Let S = x [- , ] : x ( x + x) = a, a Z . Then n(S) is equal to :

Step-by-step solution

Given the equation: x ( x + x) = a ^2 x + x x = a Using the double angle formulas ^2 x = 1 - 2x 2 and x x = 2x 2 , we get: 1 - 2x 2 + 2x 2 = a 2x - 2x = 2a - 1 We know that the range of A + B is [- A^2+B^2 , A^2+B^2 ] . Thus, the range of 2x - 2x is [- 2 , 2 ] . - 2 2a - 1 2 1 - 1.414 2a 1 + 1.414 -0.414 2a 2.414 -0.207 a 1.207 Since a Z , the possible values for a are 0 and 1 . Case 1: a = 0 x ( x + x) = 0 x = 0 or x + x = 0 For x [- , ] , x = 0 x - , 0, (3 solutions) x + x = 0 x = -1 x - 4 , 3 4 (2 solutions) Tot