Identity: (a - b)^{2} = a^{2} + b^{2} - 2ab
How is this identity obtained? Let's see how. Taking LHS of the identity: (a - b)^{2} This can also be written as: = (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} On adding like terms we get: = a^{2} - 2ab + b^{2} On rearranging the terms we get: = a^{2} + b^{2} - 2ab Hence, we obtain the identity i.e. (a - b)^{2} = a^{2} + b^{2} - 2ab Following are a few applications to this identity. Example 1: Solve (2x - 3y)^{2} Solution: This proceeds as: The given polynomial (2x - 3y)^{2} represents the identity (a - b)^{2} Where a = 2x and b = 3y On applying values of a and b on the identity i.e. (a - b)^{2} = a^{2} + b^{2} - 2ab we get: (2x - 3y)^{2} = (2x)^{2} + (3y)^{2} - 2(2x)(3y) Expand the exponential forms and we get: = 4x^{2} + 9y^{2} - 2(2x)(3y) On solving multiplication process we get: = 4x^{2} + 9y^{2} - 12xy Hence, (2x - 3y)^{2} = 4x^{2} + 9y^{2} - 12xy Example 2: Solve (6m - 9n)^{2} Solution: This proceeds as: The given polynomial (6m - 9n)^{2} represents the identity (a - b)^{2} Where a = 6m and b = 9n On applying values of a and b on the identity i.e. (a - b)^{2} = a^{2} + b^{2} - 2ab we get: (6m - 9n)^{2} = (6m)^{2} + (9n)^{2} - 2(6m)(9n) Expand the exponential forms and we get: = 36m^{2} + 81n^{2} - 2(6m)(9n) On solving multiplication process we get: = 36m^{2} + 81n^{2} - 108mn Hence, (6m - 9n)^{2} = 36m^{2} + 81n^{2} - 108mn |