The question in my book says:
Find the necessary conditions in which the quadratic equation $ax^2+bx+c$ would have roots (m, n) such that (in individual cases):
(i) m=2n
(ii) m=n+3
It's really annoying because whatever knowledge I had about quad. equ.'s didn't work and the answer the book gave was:
(i) $9AC=2B^2$
and (ii) $B^2=9A^2+4AC$
I gave this 10 minutes of thought; none.
The question form is obviously the problem; I don't have a teacher so I thought that someone might have a better explanation for "conditions" generally and specifically in quadratic roots.
$ m=2n \\ ax^2(x-m)(x-n) \\ a(x-[n+3] \hspace{3pt})(x-n) \\ ax^2-2anx-3ax+an^2+3an \\ \hspace{-.9cm} = \hspace{.3cm} ax^2-ax(2n-3)+an(n+3) \\ *n=\frac {c} {a(n+3)} \\ *-ax \left[ \frac{2c}{2an+6a} \right] = b \\ x \left( \frac {c}{n+3}+3a \right)=b$