Prove that $T_n = 3n^2 -60n + 301$ is positive for every $n$ I recently did a Mathematics exam from a previous year, and I stumbled across a question's answer I struggled to fully understand.
It is given: The quadratic pattern $244 ;~ 193 ;~ 148 ;~ 109;~ \ldots$
I've determined the $n$-$\textrm{th}$ term as $T_n = 3n^2 -60n + 301$.
Now the questions asks:

Show that all the terms of the quadratic pattern is positive.

Our teacher explained completing the square of the formula, but I couldn't catch what she said, and rather, I didn't understand why you would complete the square. I do however see that for $3n^2 - 60n + 301$ we have $\Delta < 0.$
I can deduce that you would go in the direction of an inequality whereas you have $n >0$ (incomplete).
Perhaps someone with a higher understanding can explain this to me. 

Completing the square enable you to see the vertex clearly. For $f(x)=ax^2+bx+c$ where $a>0$, we can see the minimal value that it can attain.
Alternatively, just see that the discriminant is negative, that is the function doesn't intersect the $x$-axis and it doesn't change sign. Since one of the term is positive, every term is positive.
