Mathematics: Class 9 — Chapter 4: Exploring Algebraic Identities
4.1 Introduction
Algebraic Identity: An equation that is true for all values of the variables occurring in it.
Equation: A mathematical statement of equality that is only true for specific values of the variables (e.g., x2 - 1 = 24 is only true for x = 5 or -5).
4.2 Visualising Identities
Identity I: (a + b)2 = a2 + 2ab + b2
Geometrical Model: A square of side (a + b) units partitioned into smaller squares of areas a2 and b2, and two rectangles of area ab each.
4.3 Factorisation Using Identities
Identity II: (a - b)2 = a2 - 2ab + b2
Factorisation: The process of finding the factors of an algebraic expression using identities in reverse.
Common Factor: Simplifying expressions by factoring out common numerical or literal terms before applying identities (e.g., 50p2 + 60pq + 18q2 = 2(25p2 + 30pq + 9q2) = 2(5p + 3q)2).
4.4 More Identities
Identity III: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Identity IV: a2 - b2 = (a + b)(a - b)
Śhrīdharāchārya's Method: Rewriting the difference of squares as a2 = (a + b)(a - b) + b2 to quickly compute squares of numbers (e.g., 552 = (60 × 50) + 25 = 3025).
4.5 & 4.6 Factorisation with and without Algebra Tiles
Algebra Tiles: Geometrical tools used to represent and factor quadratic expressions like x2 + 7x + 12 into (x + 3)(x + 4).
Splitting the Middle Term: To factor x2 + (a + b)x + ab, find two numbers whose sum is the coefficient of x and whose product is the constant term.
General Quadratic Form: (px + a)(qx + b) = pqx2 + (pb + aq)x + ab.