more direct and simpler solution: since we know that x must be complex, let x = a + bi, -x = -(a + bi) sqrt(a + bi) + sqrt(-(a + bi)) = sqrt(a + bi) + i sqrt(a + bi) = sqrt(a + bi) (1 + i) = 32 = 2^5 squaring both sides: 2^10 = (a + bi) 2i = 2ai - 2b since 2^10 is real, 2ai = 0, so a = 0 thus 2^10 = -2b, or b = - 2^9, whereupon x = - 2^9 i = -512 i because of the symmetry in the original equation, x is +/- 512 i
more direct and simpler solution:
since we know that x must be complex, let x = a + bi, -x = -(a + bi)
sqrt(a + bi) + sqrt(-(a + bi)) = sqrt(a + bi) + i sqrt(a + bi) = sqrt(a + bi) (1 + i) = 32 = 2^5
squaring both sides: 2^10 = (a + bi) 2i = 2ai - 2b
since 2^10 is real, 2ai = 0, so a = 0
thus 2^10 = -2b, or b = - 2^9, whereupon x = - 2^9 i = -512 i
because of the symmetry in the original equation, x is +/- 512 i