Question 6
Suppose that a parabola has vertex $\left(\frac{1}{4}, -\frac{9}{8}\right)$ and equation $y = ax^2 + bx + c,$ where $a \gt 0$ and $a + b + c$ is an integer. The minimum possible value of $a$ can be written in the form $\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$