In the book Introduction to Algorithms (the renowned CLRS), the author writes in the 3rd Chapter, Growth of functions :
"Consider any quadratic function: $f\left(n\right)=an^2+bn+c$, where a, b and c are constants and $a>0$. Throwing away the lower-order terms and ignoring the constant yields: $f(n)=\Theta(n^2)$. Formally we take the constants $c_1=a/4$, $c_2=7a/4$ and $n_0=2\cdot\max\left(\frac{\left|b\right|}{a},\sqrt{\frac{\left|c\right|}{a}}\right)$. You may verify that $0\le c_1n^2\le an^2+bn+c\le c_2n^2$ for all $n\ge n_0$."
I'm having difficulty in discovering the term: $n_0=2\cdot\max\left(\frac{\left|b\right|}{a},\sqrt{\frac{\left|c\right|}{a}}\right)$.
I tried to calculate $f\left(n\right)$ for $ n = \frac{\left|b\right|}{a}$ and got: $$f(n)= c,b<0$$ $$f(n)=\frac{2b^2}{a}+c, b>0$$ and for $n=\sqrt{\frac{\left|c\right|}{a}}$ and got: $$f\left(n\right)=b\sqrt{\frac{\left|c\right|}{a}}, c<0$$ $$f\left(n\right)=2c+b\sqrt{\frac{\left|c\right|}{a}}, c>0$$ Nevertheless, I have not got any clue as to how these constants were obtained and how can I derive its significance.