Let ABC be non-zero real numbers which are not equal to each other, and prove three equations It is impossible to prove that three equations ax ^ 2 + 2bx + C = 0, BX ^ 2 + 2cx + a = 0, CX ^ 2 + 2aX + B = 0 have two equal real roots

Let ABC be non-zero real numbers which are not equal to each other, and prove three equations It is impossible to prove that three equations ax ^ 2 + 2bx + C = 0, BX ^ 2 + 2cx + a = 0, CX ^ 2 + 2aX + B = 0 have two equal real roots

If these three equations all have two equal real roots, then 4B ^ 2-4ac = 04C ^ 2-4ab = 04A ^ 2-4bc = 0, the three equations add up to 4 (a ^ 2 + B ^ 2 + C ^ 2-ab-bc-ac) = 0, and a ^ 2 + B ^ 2 + C ^ 2-ab-ac-ab = (a-b) ^ 2 / 2 + (A-C) ^ 2 / 2 + (B-C) ^ 2 / 2 = 0, so a = B, a = C, B = C and ABC are not equal numbers, so the three equations ax