Question 895796
{{{x+y=64}}}.  {{{xy<=1024}}} ?


{{{y=64-x}}}.
{{{x(64-x)=f}}}-----the product of x and y.
{{{f=64x-x^2}}}
{{{f=-x^2+64x}}}


Complete The Square.    Use {{{(64/2)^2=32^2}}}.
{{{f=-1(x^2-64x)}}}
{{{f=-1(x^2-64x+32^2-32^2)}}}
{{{f=-1((x-32)^2-32^2)}}}
{{{highlight_green(f=-(x-32)^2+32^2)}}} ------ standard form, parabolic function.


The maximum, the vertex of f is at (32,32^2) and {{{32^2=1024}}}.



(See my lesson on Completing The Square)