Given an equation ax^2+bx+c=0
what values a,b cause the roots of the equation to be a and b and the discriminant to be equal to c?

Well yes, if the roots are a and b, then (x-a)(x-b) = 0

Multiplying by a gives

ax^{2} -a(a+b)x +a^{2}b = 0

Comparing the two equations, gives

(1) b = -a(a+b) which leads to b = -a^{2}/(a+1), just as Dej Mar says

Also,

(2) c = a^{2}b

and

(3) c = the discriminant = b^{2 }-4ac

Solving (1), (2), and (3) should lead to discrete a, b and c values which constitute the solution