If S = [- , ] : 5 2 = 7 7 2 , then n(S) is equal to _______.

Step-by-step solution

Given equation: 5 2 = 7 7 2 Multiplying both sides by 2 : 2 5 2 = 2 7 7 2 Using the identity 2 A B = (A+B) + (A-B) : ( + 5 2 ) + ( - 5 2 ) = (7 + 7 2 ) + (7 - 7 2 ) ( 7 2 ) + ( -3 2 ) = ( 21 2 ) + ( 7 2 ) Since (-x) = x , we get: ( 7 2 ) + ( 3 2 ) = ( 21 2 ) + ( 7 2 ) ( 3 2 ) = ( 21 2 ) ( 21 2 ) - ( 3 2 ) = 0 Using the identity C - D = -2 ( C+D 2 ) ( C-D 2 ) : -2 ( 21 2 + 3 2 2 ) ( 21 2 - 3 2 2 ) = 0 -2 (6 ) ( 9 2 ) = 0 This gives (6 ) = 0 or ( 9 2 ) = 0 . Case 1: (6 ) = 0 6 = n = n 6 for n Z . Since [- , ] , we ha