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I'm a little embarrassed to be asking this question as I feel it is extremely easy, but I will do so anyway. I'm self studying some algorithms, and the book that I'm using claims that

[itex]a n^2 + b n + c \in \Omega(n^2)[/itex]

Of course, this is equivalent to saying that there exists [itex] c_1, c_2 \in ℝ^+[/itex] such that

[itex] 0 \leq c_1 n^2 \leq a n^2 + b n + c \leq c_2 n^2 [/itex] for all [itex] n \geq n_0^* [/itex]

In order to verify this, the book says that I should let [itex] c_1 = a/4 [/itex], [itex] c_2 = 7a/4[/itex] and [itex] n \geq 2 \cdot max(|b|/a, \sqrt{|c|/a})[/itex]. This would imply that

[itex] 0 \leq a n^2/4 \leq a n^2 + b n + c \leq 7a n^2/4 [/itex] for all [itex] n \geq n_0^* [/itex]

I understand that they picked [itex] c_1 [/itex] and [itex] c_2 [/itex] (almost) arbitrarily. However, I just can't see how they picked their lower limit on [itex] n[/itex] given these two constants. I've been messing around with the quadratic equation for a little while now and I just fail to see it.

Any help would be appreciated! I just want to understand the logic of their selection of [itex] n_0^*[/itex]. THanks!

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# Solving a simple inequality arising from the study of algorithms -asymptotic notation

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