How are positive intervals calculated?

Question

Give these equations

$N=p^4-p^3+161*p$

$N+(n/2)^2=M^2$

$p*(p+n)=161$

How to calculate the intervals in which $n>0$ and $p>0$ and $M>0$ and $N>0$ ?

$n$ and $N$ and $p$ and $M$ not imaginary

Answer 1

All you need, in fact, is that $M>\frac{n}{2}$.

First, the $3^{rd}$ line gives us $p^2 + np = 161$. This is always satisfied for $p>0$, $n>0$.

Now, substituting this in the $1^{st}$ line, we get - $$N = p^4 + np^2 = p^2(p^2+n)$$ So, if $n>0$, $N>0$.

Thus, considering the $2^{nd}$ equation, we see that $N>0$ only if $M>\frac{n}{2}$.