Let Q = x² + bx + c. If the sum of the roots of the equation Q = 0 is equal to the product of these roots, then which of the following is/are correct? I. Q can be a perfect square. II. Q is positive for all real values of x.

Sum of roots = –b, product of roots = c. Given –b = c, so b = –c. Then Q = x² – cx + c. For Q to be a perfect square, discriminant must be 0: c² – 4c = 0, so c = 0 or c = 4. When c = 0, Q = x² (perfect square). When c = 4, Q = x² – 4x + 4 = (x–2)² (perfect square). So I is correct. For II: when c = –1, Q = x² + x – 1, which is negative for some x. So II is not always correct. Only I.