Algebraic Identities & Factorisation

An identity is an equality that is true for every value of the variable -- (a + b)^2 = a^2 + 2ab + b^2 holds whatever a and b are, which is why it is a tool for expanding and factorising rather than something you 'solve'. The misconception CTET tests here is famous enough to have a nickname, the 'freshman's error': a child writes (a + b)^2 = a^2 + b^2, forgetting the middle term 2ab. A good teacher counters it with a concrete area model -- a square of side (a + b) splits into an a-by-a square, a b-by-b square and two a-by-b rectangles, so the missing 2ab becomes visible. The three identities to know cold are (a + b)^2 = a^2 + 2ab + b^2, (a - b)^2 = a^2 - 2ab + b^2, and (a + b)(a - b) = a^2 - b^2. Factorisation runs the identities backwards: spotting that x^2 - 9 is a difference of squares and so factors as (x + 3)(x - 3), or that a quadratic of the form x^2 + (a+b)x + ab factors as (x + a)(x + b). CTET items mix a straight expansion or factorisation with a misconception-spotting question about the dropped middle term.

✅ Solved examples

✏️ Practice — try these, take hints as needed

📝 Topic test — 8 questions

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