1. The quadratic equation can be rearranged to where is the vertex of the corresponding parabola.

2. For symmetry's reason you know that the mean of the zeros equals . From you can get two different possible zeros: or

Calculating the mean value you'll get:

which is definitely no rational number;

or

which is a valid number for according to the text of your question.

3. The maximum value occurs at the vertex. Therefore you know now:

4. Plug in the value of the known zero and solve for a:

I've got