students significantly improve calculation speed, accuracy, mental agility, and confidence in school, competitive, and entrance examinations. As an UrbanPro Super Tutor, I am recognised for consistently delivering high-quality teaching and maintaining excellent student and parent satisfaction. Over the years, I have helped hundreds of students achieve outstanding academic results while developing confidence and a genuine interest in Mathematics. My passion for teaching, commitment to academic excellence, and student-centred approach enable me to create an engaging, motivating, and result-oriented learning environment where every learner can achieve their highest potential. If you are looking for a dedicated and experienced Mathematics tutor who can help you achieve excellent results and build strong fundamentals, I invite you to book a demo class today. Let’s work together to make Mathematics simple, enjoyable, and rewarding for you!
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
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
