# Construct 2: Case 4

$\ y = [LuxR] + [A] \Rightarrow [LuxR] = y - [A]\cdots\cdots(1)$
Likewise
$\ x = [AHL] + [A] \Rightarrow [AHL] = x - [A]\cdots\cdots(2)$

Substitute (1) and (2) into $\ [A] = K_{\alpha}[AHL][LuxR]$

$\therefore [A] = K_{\alpha}(x - [A])(y - [A])\cdots\cdots(3)$

Expand (3)

$[A]^2 - \left((x+y) + \frac{1}{K_{\alpha}}\right) [A] + xy = 0$

More will follow shortly, concerning the nature of roots and stability

$\Delta = \left((x+y) + \frac{1}{K_{\alpha}}\right)^2 - 4xy$