All Algebraic Identities for Class 8, 9 & 10 (Printable List)

This algebraic identities list puts every identity a Class 8, 9 or 10 student needs in one place, grouped by class with a quick tip on when to use each one, plus a one-page summary table at the end.

An identity is an equation that is true for every value of its variables — (a + b)² = a² + 2ab + b² works whether a and b are 1 and 2, or 100 and −7.

Class 8 Identities (Algebraic Expressions and Identities)

#	Identity	When to use it
1	(a + b)² = a² + 2ab + b²	Squaring a sum, or numbers like 102 or 51
2	(a − b)² = a² − 2ab + b²	Squaring a difference, or numbers like 98 or 49
3	(a + b)(a − b) = a² − b²	Multiplying a "sum × difference" pair, e.g. 103 × 97
4	(x + a)(x + b) = x² + (a + b)x + ab	Multiplying two brackets sharing the same first term

Quick examples: 103 × 97 = (100 + 3)(100 − 3) = 10000 − 9 = 9991; (x + 4)(x + 6) = x² + 10x + 24; 47² = (50 − 3)² = 2500 − 300 + 9 = 2209

Read more in our (a + b)² formula guide.

Class 9 Identities (Polynomials)

#	Identity	When to use it
5	(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca	Squaring a three-term expression
6	(a + b)³ = a³ + b³ + 3ab(a + b)	Cubing a sum, or finding a³ + b³ from a + b and ab
7	(a − b)³ = a³ − b³ − 3ab(a − b)	Cubing a difference
8	a³ + b³ = (a + b)(a² − ab + b²)	Factorising a sum of two cubes
9	a³ − b³ = (a − b)(a² + ab + b²)	Factorising a difference of two cubes
10	a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)	Three-cube factorisation
11	If a + b + c = 0, then a³ + b³ + c³ = 3abc	Shortcut questions with three cubes

Quick example: 15³ + (−8)³ + (−7)³ — since 15 − 8 − 7 = 0, the answer is 3 × 15 × (−8) × (−7) = 2520

For detailed working, see (a + b)³ and (a − b)³ formulas.

Class 10: Where Identities Are Used (and Trigonometric Identities)

Class 10 algebra mostly uses the Class 8 and 9 identities — completing the square (x² + 6x + 5 = (x + 3)² − 4), quadratic factorisation, and relations between zeroes (α² + β² = (α + β)² − 2αβ). Class 10 also introduces three trigonometric identities:

#	Identity
12	sin²θ + cos²θ = 1
13	1 + tan²θ = sec²θ
14	1 + cot²θ = cosec²θ

Tip: identities 13 and 14 come from identity 12. Divide sin²θ + cos²θ = 1 by cos²θ to get 13, or by sin²θ to get 14.

Useful Rearranged Forms

Rearranged form	Comes from
a² + b² = (a + b)² − 2ab	Identity 1
a² + b² = (a − b)² + 2ab	Identity 2
(a + b)² − (a − b)² = 4ab	Identities 1 and 2
x² + 1/x² = (x + 1/x)² − 2	Identity 1 with b = 1/x

Example: if x + 1/x = 4, then x² + 1/x² = 16 − 2 = 14.

How to Choose the Right Identity

Ask three questions: how many terms are inside the bracket (two → identities 1–4/6–9; three → identity 5/10)? Is it a square or a cube? Am I expanding (read left to right) or factorising (read right to left)?

Frequently Asked Questions

How many algebraic identities are there in Class 9?
About eight main ones: the three Class 8 square identities, (a + b + c)², the two cube identities, the sum/difference of cubes, and the a³ + b³ + c³ − 3abc identity.

What is the difference between an identity and an equation?
An identity is true for all values of the variable; an equation like 2x + 3 = 7 is true only for particular values.

Which identity finds 103 × 97 quickly?
(a + b)(a − b) = a² − b², so 103 × 97 = 100² − 3² = 9991.