If $a(p+q)^2+2bpq+c=0$ and $a(p+r)^2+2bpr+c=0$ then find $qr$

Question

If it is given that: $$a(p+q)^2+2bpq+c=0\\a(p+r)^2+2bpr+c=0$$ then find $qr$.

I dont have any clues on how to begin. I tried subtracting, but won't take it anywhere. This looks like a hidden quadratic but i cannot figure it out! Please give me some clue!

Answer 1

HINT:

Clearly, $q,r$ are the two roots of $$0=a(p+t)^2+2bpt+c=at^2+2tp(a+b)+ap^2+c$$

Answer 2

I think it's impossible to calculate $qr$. Try $a=b=c=0$.

Answer 3

Just for completeness:

If $a\not=0$, then lab bhattacharjee's answer gives

$$qr={ap^2+c\over a}$$

If $a=0$ and $bp\not=0$, then we have $q=r=-c/(2bp)$, and thus

$$qr={c^2\over4b^2p^2}$$

Finally, if $a=bp=0$, then we must have $c=0$ as well and, as in Michael Rozenberg's answer, $q$ and $r$ can be anything, so $qr$ cannot be calculated.