Completing The Square

Completing the Square: Level 3 Challenges


When buying Brilliant T-shirts, Bradan determines that the total cost of buying NN shirts is given by the function C(N)=0.04N236N+9000 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 xx such that 3620+322x4x2 \sqrt{ 3620 + 322x - 4x^2 } is a real number.

What will be the least value of the expression (x2+2xy+2y2+4y+7)?\left( x^2+2xy+2y^2+4y+7 \right) ?

Note : x,yRx,y \in \mathbb{R}

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

(2x3y4)2+(2x3y+10)2? (2x-3y-4)^2 + (2x-3y+10)^2 ?

How many positive integral value(s) of nn exist such that n2+96n^2+96 is a perfect square?


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