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Identity: (a - b)2 = a2 + b2 - 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) = a2 - ab - ab + b2 On adding like terms we get: = a2 - 2ab + b2 On rearranging the terms we get: = a2 + b2 - 2ab Hence, we obtain the identity i.e. (a - b)2 = a2 + b2 - 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 = a2 + b2 - 2ab we get: (2x - 3y)2 = (2x)2 + (3y)2 - 2(2x)(3y) Expand the exponential forms and we get: = 4x2 + 9y2 - 2(2x)(3y) On solving multiplication process we get: = 4x2 + 9y2 - 12xy Hence, (2x - 3y)2 = 4x2 + 9y2 - 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 = a2 + b2 - 2ab we get: (6m - 9n)2 = (6m)2 + (9n)2 - 2(6m)(9n) Expand the exponential forms and we get: = 36m2 + 81n2 - 2(6m)(9n) On solving multiplication process we get: = 36m2 + 81n2 - 108mn Hence, (6m - 9n)2 = 36m2 + 81n2 - 108mn |