Completing The Square

Completing the Square: Level 3 Challenges


When buying Brilliant T-shirts, Bradan determines that the total cost of buying \(N\) shirts is given by the function \[ C(N) = 0.04 N^2 - 36 N + 9000 \] If he wants to minimize the total cost of buying shirts, how many shirts should he buy?

Find the largest positive integer \(x\) such that \[ \sqrt{ 3620 + 322x - 4x^2 } \] is a real number.

What will be the least value of the expression \[\left( x^2+2xy+2y^2+4y+7 \right) ? \]

Note : \(x,y \in \mathbb{R} \)

As \((x,y) \) ranges over all pairs of real values, what is the smallest value of

\[ (2x-3y-4)^2 + (2x-3y+10)^2 ?\]

How many positive integral value(s) of \(n\) exist such that \(n^2+96\) is a perfect square?


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