GANITA MANJARI • CLASS 9 • CHAPTER 4
Special rules that make algebraic calculations simple and powerful
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Ganita Manjari • Class 9 • Chapter 4
Akash Srivastva | www.akashmaths.online
4.1 EQUATION vs IDENTITY
Equation
True only for specific values.
x² − 1 = 24
Only when x = 5 or −5
Identity
True for all values of the variables.
(x + y)² = x² + 2xy + y²
Always true
Book p.70
Akash Srivastva | www.akashmaths.online
4.1 THE THREE SQUARES PATTERN
Take any three consecutive squares. Add the first and last, subtract twice the middle. Result is always 2 .
1, 4, 92
9, 16, 252
25, 36, 492
Think & Reflect
Can you find a similar pattern for 4 consecutive squares?
Book p.68
Akash Srivastva | www.akashmaths.online
4.1 ALGEBRAIC PROOF
Let the numbers be (n−1), n, (n+1)
[(n−1)² + (n+1)²] − 2n²= 2
Algebraic Proof
Akash Srivastva | www.akashmaths.online
4.2 (a + b)² — GEOMETRIC MODEL
A square of side (a+b) is divided into:
• Yellow square → a²
• Two green rectangles → ab each
• Purple square → b²
Total area = a² + 2ab + b²
a²
ab
ab
b²
Book p.69–70
Akash Srivastva | www.akashmaths.online
4.2 (a − b)²
Just replace b with (−b) in the previous identity.
(a + (−b))² = a² + 2a(−b) + (−b)²
Think & Reflect
When is (a+b)² greater than a² + b²? When are they equal?
Book p.70–71
Akash Srivastva | www.akashmaths.online
4.2 (a + b + c)²
a² + b² + c² + 2ab + 2bc + 2ca
A square of side (a+b+c) is divided into 9 parts (3 squares + 6 rectangles).
Example: 119² = (100+10+9)² = 14161
Book p.72
Akash Srivastva | www.akashmaths.online
4.3 FACTORISATION USING SQUARES
x² + 4x + 4
36x² + 12x + 1
Always check if the expression matches a² ± 2ab + b² form.
Book p.73
Akash Srivastva | www.akashmaths.online
4.3 TAKE COMMON FACTOR FIRST
Example: 50p² + 60pq + 18q²
1. Take out 2 → 2(25p² + 30pq + 9q²)
2. 25p² + 30pq + 9q² = (5p + 3q)²
Final: 2(5p + 3q)²
Book p.74
Akash Srivastva | www.akashmaths.online
4.4 DIFFERENCE OF SQUARES
A square of side a with a smaller square of side b removed can be rearranged into a rectangle of sides (a+b) and (a−b).
Example: 16y² − 9 = (4y+3)(4y−3)
Geometric Idea
Big square − Small square
Book p.75
Akash Srivastva | www.akashmaths.online
4.4 ลHRฤชDHARฤCHฤRYA’S METHOD
a² = (a+b)(a−b) + b²
Example: 55² = (55+5)(55−5) + 25 = 60×50 + 25 = 3025
Think & Reflect
Try 35², 65², 85² using this method. What pattern do you notice?
Book p.76
Akash Srivastva | www.akashmaths.online
4.5 PRODUCT OF BINOMIALS
Can be understood using algebra tiles or by direct expansion.
Example: (x+3)(x+4) = x² + 7x + 12
Book p.77
Akash Srivastva | www.akashmaths.online
4.6 SPLITTING THE MIDDLE TERM
For x² + Sx + P, find two numbers whose sum = S and product = P.
x² + 11x + 30 = (x+5)(x+6)
x² − 5x + 6 = (x−2)(x−3)
Book p.78–79
Akash Srivastva | www.akashmaths.online
4.7 CUBE OF A BINOMIAL
(a+b)³ = a³ + 3a²b + 3ab² + b³
(a−b)³ = a³ − 3a²b + 3ab² − b³
These can be visualised as volume of a cube of edge (a±b).
Book p.80–81
Akash Srivastva | www.akashmaths.online
4.7 SUM & DIFFERENCE OF CUBES
Difference
x³ − y³ = (x−y)(x² + xy + y²)
Sum
x³ + y³ = (x+y)(x² − xy + y²)
Book p.82
Akash Srivastva | www.akashmaths.online
4.7 THREE VARIABLE IDENTITY
(x+y+z)(x² + y² + z² − xy − yz − zx)
Special case: If x + y + z = 0, then x³ + y³ + z³ = 3xyz
Book p.83
Akash Srivastva | www.akashmaths.online
MASTER LIST OF IDENTITIES
(x+y)² = x²+2xy+y²
(x−y)² = x²−2xy+y²
(x+y+z)² = ... + 2xy+2yz+2zx
x²−y² = (x+y)(x−y)
(x+a)(x+b) = x²+(a+b)x+ab
(x+y)³ = x³+3x²y+3xy²+y³
(x−y)³ = x³−3x²y+3xy²−y³
x³−y³ = (x−y)(x²+xy+y²)
x³+y³ = (x+y)(x²−xy+y²)
x³+y³+z³−3xyz = ...
Chapter Summary
Akash Srivastva | www.akashmaths.online
GANITA MANJARI • CLASS 9 • CHAPTER 4
Identities turn complicated expressions into simple and elegant forms.
“Algebra is but written geometry and geometry is but figured algebra.”
— Sophie Germain
Designed with Dedication & Love For Mathematics
Akash Srivastva, TGT (Mathematics)
Ganita Manjari • Class 9
Akash Srivastva | www.akashmaths.online
