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CBSE Class 9 — Notes, Chapters & Practice Quizzes

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Mathematics · 8 chapters
Summary, key terms, extra questions with answers and a practice quiz with AI diagnosis for each.

Chapter 4: Exploring Algebraic IdentitiesClass 9 Mathematics — summary, notes, extra questions & NCERT solutions

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An identity is an equation that is true:

Summary

This chapter develops algebraic identities — equations true for all values of the variables — and shows how to verify them geometrically with squares, rectangles and cubes, and using algebra tiles. The basic square identities are \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\), extended to three terms by \((a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca\). The difference of squares is \(a^2 - b^2 = (a+b)(a-b)\), and the product pattern \((x+a)(x+b)=x^2+(a+b)x+ab\) leads to factorising quadratics by splitting the middle term. Cubic identities include \((a+b)^3 = a^3+3a^2b+3ab^2+b^3\), \((a-b)^3 = a^3-3a^2b+3ab^2-b^3\), the sum and difference of cubes \(a^3 \pm b^3\), and \(a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\). These identities speed up numerical squaring and cubing, support factorisation, and let us simplify rational algebraic expressions by cancelling common factors.

Identities vs equationsSquare identitiesCubic identitiesFactorisationSimplifying rational expressions

Key terms

Identity
An equation that is true for every value of the variables involved.
Equation
A statement that holds only for particular values of the variable, unlike an identity.
Factorisation
Writing an expression as a product of its factors, e.g. \(x^2+4x+4=(x+2)^2\).
Difference of squares
The identity \(a^2 - b^2 = (a+b)(a-b)\).
Splitting the middle term
Factorising \(x^2+px+q\) by finding \(a,b\) with \(a+b=p\) and \(ab=q\).
Algebra tiles
Square and rectangular pieces used to visualise products and factors of expressions.

Extra questions & answers

Think and Reflect

Think and Reflect

Exercise Set 4.1

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Exercise Set 4.2

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Exercise Set 4.3

Think and Reflect

Think and Reflect

Think and Reflect

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Exercise Set 4.4

Think and Reflect

Think and Reflect

Think and Reflect

Exercise Set 4.5

End-of-Chapter Exercises

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Identity
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An equation that is true for every value of the variables involved.
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Practice quiz · Exploring Algebraic Identities

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