Identity: a^{2} - b^{2} = (a + b) (a – b)
How is this identity obtained? Let's see how. Taking RHS of the identity: (a + b) (a – b) Multiply as we do multiplication of two binomials or use F.O.I.L. method and we get: = a(a - b) + b(a - b) = a^{2} - ab + ab - b^{2} Solve like terms and we get: = a^{2} - b^{2} Hence, in this way we obtain the identity i.e. a^{2} - b^{2} = (a + b) (a – b) Following are a few applications to this identity. Example 1: Solve 9a^{2} - 4b^{2} Solution: This proceeds as: Given polynomial 9a^{2} - 4b^{2} represents the identity a^{2} - b^{2} = (a + b) (a – b) Where a = 3a and b = 2b On applying values of a and b on the identity a^{2} - b^{2} = (a + b) (a - b) and we get: (3a)^{2} - (2b)^{2} = (3a + 2b) (3a - 2b) Hence, 9a^{2} - 4b^{2} = (3a + 2b) (3a - 2b) Example 2: Solve (6m + 3n) (6m – 3n) Solution: This proceeds as: Given polynomial (6m + 3n) (6m – 3n) represents the identity a^{2} - b^{2} = (a + b) (a – b) Where a = 6m and b = 3n Now apply values of a and b on the identity i.e. a^{2} - b^{2} = (a + b) (a - b) and we get: (6m + 3n) (6m – 3n) = (6m)^{2} - (3n)^{2} Expand the exponential forms on the LHS and we get: = 36m^{2} - 9n^{2} Hence, (6m + 9n) (6m – 9n) = 36m^{2} - 81n^{2} |