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
Observe that this is an equation in the one unknown X.