📘 About the new NCERT Class 9 book Ganita Manjari
The new NCERT Maths book “Ganita Manjari” ✨ is designed to make mathematics more interactive, meaningful, and connected to real life 🧠🌍. Instead of only focusing on formulas, it:
- Uses stories and situations (like Reiaan’s room 🏠) to explain concepts
- Encourages thinking, reasoning, and exploration 🤔
- Includes “Think and Reflect” questions 💭 to build deeper understanding
- Connects maths with history and practical use 📜📏
👉 Overall, this new book focuses less on rote learning and more on understanding concepts clearly and applying them in real situations 🌐.
📘 Chapter 4: Exploring Algebraic Identities
Algebraic identities are special mathematical equations that are true for all values of the variables. They help us simplify algebraic expressions, factorise expressions and perform calculations quickly. 🧮✨
🎯 Learning Objectives
- Understand the meaning of an algebraic identity.
- Differentiate between an equation and an identity.
- Visualise identities using geometrical models.
- Use identities for expansion and factorisation.
- Use algebra tiles to understand quadratic factorisation.
- Develop and use new algebraic identities.
- Simplify rational algebraic expressions using factorisation.
- Use identities to perform calculations quickly.
🔍 1. What is an Algebraic Identity?
An algebraic identity is an equation that is true for all possible values of the variables involved.
(x + y)2 = x2 + 2xy + y2
This is an identity because it remains true for every value of x and y.
⚖️ Equation vs Identity
| Equation | Identity |
|---|---|
| May be true only for certain values. | True for all values of the variables. |
| x2 - 1 = 24 | (x + y)2 = x2 + 2xy + y2 |
📐 2. Visualising Algebraic Identities
The chapter explains identities geometrically using squares and rectangles. For example, a square with side (a + b) can be divided into smaller squares and rectangles.
The four parts of the square represent:
- 🟨 One square of area a2
- 🟩 Two rectangles of area ab each
- 🟦 One square of area b2
Therefore, the total area is a2 + 2ab + b2.
⭐ 3. Important Algebraic Identities
🔹 (a + b)2 = a2 + 2ab + b2
🔹 (a - b)2 = a2 - 2ab + b2
🔹 (a + b)(a - b) = a2 - b2
🔹 (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
🔹 (x + a)(x + b) = x2 + (a + b)x + ab
🔹 (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
🧮 4. Using Identities for Calculations
Identities can make difficult calculations much faster.
43 = 40 + 3
432 = (40 + 3)2
= 402 + 2(40)(3) + 32
= 1600 + 240 + 9
= 1849 ✅
Similarly, numbers close to a convenient number can be handled using (a - b)2.
29 = 30 - 1
292 = (30 - 1)2
= 302 - 2(30)(1) + 12
= 900 - 60 + 1
= 841 ✅
🔢 5. Factorisation Using Identities
Identities can also be used in the reverse direction to factorise algebraic expressions.
x2 + 4x + 4
= x2 + 2(x)(2) + 22
= (x + 2)2
Therefore, (x + 2) is a factor of x2 + 4x + 4.
36x2 + 12x + 1
= (6x)2 + 2(6x)(1) + 12
= (6x + 1)2
🟦 6. Difference of Squares
One of the most useful identities is:
It can also be rearranged as:
The chapter uses this identity as a quick method for calculating squares. For example:
= 60 × 50 + 25
= 3025 ✅
🧩 7. Algebra Tiles and Factorisation
Algebra tiles provide a visual way to understand multiplication and factorisation of algebraic expressions.
(x + 3)(x + 4)
= x2 + 3x + 4x + 12
= x2 + 7x + 12
Therefore:
In the algebra-tile model, the 7x term is split into 3x + 4x, while the constant 12 is arranged as a 3 × 4 rectangle.
🧠 8. Factorisation Without Algebra Tiles
For an expression such as:
We look for two numbers a and b such that:
- a + b = 7
- ab = 12
The required numbers are 3 and 4.
For expressions with a negative middle term, the signs of the factors must be chosen accordingly.
= (x - 2)(x - 3)
🚀 9. More Identities
🔹 Square of Three Terms
This identity can be used to calculate squares of numbers by breaking them into convenient parts.
= (100 + 10 + 9)2
= 1002 + 102 + 92 + 2(100)(10) + 2(100)(9) + 2(10)(9)
= 14161
🎲 10. Cubic Identities
The chapter develops identities involving cubes.
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
The cube identity can also be visualised geometrically by dividing a cube of side (a + b) into smaller cubes and cuboids. 📦
🔺 11. Sum and Difference of Cubes
a3 - b3 = (a - b)(a2 + ab + b2)
a3 + b3 = (a + b)(a2 - ab + b2)
These identities are especially useful when factorising expressions involving cubes.
🌟 12. A Special Identity Involving Three Variables
This identity can be used to solve problems involving the sum, squares and product of three numbers.
If x + y + z = 10, xyz = 25 and x2 + y2 + z2 = 38, then the chapter uses the identities to obtain:
x3 + y3 + z3 = 145
🧮 13. Simplifying Rational Algebraic Expressions
Rational algebraic expressions can often be simplified by factorising the numerator and denominator and then cancelling common factors.
- Factorise the numerator.
- Factorise the denominator.
- Identify common factors.
- Cancel the common factors.
- Remember that a denominator cannot be zero.
For example, the chapter factorises x2 - 7x + 12 as (x - 3)(x - 4), and 5x2 + 5x - 100 as 5(x - 4)(x + 5).
The common factor can then be cancelled, subject to the restriction that the denominator is not zero. ⚠️
📋 14. Formula Cheat Sheet
| 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) |
| Square of three terms | (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca |
| Product of two binomials | (x+a)(x+b) = x2+(a+b)x+ab |
| Cube of sum | (a+b)3 = a3+3a2b+3ab2+b3 |
| Cube of difference | (a-b)3 = a3-3a2b+3ab2-b3 |
| Difference of cubes | a3-b3 = (a-b)(a2+ab+b2) |
| Sum of cubes | a3+b3 = (a+b)(a2-ab+b2) |
| Three-variable identity | x3+y3+z3-3xyz = (x+y+z)(x2+y2+z2-xy-yz-zx) |
💡 Quick Revision Points
- 🧠 An identity is true for all values of its variables.
- 📐 Identities can be visualised using geometrical models.
- 🧩 Algebra tiles help visualise multiplication and factorisation.
- 🔄 Identities can be used both for expansion and factorisation.
- ⚡ Identities make numerical calculations faster.
- 🔺 Cubic identities help factorise expressions involving cubes.
- ➗ Rational expressions can be simplified by cancelling common factors.
- ⚠️ A factor may be cancelled only when the original denominator is non-zero.
❓ Frequently Asked Questions (FAQs)
1️⃣ What is an algebraic identity?
An algebraic identity is an equation that is true for all values of the variables involved.
2️⃣ What is the difference between an equation and an identity?
An equation may be true only for particular values of its variables, whereas an identity is true for every value of the variables for which the expressions are defined.
3️⃣ What is the identity for (a + b)2?
(a + b)2 = a2 + 2ab + b2
4️⃣ What is the identity for (a - b)2?
(a - b)2 = a2 - 2ab + b2
5️⃣ How can identities help in factorisation?
We can recognise an algebraic expression as one side of a known identity and write it in factorised form. For example, x2 + 4x + 4 = (x + 2)2.
6️⃣ What is the difference of squares identity?
a2 - b2 = (a + b)(a - b)
7️⃣ What are algebra tiles?
Algebra tiles are visual models used to represent algebraic expressions, multiplication and factorisation using shapes such as squares, rectangles and unit tiles.
8️⃣ What are the identities for the sum and difference of cubes?
a3 + b3 =
(a+b)(a2-ab+b2)
a3 - b3 =
(a-b)(a2+ab+b2)
9️⃣ How do we factorise x2 + 7x + 12?
Find two numbers whose sum is 7 and whose product is 12. The numbers are 3 and 4. Therefore: x2 + 7x + 12 = (x+3)(x+4).
🔟 How are rational algebraic expressions simplified?
First factorise the numerator and denominator, then cancel their common factors, making sure that the original denominator is not zero.
🏆 Final Takeaway
Exploring Algebraic Identities shows how algebraic patterns can be converted into useful mathematical tools. By mastering these identities, students can expand expressions, factorise polynomials, simplify rational expressions and perform numerical calculations much faster. 🚀📚