The jacksons wish to build a rectangular enclosure for their dog , fido. they purchased 100 feet of fencing for the enclosure. the area of the enclosure , a , is related to the length of one of the sides , x (measured in feet) , by the formula f ( x ) = − x^2 + 50 x . identify the vertex of this situation and explain its meaning in the context of the problem?

The vertex can be found by completing the square:
f(x) = -(x^2 - 50x)
      = - (x^2 + 625 - 625 - 50x)    (take half the coeff of x, square
      = - (x^2 + 625 - 50x ) + 625            it then add and subtract it)
      = - (x^2 - 25x - 25x + 625) + 625
      = - (x(x- 25) -25(x + 25)) + 625
      = - ((x- 25) (x + 25)) + 625
      = - ((x + 25)^2 + 625
so the vertex is (25, 625)
in this context this is the 'optimized' area or the largest possible area with the given perimeter (so it is basically a square, 25 x 4 = 100)