Let x [0, 6 ] satisfy the equation x - x = -1 . If x = k ( 3 ) where k N , then find the number of possible values of k is .....

Step-by-step solution

Given equation: x - x = -1 x + 1 = x Squaring both sides: ^2 x + 2 x + 1 = ^2 x ^2 x + 2 x + 1 = 1 - ^2 x 2 ^2 x + 2 x = 0 2 x( x + 1) = 0 x = 0 or x = -1 If x = 0 , substituting in the original equation gives x = 1 . x = 2n + 2 = (4n+1) 2 Equating with x = k ( 3 ) , we get k ( 3 ) = (4n+1) 2 k = 3(4n+1) 2 Since k N , this yields no integer solutions for k . If x = -1 , substituting in the original equation gives x = 0 . x = (2n+1) Equating with x = k ( 3 ) , we get k ( 3 ) = (2n+1) k = 3(2n+1) Since x [0, 6 ] , th