# Find roots of characteristic equation $p(a,k)=0$

I need help understanding a derivation I've seen in a paper.

I have a characteristic equation expressed as a polynomial $p(a,k)$, which means I can represent the characteristic equation as

$p_a(a,k) = c_0(k) + c_1(k)a + ... + c_n(k)a^n$

or

$p_k(a,k) = d_0(a) + d_1(a)k + ... + d_m(a)k^m$

Assume that by looking at the coefficients of $p_a$ one observes that a root $a$ with positive real part exists. Assume also that $p_k(a,k)$ is quadratic in $k$, that $k>0$ and that by looking at $k\rightarrow 0$ and $k\rightarrow \infty$ one observes that in those extreme cases the roots are either zero or have a negative real part.

Given the previous observations, the next step in the derivation is concluding that there exists a value of $k=k_m$ for which the corresponding $a_m>0$ is a maximum and satisfies $\frac{\partial a_m}{\partial k_m}=0$

Afterwards (and this is where I got lost) it is claimed that the value of $k_m$ will be given by

$k_m=-\frac{d_1}{2d_2}$

considering that $p_k=d_0+d_1k+d_2k^2$. This seems like the maximum of $p_k$ with respect to $k$. Is there a correlation between the partial of $p$ wrt $k$ and the partial of $a$ wrt $k$? Something like

$\frac{\partial p}{\partial k}\bigg|_{k_m}=0 \Rightarrow \frac{\partial a}{\partial k}\bigg|_{k_m}=0$ ?

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