If $\frac{1}{2-i}$ is a root of $ax^2+bx+c=0$ and $\frac{1}{3-2\sqrt 2} $ is a root of $px^2+dx+q=0$

Question

find the inequality relation between $a, b, c$ and $d$

The roots can be written as $$\frac{2+i}{5}$$ and $$3+2\sqrt 2$$ Then $$\frac{-b}{a}=\frac 45, \frac ca = \frac 15$$ And $$\frac{-d}{p}=6, \frac qp=1$$

It’s clear that $a>c>b$ but I can’t find a relation for $d$. Please help thanks!

Answer 1

To reach your conclusions so far you are assuming that $a,b,c,d,p,q$ are real - is that condition actually given in the question ?

Without further constraints there is no relation between $a,b,c$ and $d$. This is because

$px^2+dq+q=0 \Rightarrow kp^2 + kd + kq = 0 \forall k \ne 0$

which means we can multiply or divide $p,d,q$ by any non-zero number to get alternate values

$p'=kp, d'=kd', q'=kq$

So we can create a $d'$ that is greater than $\max(a,b,c)$ or less than $\min(a,b,c)$ or anywhere in between.