Revision history [back]

Your question is unclear: do you want to solve the inequality, or do you want to express $d$ in terms of the other variables, knowing $f(a,b,c,d) > 0$ and $c < 0$?

In the first case, you have a polynomial expression $f(a,b,c,d)$ which you want to be positive. The usual way to solve that is to understand where the expression is zero.

Since $f$ is continuous, it has a fixed sign on each connected component of the complement of this zero locus, so all that remains to understand is on which components $f$ is positive.

In the second case, you should notice that, calling $\Delta$ the fixed value of $f(a,b,c,d)$, the equation $f(a,b,c,d) = \Delta$ can be rewritten as $A * d^2 + B * d + C = 0$, where $A$, $B$, $C$ are expressions in $a$, $b$, $c$, $d$, $\Delta$.

Your question is unclear: do you want to solve the inequality, or do you want to express $d$ in terms of the other variables, knowing $f(a,b,c,d) > 0$ and $c < 0$?

In the first case, you have a polynomial expression $f(a,b,c,d)$ which you want to be positive. The usual way to solve that is to understand where the expression is zero.

Since $f$ is continuous, it has a fixed sign on each connected component of the complement of this zero locus, so all that remains to understand is on which components $f$ is positive.

In the second case, you should notice that, calling $\Delta$ the fixed value value of $f(a,b,c,d)$, $f(a,b,c,d)$, notice that the equation $f(a,b,c,d) = \Delta$ can be rewritten as $A * d^2 + B * d + C = 0$, where $A$, $B$, $C$ are expressions in $a$, $b$, $c$, $d$, $\Delta$.