🌎 Real-World Hook: A shopkeeper sells 43 items per day. How many items in a week? Instead of multiplying 43 × 7 the hard way, a clever student writes: 43 = 40 + 3, so 43² = (40 + 3)² = 1600 + 240 + 9 = 1849. That is an algebraic identity doing real work! Architects use identities to compute areas. Engineers use them to simplify complex designs. Algebraic identities are shortcuts that always work — for every number, every time.
A big square of side (a + b) split into a², two ab rectangles, and b².
A square of side a split into (a−b) and b parts. The yellow area is (a−b)².
Take away b² from a². The remaining shape = a rectangle of (a+b) by (a−b).
Algebra tiles show x² + 7x + 12 as a rectangle with sides (x + 3) and (x + 4).
All the identities in this chapter shown as a family tree.
Identity (a + b)² = a² + 2ab + b²
Method Find a and b where a + b = 7 and a × b = 12.
Identity (a + b)² = a² + 2ab + b²
| # | Identity Name | Formula | In Simple Words |
|---|---|---|---|
| 1 | Square of Sum | (a+b)² = a²+2ab+b² | Square each, add double product |
| 2 | Square of Difference | (a−b)² = a²−2ab+b² | Same but middle is minus |
| 3 | Difference of Squares | a²−b² = (a+b)(a−b) | Two brackets: one +, one − |
| 4 | Three Terms Squared | (a+b+c)² = a²+b²+c²+2ab+2bc+2ca | 3 squares + 3 double products |
| 5 | Cube of Sum | (a+b)³ = a³+3a²b+3ab²+b³ | Cube each, add 3×each×other² |
| 6 | Cube of Difference | (a−b)³ = a³−3a²b+3ab²−b³ | Signs alternate: +−+− |
| 7 | Sum of Cubes | x³+y³ = (x+y)(x²−xy+y²) | Sum × special bracket |
| 8 | Difference of Cubes | x³−y³ = (x−y)(x²+xy+y²) | Difference × special bracket |
| 9 | Product of Binomials | (x+a)(x+b) = x²+(a+b)x+ab | x² + (sum)x + product |
| 10 | Three Cubes Identity | x³+y³+z³−3xyz = (x+y+z)(...) | Big identity with 3 variables |
| 11 | Factorising | Find a, b: a+b = middle, ab = last | Split the middle term |
1. (a + b)² = a² + 2ab + b²
2. (a − b)² = a² − 2ab + b²
3. a² − b² = (a + b)(a − b)
These help you expand and factor.
They also help you square big numbers fast!
4. (a + b)³ = a³ + 3a²b + 3ab² + b³
5. (a − b)³ = a³ − 3a²b + 3ab² − b³
6. x³ + y³ = (x + y)(x² − xy + y²)
7. x³ − y³ = (x − y)(x² + xy + y²)
Step 1: Find two numbers a and b.
Step 2: They must add to give the middle number.
Step 3: They must multiply to give the last number.
Step 4: Write (x + a)(x + b).
43²: (40+3)² = 1600 + 240 + 9 = 1849
29²: (30−1)² = 900 − 60 + 1 = 841
23×17: (20+3)(20−3) = 400 − 9 = 391
Equation: true for some values only.
x² − 1 = 24 works only for x = 5 or −5.
Identity: true for ALL values.
(a + b)² = a² + 2ab + b² always works!
💡 Verify: 97 × 97 = 9409. Quick mental check: 97² is just under 100² = 10000, so 9409 is reasonable!
Orange side = question. Dark side = answer.
Front
What is (a + b)²?
Back
a² + 2ab + b²
Square each, add double product.
Front
What is (a − b)²?
Back
a² − 2ab + b²
Same as (a+b)² but minus in middle.
Front
What is a² − b²?
Back
(a + b)(a − b)
Two brackets: one plus, one minus.
Front
Factorise x² + 7x + 12.
Back
(x + 3)(x + 4)
3 + 4 = 7 and 3 × 4 = 12.
Front
What is (a + b)³?
Back
a³ + 3a²b + 3ab² + b³
Think: 2 cubes + 6 cuboids.
Front
x³ + y³ = ?
Back
(x + y)(x² − xy + y²)
Sum bracket × special bracket.
Front
x³ − y³ = ?
Back
(x − y)(x² + xy + y²)
Difference bracket × special bracket.
Front
Identity or Equation?
(a+b)² = a²+2ab+b²
Back
Identity!
It is true for ALL values of a and b.
Mind map: all key ideas of Chapter 4 connected together.
Maths Class 9 NCERT Lesson 4 — Exploring Algebraic Identities
Special Edition: Different Learner Friendly Learning Materials
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