Why doesn't chemical equilibrium lead to uncertainties? Let's consider a reaction

$$\ce{a A + b B &lt;=&gt; c C + d D}$$

We will also assume that the concentrations are low and the activity coefficient is one. Now let's say initially one has $p_0$, $q_0$, $r_0$, $s_0$ as the concentrations of $\ce{A}$, $\ce{B}$, $\ce{C}$, $\ce{D}$, respectively. Let the equilibrium constant be $K$. Let's investigate about the equilibrium composition.

If $x$ equivalents of reactants reacted, then the equation for $x$ would be: 

$$K = \frac{(r_0 + cx)(s_0 + dx)}{(p_0 - ax)(q_0 - bx)}.$$

Now, my question is how do we know all the time such equations have proper roots for $x$? I mean all the conditions $x$ needs to satisfy, like $ax &lt; p_0$ and first of all, how do we know the equation has roots in first place? It's a quadratic and it need not have real roots necessarily. This is just a simple case of 2 reactants and 2 products.

I would like to see some rigorous proof that such equations always contain proper (roots which satisfy) all the conditions or a counter example of such a chemical equation. If such counter examples exist, how does the chemical reaction proceed?
The reaction quotient Q approaches zero when at least one of the product concentration approaches zero (as the reaction goes to completion in the reverse direction and the extent of reaction x is at the lowest possible value), and it approaches infinity when at least one of the reactant concentrations approaches zero (as the reaction goes to completion in the forward direction and the extent of reaction x is at the highest possible value). Also, the reaction quotient increases monotonically with x.

No matter what K is, there is always a Q that matches K because Q assumes all values between zero and infinity.

The only real case where this argument does not work (real as in concentrations have to be non-negative real numbers) is when at least one reactant and one product concentration is zero. In that case, there will be no reaction at all, and Q is not defined.