Algebra

# 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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