Which of the following equations has only one solution?

To determine which equation has only one solution, we need to examine the given options. An equation has only one solution if it can be expressed in the form of a perfect square trinomial, where the discriminant (b² – 4ac) is equal to zero.

Let’s analyze the equations:

1. x² – 9 = 0: This can be rewritten as (x – 3)(x + 3) = 0, yielding two solutions: x = 3 and x = -3.
2. x² + 9x + 1 = 0: The discriminant is 9² – 4(1)(1) = 81 – 4 = 77, which is greater than 0. This means this equation has two solutions.
3. x² – 6x + 9 = 0: This can be factored as (x – 3)² = 0, resulting in only one solution: x = 3.

Thus, the equation that has only one solution is x² – 6x + 9 = 0.