asymptotic tight bound for quadratic functions

Question

In CLRS (Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein), for a function

f(n) = an² + bn

Answer 1

The idea is to (for big enough n) "trap" the function of interest between two "pure" growth functions (that have only a single constant of proportionality). In this figure two quadratic functions (drawn in red and blue) are trapped between two pure growth functions (drawn in black), and the minimum possible value of n₀ in each case is indicated.

So once you've picked your values of c₁ and c₂, you can find the value of n₀ by intersecting your function with the two pure growth functions and taking the rightmost intersection.

But you don't care about the getting smallest possible value for n₀ — we're doing asymptotics here, so any big enough value will do — so you can use approximations to get an upper bound on it.

See davin's answer for how to get the upper bound for n₀ into the form given in CLRS.