The overall question I'd like to ask has been answered here in the accepted answer: asymptotic tight bound for quadratic functions but I'd like to focus on a sub-part of the answer that I can't understand.
Specifically, it is this part: "So we can bound from above the inside of the sqrt(...) with 4b^2".
I can't figure out how assuming that |b|/a >= sqrt(|c|/a) helps us arrive at the 4b^2 bound for the b^2-3ac term. Here's what I get:
n >= 2*(sqrt(b^2-3ac)-b)/3a
There are two cases (as the original respondent said). I'd like to understand the first one:
(square both sides) b^2/a^2 >= |c|/a
(multiply across by a^2) b^2 >= a^2*|c|/a
(simplify the a^2 and a) b^2 >= a*|c|
(a is positive, so a|c| >= ac) b^2 >= ac
So if we look at the inside of the original sqrt, which was b^2-3ac, we have
b^2-3ac is >= -2ac
Not 4b^2 as indicated in the original response.
Could someone help me understand where I have gone wrong?
Thanks!
There are two assumptions made in the answer : "we have a positive leading coefficient polynomial", i.e., a>0 and |b|/a >= sqrt(|c|/a).
Here's the derivation(each step implies the next) :
|b|/a >= sqrt(|c|/a)
b^2/a^2 >= |c|/a, (squaring both sides)
b^2 >= |c|*a, (multiplying by a^2, since a^2>=0)
3b^2 >= 3*a*|c| = |3*a*c|, since a>0
b^2 + 3b^2 >= b^2 + |3*a*c| == b^2 + |-3*a*c| >= b^2 - 3*a*c, since |x| + |y| >= |x+y|
The derivation you have shown isn't incorrect. It simply doesn't give you a bound that lets you proceed.