Class 9 Mathematics Chapter 4 Exploring Algebraic Identities – Extra Questions with Answers

Expanding an expression like (x+5) squared becomes fast once you recognise the underlying identity, and these questions cover expansion, factorisation and mental-math shortcuts built on that idea.

Last Updated: September 23, 2026

Very Short Answer Questions (1 mark)

Q1. Expand (x + 5)² using the identity.
Ans: x² + 10x + 25.

Q2. Factorise x² − 16 using the difference of squares identity.
Ans: (x + 4)(x − 4).

Q3. Expand (a − 3)².
Ans: a² − 6a + 9.

Q4. State the identity used to expand (x + 2)(x + 5).
Ans: (x + a)(x + b) = x² + (a+b)x + ab.

Q5. Write the identity for a³ + b³ in factored form.
Ans: a³ + b³ = (a + b)(a² − ab + b²).

Short Answer Questions (2–3 marks)

Q6. Use a suitable identity to evaluate 998² without direct multiplication.
Ans: 998² = (1000 − 2)² = 1000² − 2×1000×2 + 2² = 1000000 − 4000 + 4 = 996004.

Q7. Factorise 4x² − 9y² using an identity, showing your working.
Ans: 4x² − 9y² = (2x)² − (3y)² = (2x + 3y)(2x − 3y), using the difference of squares identity.

Q8. Expand (2x + 3y)² using the standard identity, showing the substitution.
Ans: Using (a+b)² = a²+2ab+b² with a=2x, b=3y: (2x+3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².

Higher-Order Thinking / Application Questions

Q9. If a + b = 10 and ab = 21, find the value of a² + b² without finding a and b individually.
Ans: Using (a+b)² = a² + 2ab + b²: 10² = a² + 2(21) + b², so 100 = a² + b² + 42, giving a² + b² = 58.

Q10. A square garden of side (x + 4) metres has a square flower bed of side x metres removed from one corner. Using algebraic identities, find a simplified expression for the remaining garden area.
Ans: Total area = (x+4)² = x² + 8x + 16 (using the identity). Flower bed area = x². Remaining area = (x² + 8x + 16) − x² = 8x + 16 = 8(x + 2) square metres.

Frequently Asked Questions

How can the identity for the square of a sum be used to quickly calculate 105 squared without direct multiplication?
Writing 105 as 100 plus 5 and applying the identity gives 100 squared plus twice 100 times 5 plus 5 squared, which simplifies to 10000 plus 1000 plus 25, giving 11025 without direct multiplication.

What is the difference between the identity for the square of a sum and the identity for the difference of two squares?
The square of a sum identity expands an expression, while the difference of two squares identity factors an expression into a product, serving opposite purposes of expanding versus factorising.

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.