Choice A is the correct answer.

1. $\Delta = 17 > 0$, so roots are real and distinct.
2. 17 is not a perfect square, so $\sqrt{17}$ is irrational.
3. Therefore, the roots are irrational.

Choice B is incorrect because 17 is not a perfect square. Choice C is incorrect because $\Delta eq 0$. Choice D is incorrect because $\Delta > 0$.