In the book CLRS the authors say that every quadratic, asymptotically nonnegative function $ f(n) = an^2 + bn + c$ is an element of $ \Theta(n^2)$ . Using the following definition

\begin{align*} \Theta(n^2) = \{h(n) \,|\, \exists c_1 > 0, c_2 > 0, n_0 > 0 \,\forall n \geq n_0: 0 \leq c_1n^2 \leq h(n) \leq c_2n^2\} \end{align*}

the authors write that $ n_0 = 2*\max(|b|/a, \sqrt{|c|/a})$ .

I have difficulties proving that the value of $ n_0$ is indeed that value.

We know that $ a \ge 0$ because otherwise $ f$ would not be asymptotically nonnegative. Calculating the roots of $ f$ gives us:

\begin{align*} n_{1/2} &= \frac{-b \, \pm \, \sqrt{b^2 – 4ac} }{2a} \ &\leq \frac{|b| + \sqrt{b^2 – 4ac} }{a} \end{align*}

The case $ c \ge 0$ gives us:

\begin{align*} \frac{|b| + \sqrt{b^2 – 4ac} }{2a} \leq \frac{|b| + \sqrt{b^2} }{a} = 2\frac{|b|}{a} \end{align*}

which is two times the first argument of the $ \max$ function.

But what about the case $ c < 0$ ? How can we find an upper bound for that? Where does the value $ \sqrt{|c|/a}$ actually come from?