NCERT Solutions for Class 9th Maths Chapter 4 Exploring Algebraic Identities
Updated on 2026-09-19
About this chapter
An equation such as x² − 1 = 24 holds only for x = 5 or x = −5; an identity such as (x + y)² = x² + 2xy + y² holds for every value of x and y. That is the whole difference. A square of side (a + b) cuts into a², b² and two ab rectangles, so (a + b)² = a² + 2ab + b². Replacing b by −b — legitimate, because an identity is true for all values — gives (a − b)² = a² − 2ab + b². A square of side (a + b + c) cuts into nine pieces and gives the three-letter version. Read backwards, every identity is a factorisation rule. Recognising a² + 2ab + b² inside 9x² + 24xy + 16y² is what lets you write it as (3x + 4y)². Algebra tiles turn factorising x² + 7x + 12 into a jigsaw: the only split of 7x that makes the unit tiles fill a rectangle is 3x + 4x, because 3 + 4 = 7 and 3 × 4 = 12. Without tiles this i
- Introduction
- Visualising Identities
- Factorisation of Algebraic Expressions Using Identities
- More Identities
- Factorisation Using Algebra Tiles
- Factorisation Without Using Algebra Tiles
- Finding New Identities
- Simplifying Rational Expressions
- Chapter 4 Exploring Algebraic Identities
Quick revision
| Identity | Statement | Where it comes from | Use it for |
|---|---|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² | Fig. 4.2 — square of side a + b cut into a², b² and two ab rectangles | 64², expanding (7x + 4y)² |
| Square of a difference | (a − b)² = a² − 2ab + b² | Fig. 4.3, or put −b for b in the first identity | 79², factoring 16y² − 24y + 9 |
| Square of a trinomial | (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca | Fig. 4.4 — a 3 × 3 grid on a square of side a + b + c | 117², factoring 9a² + 4b² + c² − 12ab + 6ac − 4bc |
| Difference of two squares | a² − b² = (a + b)(a − b) | Fig. 4.5 — cut a strip off a square and slide it round | 104 × 96, factoring 36s² − 49t² |
| Śhrīdharāchārya’s form | a² = (a + b)(a − b) + b² | the same figure, read the other way | 35², 55², 105² in the head |
| Product of two binomials | (x + a)(x + b) = x² + (a + b)x + ab | Fig. 4.7 — algebra tiles forming a rectangle | factoring x² + 7x + 12, x² − 5x + 6 |
| General binomial product | (px + a)(qx + b) = pq·x² + (pb + aq)x + ab | Fig. 4.8 — tiles for (2x + 3)(3x + 1) | factoring 6x² + 7x + 2, 10x² − 11x − 6 |
| Cube of a sum | (a + b)³ = a³ + 3a²b + 3ab² + b³ | Fig. 4.10 — a cube of edge a + b in 2 cubes + 6 cuboids | 147³, factoring p³ + 6p²q + 12pq² + 8q³ |
| Cube of a difference | (a − b)³ = a³ − 3a²b + 3ab² − b³ | put −b for b in the cube of a sum | 199³, factoring 8n³ − 60n²m + 150nm² − 125m³ |
| Sum and difference of cubes | x³ − y³ = (x − y)(x² + xy + y²) x³ + y³ = (x + y)(x² − xy + y²) | multiply out with the distributive property; the middle terms cancel | factoring 27b³ − 1/(64b³), 64y³ + z³/125 |
| Three cubes | x³ + y³ + z³ − 3xyz = (x + y + z)(x² + y² + z² − xy − yz − zx) | multiply out; all nine cross terms cancel in threes | factoring p³ + 27q³ + r³ − 9pqr; proving a³+b³+c³−3abc = −25 |
Exercises
- Think and Reflect — Introduction Page 69
- Think and Reflect — Visualising Identities Page 71
- Exercise Set 4.1 — Visualising Identities Page 71–72
- Think and Reflect — Factorisation of Algebraic Expressions Using Identities Page 73
- Exercise Set 4.2 — Factorisation of Algebraic Expressions Using Identities Page 74–75
- Think and Reflect — More Identities Page 76
- Exercise Set 4.3 — More Identities Page 76–77
- In-text Questions — More Identities Page 77
- Think and Reflect — More Identities Page 78
- Think and Reflect — Factorisation Using Algebra Tiles Page 79
- Think and Reflect — Factorisation Using Algebra Tiles Page 79
- Think and Reflect — Factorisation Using Algebra Tiles Page 80
- In-text Questions — Factorisation Using Algebra Tiles Page 80
- Exercise Set 4.4 — Factorisation Without Using Algebra Tiles Page 81–82
- Think and Reflect — Factorisation Without Using Algebra Tiles Page 82
- In-text Questions — Finding New Identities Page 85
- Think and Reflect — Finding New Identities Page 85
- Think and Reflect — Simplifying Rational Expressions Page 87
- Exercise Set 4.5 — Simplifying Rational Expressions Page 87
- Chapter 4 Exploring Algebraic Identities — End-of-Chapter Exercises Page 88–90