Perfect Squares Only

Number Theory Level 5

Let $a,b,c$ be nonzero perfect squares such that $b = a+c$ . The equation $ax^2 - bx + c = 0$ has the following properties:

It has a real root $d$ that is greater than 1. (The other root may or may not be real.)
The square root of the quadratic discriminant is a prime number.

Among all such $(a,b,c,d)$ , pick the one such that the discriminant is minimum. Find $\sqrt{abcd}$ .