Square Extension!

Using the law of cosines on $\triangle{BEA}$ and $\triangle{AED} \implies$

$25 = x^2 + 4 - 4x\cos(\theta)$

$16 = x^2 + 4 + 4x\sin(\theta)$

$\implies$

$4x\cos(\theta) = x^2 - 21$

$4x\sin(\theta) = 12 - x^2$

$\implies$

$16x^2\cos^(\theta) = x^4 - 42x^2 + 441$

$16x^2\sin^2(\theta) = x^4 - 24x^2 + 144$

Adding the two equations directly above $\implies$

$2x^4 - 82x^2 + 585 = 0 \implies x^2 = \dfrac{41 \pm \sqrt{511}}{2}$

For $(+)$ $\sin(\theta) = \dfrac{12 - x^2}{4x} < -1 \therefore$ we choose $A = x^2 = \dfrac{41 -\sqrt{511}}{2} =$

$\dfrac{a - \sqrt{b}}{c} \implies a + b + c = \boxed{554}$ .