Prove that if x and y are .... 1. The problem statement, all variables and given/known data Prove that if x and y are distinct real numbers, then (x+1)2=(y+1)2 if and only if x+y=-2. How does the conclusion change if we allow x=y? 2. Relevant equations ... 3. The attempt at a solution Suppose x and y are real numbers. If x≠y then x+2≠y+2 ----> (x+2)/(y+2)≠1 ----> (x+2)/(y+2)≠ x/y or y/x (since x/y=y/x=1) ----> (x+2)x=(y+2)y or (x+2)/x=(y+2)/y ----> (x+2)x=(y+2)y (since (x+2)/x=(y+2)/y ---> x=y) ----> x2+2x=y2+2y ---->x2+2x+1=y2+2y+1 ---->(x+1)2=(y+1)2 ----> x+1 = (y+1) or (-y-1) ----> x+1=-y-1 (since x+1=y+1 ---> x=y) ----> x+y=-2 ????? But I'm not sure if I've done everything it asked. I know for "P if and only Q," it needs to be proven that P--->Q and Q--->P, but it seems here that if P is (x+1)2=(y+1)2 and Q is x+y=-2, I'm just kind of going in circles by proving the "if and only if" part. See what I'm saying?