Lets make the numbers less. Divide both sides by 2
#1/2(42-2x)(25-2x)=500/2#
#color(blue)((21-x))color(green)((25-2x))=250#
Multiply everything in the right brackets by everything in the left.
#color(green)(color(blue)(21)(25-2x)color(blue)(-x)(25-2x)=250#
#525-42x-25x+2x^2=250#
#2x^2-67x+275=0#
Compare to #y=0=ax^2+bx+c#
#x=(-b+-sqrt(b^2-4ac))/(2a)#
Where #a=2"; "b=-67"; "c=275#
Thus we have:
#x=(+67+-sqrt((-67)^2-4(2)(275)))/(2(2))#
#color(brown)("I will let you finish this off.")#
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Note that as #2x^2# is positive the graph is of form #uu#
Thus the vertex is a minimum.
#x_("vertex") =(-1/2)xxb/a" "->" "(-1/2)xx(-67)/2 = +67/4#
#y_("vertex")=2(67/4)^2-67(67/4)+275 = -286 1/8#
As #" "y_("vertex")<0# and the graph is of form #uu# then solutions exists to #y=0=2x^2-67x+275#
