📚 Chapter at a Glance
🔢 Introduction to Algebraic Identities
Algebra is a branch of mathematics in which letters and symbols are used to represent numbers and quantities. Algebraic expressions allow us to describe mathematical relationships in a short and useful form.
An algebraic identity is an equality that remains true for all permissible values of the variables involved.
⚖️ Identity vs Equation
| Identity | Equation |
|---|---|
| True for all permissible values | May be true only for particular values |
| Shows a general mathematical relationship | Usually asks us to find unknown values |
| Example: (a+b)² = a²+2ab+b² | Example: x+5=10 |
📌 Basic Algebraic Terms
Variable
A letter representing a number whose value may vary is called a variable.
Constant
A fixed numerical value is called a constant.
Coefficient
The numerical factor of a term is called its coefficient.
Term
Parts of an algebraic expression separated by addition or subtraction signs are called terms.
This expression has three terms.
⭐ Important Algebraic Identities
These identities are useful for expansion, simplification, mental calculation and factorisation.
1️⃣ Identity: (a + b)²
This identity tells us how to expand the square of a sum of two quantities.
Expand:
Using:
Here a=x and b=5.
2️⃣ Identity: (a - b)²
Expand:
3️⃣ Identity: Difference of Two Squares
Factorise:
Since:
Therefore:
4️⃣ Cube of a Sum
Expand:
5️⃣ Cube of a Difference
Expand:
6️⃣ Sum of Two Cubes
Factorise:
Since:
Therefore:
7️⃣ Difference of Two Cubes
Factorise:
8️⃣ Square of Three Terms
Expand:
9️⃣ Product of Two Binomials
Expand:
📋 Algebraic Identities – Quick Revision Table
| Identity | Formula |
|---|---|
| Square of Sum | (a+b)² = a²+2ab+b² |
| Square of Difference | (a-b)² = a²-2ab+b² |
| Difference of Squares | a²-b²=(a+b)(a-b) |
| Cube of Sum | (a+b)³=a³+3a²b+3ab²+b³ |
| Cube of Difference | (a-b)³=a³-3a²b+3ab²-b³ |
| Sum of Cubes | a³+b³=(a+b)(a²-ab+b²) |
| Difference of Cubes | a³-b³=(a-b)(a²+ab+b²) |
🧠 Solved Examples
Expand:
Using (a+b)²:
Expand:
Factorise:
Expand:
Factorise:
Factorise:
📝 Important Practice Questions
✅ Solutions to Practice Questions
🎯 Class 9 Maths Chapter 4 MCQs
Answer: B
Answer: C
Answer: C
Answer: B
Answer: A
Answer: B
Answer: A
Answer: A
Answer: B
Answer: B
Answer: C
Answer: C
Answer: A
Answer: A
Answer: B
🧠 Competency-Based Questions
A student expands (x+5)² as x²+25.
Is the solution correct?
Answer: No.
The middle term 10x cannot be omitted.
A rectangular garden has length (x+5) metres and breadth (x+3) metres.
Find its area.
A student wants to factorise x²-81.
Identify the appropriate identity.
📝 Class 9 Maths Chapter 4 Worksheet
Section A – Fill in the Blanks
1. (a+b)² = __________________.
2. (a-b)² = __________________.
3. a²-b² = __________________.
4. a³+b³ = __________________.
5. a³-b³ = __________________.
Section B – Expand
6. (x+6)²
7. (x-7)²
8. (2x+3)²
9. (3x-2)²
10. (x+4)(x+9)
Section C – Factorise
11. x²-36
12. 4x²-25
13. x³+27
14. x³-64
Section D – Application Questions
15. The side of a square is (x+3) cm. Find its area.
16. A rectangle has dimensions (x+4) cm and (x+6) cm. Find its area.
17. Simplify:
Section E – Higher Order Thinking
18. Without directly multiplying, find 102².
19. Using an identity, calculate 99².
20. Factorise:
✅ Worksheet Answer Key
- a²+2ab+b²
- a²-2ab+b²
- (a+b)(a-b)
- (a+b)(a²-ab+b²)
- (a-b)(a²+ab+b²)
- x²+12x+36
- x²-14x+49
- 4x²+12x+9
- 9x²-12x+4
- x²+13x+36
- (x+6)(x-6)
- (2x+5)(2x-5)
- (x+3)(x²-3x+9)
- (x-4)(x²+4x+16)
- x²+6x+9 cm²
- x²+10x+24 cm²
- 12x
- 10404
- 9801
- (5x+6y)(5x-6y)
⭐ Important Questions for Revision
- What is an algebraic identity?
- Differentiate between an equation and an identity.
- Write the square of sum identity.
- Write the square of difference identity.
- State the difference of squares identity.
- Write the identities for the sum and difference of cubes.
- Expand algebraic expressions using identities.
- Factorise expressions using algebraic identities.
- Apply identities to solve numerical calculations quickly.
- Use algebraic identities in real-life area and measurement problems.
⚡ One-Minute Revision
(a+b)²=a²+2ab+b²
(a-b)²=a²-2ab+b²
a²-b²=(a+b)(a-b)
(a+b)³=a³+3a²b+3ab²+b³
(a-b)³=a³-3a²b+3ab²-b³
a³+b³=(a+b)(a²-ab+b²)
a³-b³=(a-b)(a²+ab+b²)
Expansion ↔ Factorisation
🎯 Exam Tips
- Memorise the basic identities rather than trying to derive them during every question.
- Pay special attention to the sign of the middle term in (a-b)².
- For a²-b², remember the pattern: (a+b)(a-b).
- When factorising cubes, first check whether the terms are perfect cubes.
- Always substitute the correct values of a and b before using an identity.
- After factorisation, multiply the factors again to check your answer.
- Algebraic identities can also make numerical calculations much faster.
❓ Frequently Asked Questions
What is Chapter 4 of Class 9 Maths?
The chapter is titled Exploring Algebraic Identities.
What is an algebraic identity?
An algebraic identity is an equality that remains true for all permissible values of the variables.
What is the formula for (a+b)²?
What is the formula for (a-b)²?
What is the formula for a²-b²?
How are algebraic identities useful?
They are useful for expanding expressions, simplifying algebraic expressions, factorising expressions and performing numerical calculations efficiently.
📚 Class 9 Maths Chapter 4 Study Material
Students searching for Class 9 Maths Chapter 4 notes, Exploring Algebraic Identities Class 9 notes, Class 9 Maths Chapter 4 2026-27, algebraic identities Class 9, Class 9 algebraic identities formulas, Class 9 Maths Chapter 4 important questions, Class 9 Maths Chapter 4 MCQs, Class 9 Maths Chapter 4 worksheet, algebraic identities questions and answers, and Class 9 Maths identities worksheet can use this Study Notes Hub page for revision and practice.
This page provides student-friendly explanations of important algebraic identities, including squares, difference of squares, cubes, sum and difference of cubes, expansion, factorisation, solved examples, competency-based questions, MCQs and a practice worksheet.