MHB Modifying values to lie within existence domain

[Siron]

Given three parameters:
$$a= \frac{(k-3)^2 \sqrt{v}}{s}, \ \ b = \frac{v}{s}(w-10s), \ \ c = s \sqrt{v}.$$
which exact values I know (that is, I know $v,s,k$ and $w$ exactly). I need to guarantee that $a<0$ (this is always satisfied in my calculations!) and
$$0<b<1.08148a^2$$
For instance, if $b>1.08148a^2$ then I will replace $w$ by $w'$ such that $b(v,s,w') = 1.08148a^2$ and hence $b(v,s,w') - \epsilon$ satisfies the constraint. Here $\epsilon>0$ is an arbitrary small number.Consider another parameter $c = s \sqrt{v}$. I now have to guarantee that:
$$a<0, \quad 0<b<1.08148a^2, \quad q < c < 0, \qquad (*)$$
where $q$ is the greatest real root of the quartic polynomial
$$(48a^2+16b)x^4 - (40a^3+168 ab)x^3+(-45a^4+225a^2b + 72b^2)x^2+(27a^3b-162 ab^2)x+27b^3.$$
The explicit expression (which is a function of $a$ and $b$) for $q<0$ is quite horrendous. However, to guarantee that $q<0$ is real, I need that $a<0$ and $0<b<1.08148a^2$. So my question is, suppose I know the values of $a,b$ and $c$ where $b$ and $c$ do not satisfy the constraints. How can I modify $w$ and $s$ (similarly as above) **beforehand**, such that the constraints are satisfied? An ideal solution would be: replace $(s,w)$ by $(s',w')$, which gives me modified values $a',b'$ and $c'$, such that
$$0<b'<1.08148(a')^2, \qquad q(a',b')<c'<0.$$Thanks in advance!

 

[Siron]

UPDATE:

If for instance $b>1.08148a^2$ then I replace $w$ by $w'$ such that $b(v,s,w') = 1.08148a^2$ and then I extract $\epsilon>0$ to guarantee that the new value $b'(v,s,w')<1.0848a^2$ as required. On the other hand I also need to guarantee that $q(a,b)<c<0$. Since I adjusted $b$, $q(a,b)$ also changes and hence I need to guarantee that $q(a,b(v,s,w')) = q(a,1.08148a^2-\epsilon)<c<0$. However, to that end I have to adjust $s$ since $c(s,v) = s\sqrt{v}$. Since I adjust $s$ the value for $a$ and $b$ will be different as well, because they are functions of $s$. Therefore, replacing $(s,w)$ beforehand by $(s',w')$ such that the constraints are satisfied, comes down to solving the following system of equation for $(s',w')$
$$\begin{cases} b(v,s',w') = 1.08148(a(v,s',k))^2 \\ q(a',1.08148(a')^2-\epsilon )= s' \sqrt{v} \end{cases},$$
which is equivalent with
$$\begin{cases} \frac{v}{s'}(w'-10s') = 1.0848 \frac{(k-3)^2 v}{(s')^2} \\ q(a',1.0848(a')^2-\epsilon) = s' \sqrt{v} \end{cases}$$

Now, solving the first equation of the system for $w'$ is straightforward. The value for $s'$ should be computed from second equation. However, the expression for $q(a',1.08148(a')^2-\epsilon)$ is probably quite ugly (but it is possible to compute it). Maybe I should use a numerical algorithm here? Any suggestions?

Thanks!