we know that

We want to minimize A

subbing the first into the second we get...

Since we want the minimum value we need to find the vertex of the parabola.

There are a few different ways. Since it is symmetric about its vertex the vertex is half way between its intercepts -24 and 0 so at -12

or we could use the vertex fromula

Now that we know y we can solve for x

so the minimum product is