As you didn't show your results, I'll redo the computation.
Let us substitue the RHS of the two middle equations in the fourth, squared:
$$d^2(x^2+b)y^2=(y^2+c)x^2,$$ or
$$(d^2b-x^2)y^2=(c-d^2)x^2.$$
Then, from the first equation,
$$y^2=(a-x)^2$$
and finally, eliminating $y$,
$$(c-d^2)x^2=(d^2b-x^2)(a-x)^2.$$
This confirms that the problem is of the quartic degree and has an analytical solution, but the formulas are terrible and hardly manageable by hand.
And as the polynomial has three independent coefficients, there is no reason that any simplification occurs and you won't find any simpler analytical resolution (otherwise, you would revolutionize the Galoi's theory).
The other option is to resort to numerical methods, but whether this is feasible by hand is unsure, and it is only doable for particular values of the parameters $a,b,c,d$.
Best Answer
Obviously $A\neq 0$, then $B=\frac{2}{A}$. Subtitute into the first equation:
$A^2+\frac{4}{A^2}=5 \implies 0=A^4-5A^2+4=(A^2-1)(A^2-4)$
Here we can deduce 4 values of $A$ like your solutions, find $B=\frac{2}{A}$ and it's done