A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 2

To determine the range of possible values for $k$ that allows the equation $x^2 - 6x + k = 0$ to have two distinct positive roots, we consider two conditions:

1. Condition for Distinct Roots: The discriminant must be positive. Thus,

   36 - 4k > 0\ k < 9$$

2. Condition for Positive Roots: The value of $k$ must also ensure that the roots are positive. The vertex $(3, k - 9)$ indicates that for the roots to be positive (the parabola opens upwards), we must have:

   1. The y-intercept $k > 0$ (the parabola must cross the y-axis above 0).
   2. Ensuring the parabola intersects the x-axis positively requires the vertex' vertical position to be such that its height keeps the roots in the positive region, specifically:
   $$$$

ightarrow k < 9$$

Therefore, combining both conditions gives:

$0 < k < 9$

Thus, the final range of possible values for $k$ is:

Answer: $0 < k < 9$.