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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!

Harsha
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Aditya
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    Multiplying the quadratic by $-1$ doesn't change the roots but reverses the sign of coefficients. Notice $a=-5 \Rightarrow b=4, c = -1$. So you can't even conclude $a\gt c\gt b$. – AgentS Nov 14 '19 at 12:42

1 Answers1

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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.

gandalf61
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