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Algebraic identities are the first place in school maths where you can prove something true for every number at once. They also make mental arithmetic quicker, and they come back in factorisation, quadratic equations and nearly every chapter of Class 9 and 10. This guide covers where they come from, how to check them, four worked examples and the errors that appear in almost every class.
Where is this topic in the 2026-27 NCERT books?
The new home is Ganita Prakash Grade 8 Part I, Chapter 6, "We Distribute, Yet Things Multiply". The older Class 8 chapter "Algebraic Expressions and Identities" (2024-25 reprint) focuses on multiplying monomials and polynomials. Identities return in more depth in Class 9 Ganita Manjari, Chapter 4, "Exploring Algebraic Identities", which adds algebra tiles and factorisation.
Where do identities come from?
Everything starts with the distributive property: a(b + c) = ab + ac. Ganita Prakash draws it as a rectangle of dots with a rows and b + c columns, split into two parts, ab and ac.
Apply it twice and you get the chapter's first identity:
(a + m)(b + n) = ab + an + mb + mn (Identity 1)
Every term in the first bracket multiplies every term in the second. The book uses this to ask a neat question: if you add 1 to both numbers in a product ab, by how much does it grow? (a + 1)(b + 1) = ab + a + b + 1, so it grows by a + b + 1. For 23 × 27, increasing both by 1 adds 23 + 27 + 1 = 51.
From Identity 1, the chapter derives three special cases:
(a + b)² = a² + 2ab + b² (Identity 1A)
(a − b)² = a² − 2ab + b² (Identity 1B)
(a + b)(a − b) = a² − b² (Identity 1C)
An identity is different from an equation. An equation such as 2x + 1 = 7 is true only for x = 3. An identity is true for every value of its letters.
Why is (a + b)² not a² + b²?
Draw it. A square of side a + b splits into four parts: a square of area a², a square of area b², and two rectangles each of area ab. Ganita Prakash does this for 65² by splitting 65 into 60 + 5:
65² = 60² + 2 × 60 × 5 + 5² = 3600 + 600 + 25 = 4225
The two rectangles, 2ab, are exactly what a² + b² leaves out.
Four worked examples
Example 1: a square near 100. 104² = (100 + 4)² = 10,000 + 800 + 16 = 10,816. 97² = (100 − 3)² = 10,000 − 600 + 9 = 9,409.
Example 2: the difference of squares (the book asks for these). 98 × 102 = (100 − 2)(100 + 2) = 10,000 − 4 = 9,996. 45 × 55 = (50 − 5)(50 + 5) = 2,500 − 25 = 2,475.
Example 3: an expansion with coefficients. Expand (3x − 2y)². Use Identity 1B with a = 3x and b = 2y: (3x)² − 2 × 3x × 2y + (2y)² = 9x² − 12xy + 4y². Check with x = 1, y = 1: (3 − 2)² = 1, and 9 − 12 + 4 = 1. Correct.
Example 4: two different brackets. Expand (x + 3)(x + 5). By Identity 1: x² + 5x + 3x + 15 = x² + 8x + 15. Check with x = 1: 4 × 6 = 24, and 1 + 8 + 15 = 24. Correct.
What mistakes should you avoid?
Ganita Prakash has a section called "Mind the Mistake, Mend the Mistake" with wrong expansions to fix. These are the ones to look for in your own work:
Dropping the middle term: (a + b)² written as a² + b².
Sign of the middle term in (a − b)²: it is −2ab, but b² stays positive.
Squaring only the letter: (3x)² is 9x², not 3x².
Distributing a minus sign to the first term only: −(2a − 3b) = −2a + 3b.
Missing a term in Identity 1: two brackets of two terms give four products before you combine like terms.
How should you practise?
Practice that makes you recall the identity from memory works better than reading it again. A review of ten study techniques by Dunlosky and colleagues (2013) rated practice testing and distributed practice as the two of "high utility", while rereading and highlighting rated low.
Write all four identities from memory at the start of each session, then check them with a = 1, b = 2.
Five expansions a day, one with coefficients and one with a minus sign.
Two mental calculations a day using the identities, such as 99², 51² or 103 × 97.
One "mend the mistake" item: write a wrong expansion on purpose, then find and fix the error a day later.
Identities run backwards too: that is factorisation, the subject of our guide to factorisation in Class 8, and later of polynomials in Class 9. If a step in an expansion keeps going wrong, the Fermi Problem Library has a step-by-step solver that explains the reasoning rather than printing an answer.
Try it now: work out 49 × 51 and 49² in your head using the identities, then check on paper.
