If + =0= + , then what is a value of 2 + 2 +2 ( + ) ?

Step-by-step solution

Let z₁ = + i and z₂ = + i . From the given conditions: z₁ + z₂ = ( + ) + i( + ) = 0 Squaring both sides: (z₁ + z₂)^2 = 0 z₁^2 + z₂^2 + 2z₁ z₂ = 0 Using De Moivre's theorem, z₁^2 = 2 + i 2 and z₂^2 = 2 + i 2 . The product term is 2z₁ z₂ = 2( ( + ) + i ( + )) . Substituting these into the squared equation: ( 2 + 2 + 2 ( + )) + i( 2 + 2 + 2 ( + )) = 0 Equating the real part to zero: 2 + 2 + 2 ( + ) = 0 Answer: 0