NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4: Exploring Algebraic Identities

Algebraic identities can be challenging in Class 9 Maths when you need to expand expressions, factorise polynomials, and solve problems involving squares and cubes. Remembering different formulas and applying them correctly in numerical calculations or algebraic expressions can also be difficult. Word problems that require you to use identities to find unknown values may need additional practice.

NCERT Solutions for Chapter 4: Exploring Algebraic Identities provide exercise-wise answers with step-by-step explanations. These solutions help you understand algebraic formulas, practise factorisation, simplify expressions, and solve textbook problems involving numerical calculations and real-life situations. Regular practice can strengthen your mathematical skills and improve your accuracy when solving examination questions.

NCERT Solutions for Chapter 4: Exercise-Wise Solutions

Chapter 4 covers algebraic identities through four exercises and a lecture section. The solutions below provide selected questions with step-by-step answers to help you understand the concepts and practise applying the identities.

Exploring Algebraic Identities Exercise 4.1

Exercise 4.1 focuses on applying algebraic identities to expand expressions, calculate squares, and simplify algebraic calculations.

1. Write the algebraic expression: A number increased by 7.

Answer: x + 7

2. Write the algebraic expression: Twice a number decreased by 5.

Answer: 2x − 5

3. Write the algebraic expression: Product of two consecutive numbers.

Answer: x(x + 1) = x² + x

4. Write the algebraic expression: Area of a rectangle whose length is (x + 2) and breadth is (x + 

Answer: (x + 2)(x + 5)

5. Expand (x + 2)².

Answer: (x + 2)² = x² + 4x + 4

Exploring Algebraic Identities Exercise 4.2

Exercise 4.2 focuses on expanding algebraic expressions, factorising polynomials, and identifying perfect square expressions using algebraic identities.

1. Expand (x − 2)².

Answer: (x − 2)² = x² − 4x + 4.

2. Expand (a − 5)².

Answer: (a − 5)² = a² − 10a + 25.

3. Expand (7x − 4y)².

Answer: (7x − 4y)² = 49x² − 56xy + 16y².

4. Expand (5p − 2q)².

Answer: (5p − 2q)² = 25p² − 20pq + 4q².

5. Find mentally: 79².

Answer: 79² = (80 − 1)² = 6400 − 160 + 1 = 6241.

Exploring Algebraic Identities Exercise 4.3

Exercise 4.3 focuses on factorising algebraic expressions, applying identities involving squares and three variables, and simplifying calculations using algebraic formulas.

1. Factorise 169x² − 26xy + y².

Answer: 169x² − 26xy + y² = (13x − y)² = (13x − y)(13x − y).

2. Factorise 225p² − 90pq + 9q².

Answer: 225p² − 90pq + 9q² = (15p − 3q)² = (15p − 3q)(15p − 3q).

3. Factorise 400a² + 120ab + 9b².

Answer: 400a² + 120ab + 9b² = (20a + 3b)² = (20a + 3b)(20a + 3b).

4. Factorise 324a² − 180ab + 25b².

Answer: 324a² − 180ab + 25b² = (18a − 5b)² = (18a − 5b)(18a − 5b).

5. Factorise 0.04x² + 0.4xy + y².

Answer: 0.04x² + 0.4xy + y² = (0.2x + y)² = (0.2x + y)(0.2x + y).

Exploring Algebraic Identities Exercise 4.4 

Exercise 4.4 focuses on applying algebraic identities involving three variables to expand and factorise expressions and simplify square calculations.

1. Without expanding, find ab + bc + ca, if a + b + c = 10 and a² + b² + c² = 38.

Answer: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Therefore, 100 = 38 + 2(ab + bc + ca), so ab + bc + ca = 31.

2. Expand (x + y + z)² + (x − y − z)².

Answer: (x + y + z)² + (x − y − z)² = 2x² + 2y² + 2z² + 4yz.

3. Is this an identity? (a + b − c)² + (a − b + c)² + (a − b − c)² = 2a² + 2b² + 2c².

Answer: Expanding the left-hand side gives 3a² + 3b² + 3c² − 2ab − 2ac − 2bc, which is not equal to 2a² + 2b² + 2c² in general. Therefore, it is not an identity.

4. Factorise 9a² + 4b² + c² − 12ab + 6ac − 4bc.

Answer: 9a² + 4b² + c² − 12ab + 6ac − 4bc = (3a − 2b + c)² = (3a − 2b + c)(3a − 2b + c).

5. Factorise 16x² + 25y² + 9z² + 40xy − 24xz − 30yz.

Answer: 16x² + 25y² + 9z² + 40xy − 24xz − 30yz = (4x + 5y − 3z)² = (4x + 5y − 3z)(4x + 5y − 3z).

How to Prepare with Class 9 Maths Chapter 4 NCERT solution by PW 

To prepare for Chapter 4, Exploring Algebraic Identities, focus on understanding the formulas and practising their applications. Follow these steps for effective preparation:

• Learn the important identities: Memorise the formulas involving squares, cubes, and three variables.

• Practise each exercise: Solve questions from Exercises 4.1, 4.2, 4.3, and 4.4 to understand different applications of identities.

• Show each step: Write the formula first, substitute the given values, and simplify carefully to avoid calculation errors.

• Revise factorisation: Practise identifying perfect squares and factorising expressions using suitable identities.

• Solve word problems: Apply algebraic identities to numerical calculations and practical situations.

Practising the NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 4 helps you understand algebraic identities, expand expressions, factorise polynomials, and solve numerical problems. Work through the exercises and Lecture 10 step by step, and practise similar questions to improve your accuracy. Regular revision of the identities will also help you apply them confidently in exams.