Find the best tutors and institutes for Class 10 Tuition
Q15(i):
Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given:
(i) Area : $25a^2 – 35a + 12$
Solution :
Initial Setup & Theoretical Foundation
The area of a rectangle is geometrically defined as the product of its two adjacent spatial dimensions: length ($l$) and breadth ($b$). This relationship is expressed by the formula:
$\text{Area} = l \times b$
We are given the area of the rectangle in the form of a quadratic polynomial in terms of the variable $a$:
$\text{Area} = 25a^2 - 35a + 12$
[Per the Fundamental Theorem of Algebra and polynomial factorization principles], to find the possible expressions for the length and breadth, we must factorize this quadratic polynomial into the product of two linear binomials. These two resulting factors will represent the possible dimensions of the rectangle.
Step 1: Identifying the Factorization Method
We will factorize the quadratic polynomial $P(a) = 25a^2 - 35a + 12$ using the method of splitting the middle term. We compare the given polynomial to the standard quadratic form $Ax^2 + Bx + C$ to identify the coefficients:
- $A = 25$ (Coefficient of $a^2$)
- $B = -35$ (Coefficient of $a$)
- $C = 12$ (Constant term)
Step 2: Splitting the Middle Term
To split the middle term, we must find two real numbers, let us call them $p$ and $q$, that satisfy two specific conditions simultaneously:
- Their product must equal $A \times C$:
$p \times q = 25 \times 12 = 300$ - Their sum must equal $B$:
$p + q = -35$
Because the product ($300$) is positive and the sum ($-35$) is negative, [by the rules of integer arithmetic], both numbers $p$ and $q$ must be negative.
Let us evaluate the factor pairs of $300$ to find the correct combination:
| Factor Pair ($p, q$) | Product ($p \times q$) | Sum ($p + q$) |
|---|---|---|
| $-10, -30$ | $300$ | $-40$ |
| $-12, -25$ | $300$ | $-37$ |
| $-15, -20$ | $300$ | $-35$ |
The numbers $-15$ and $-20$ satisfy both conditions perfectly. Therefore, we will split the middle term $-35a$ into $-20a$ and $-15a$.
Step 3: Factorization by Grouping
We substitute the split terms back into the original polynomial:
$P(a) = 25a^2 - 20a - 15a + 12$
Next, we group the terms into pairs to extract the greatest common monomial from each group:
$P(a) = (25a^2 - 20a) - (15a - 12)$
Factor out the greatest common divisor (GCD) from the first group ($5a$) and the second group ($3$):
$P(a) = 5a(5a - 4) - 3(5a - 4)$
Notice that the binomial $(5a - 4)$ is now a common factor in both terms. We factor out $(5a - 4)$ [by the Distributive Property of Multiplication over Addition]:
$P(a) = (5a - 4)(5a - 3)$
Geometric Visualization
The factorization proves that a rectangle with an area of $25a^2 - 35a + 12$ can be constructed with sides measuring $(5a - 4)$ and $(5a - 3)$. Below is the geometric representation of this relationship.
Final Conclusion
Because multiplication is commutative ($l \times b = b \times l$), either of the two binomial factors can represent the length, and the remaining factor will represent the breadth.
Final Solution: The possible expressions for the length and breadth of the rectangle are $(5a - 3)$ and $(5a - 4)$.
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(ii): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (ii) Area : $35y^2 + 13y –12$
- 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
Pandeypur, Varanasi
10 yrs of
Exp
14 students
1971 hrs of classes completed
500
per hour
Online Classes
I am Mathematics teacher teaching in a reputed school.
Super Tutor
Shahdara Chitrakoot, Delhi
10 yrs of
Exp
72 students
3975 hrs of classes completed
350
per hour
Online Classes
With extensive experience in providing Class 10 tuition, I specialize in teaching mathematics, science, and computer science. My approach emphasizes a thorough understanding of key concepts and practical applications. In mathematics, I cover topics such as algebra, geometry, and trigonometry. My science classes include physics, chemistry, and biology, focusing on fundamental principles and real-world relevance. In computer science, I teach programming basics, algorithms, and data management. I use interactive tools, clear presentations, and regular assessments to ensure comprehensive learning. My goal is to build strong foundational knowledge, boost confidence, and prepare students for their board exams and future academic pursuits.
Rahul sir has an outstanding knowledge of computer html/javascript...He became my saviour during my school exams and helped me to prepare and cover my syllabus in a very short span of time...His guidance helped me to score full marks in my computer exams..thank you for all your guidance and support sir.
Top Tutor
Mira Road East, Mumbai
20 yrs of
Exp
11 students
350
per hour
Online Classes
Student's Home
Tutor's home
25 years of private teaching experience in maths and science at Mumbai western line.
Top Tutor
Venkatraman Nagar Chennai, Chennai
10 yrs of
Exp
27 students
3790 hrs of classes completed
320
per hour
Online Classes
Tutor's home
im having good experince in teaching with make them understand the concepts better
Classes are much interactive so useful to assess the students understanding capacity during the class itself.
Top Tutor
Gharoli Gharoli, Delhi
16 yrs of
Exp
1 student
42 hrs of classes completed
400
per hour
Online Classes
I bring with me 17 years of diverse experience in the pharmaceutical industry along with academic expertise, combining strong practical knowledge with a passion for teaching. I am dedicated to simplifying complex concepts and delivering them in an engaging manner, ensuring that students not only gain subject knowledge but also develop the skills, confidence, and potential required to achieve academic excellence and become industry-ready professionals.
"The teacher explains Math and Science in such a brilliant way. Her teaching style is so simple that even difficult topics become easy to understand. She is a truly wonderful teacher!"
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
