Two sets S₁ and S₂ are defined for [0, 3 ] as follows: S₁ = [0, 3 ] : ^2 - 3 + 1 = 0 S₂ = [0, 3 ] : 2 = 0 The sum of all elements in S₁ S₂ is

Step-by-step solution

For the set S₁ , we have the equation: ^2 - 3 + 1 = 0 Using the identity ^2 = 1 + ^2 , this becomes: ^2 + 1 - 3 + 1 = 0 ^2 - 3 + 2 = 0 ( - 1)( - 2) = 0 Thus, = 1 or = 2 . For the set S₂ , we have the equation: 2 = 0 Using the identity 2 = 1 - ^2 1 + ^2 , we get: 1 - ^2 1 + ^2 = 0 ^2 = 1 Thus, = 1 or = -1 . The elements in S₁ S₂ must satisfy the conditions of both sets simultaneously. The only common condition is: = 1 We need to find all solutions for = 1 in the interval [0, 3 ] . The solutions are = 4 , + 4 , 2 + 4