It is known that z = I-1 is a root of the equation Z ^ 2 + ax + B = 0. 1) find the value of real numbers a and B. 2) conjecture the equation by combining WIDA's theorem And prove it

It is known that z = I-1 is a root of the equation Z ^ 2 + ax + B = 0. 1) find the value of real numbers a and B. 2) conjecture the equation by combining WIDA's theorem And prove it

From z = I-1, (i-1) ^ 2 + a (i-1) + B = 0, i.e. I ^ 2-2i + 1 + ai-a + B = 0, (A-2) I + (B-A) = 0, because a and B are real numbers, A-2 = 0, A-B = 0, so a = 2, B = 2, let the other root be X. according to Weida's theorem, Z + x = - A, x = - 1-I, substitute x = - 1-I into the original equation, the left and right sides are equal, so the other root is - 1-i