The first polynomial equation is a
(Ricatti-Lyapunov) equation in 
.
Numerical methods for solving Ricatti equations
are common.
For this reason assuming that a Ricatti equation has a solution is a
socially acceptable
necessary condition throughout control
engineering.
Thus we can consider to be known.
A first glance at the second equation reveals
that the same products of unknowns appear over and over.
Also we can see that this equation is symmetric. It would not take an
experienced person long to realize that by multiplying this
equation on the left by 
and
on the right by 
,
we will have an equation in one unknown.
Now we can replace 
with a new
variable X. This yields
X
+
X
B
C
+
C
B
X
-
X
B
B
X
+
X
B
B
X
Observe that this is an equation in the one unknown X.