Question 816221
{{{drawing(400,200,-72,72,-18,18,
line(-63,0,63,0),line(0,-16,0,16),
line(-63,0,0,16),line(63,0,0,16),
line(-63,0,0,-16),line(0,-16,63,0),
locate(-33,0,x),locate(30,0,x),
locate(1,9,y),locate(1,-6,y),
rectangle(0,0,4,2),locate(30,11,side)
)}}}
{{{area=2xy}}}
{{{side^2=x^2+y^2}}}
{{{area+side^2=x^2+2xy+y^2=(x+y)^2}}}
Making
{{{b=-sqrt(area+side^2)=-(x+y)}}} and
{{{c=area/2=xy}}} ,
we can write the equation
{{{z^2+bz+c=0}}} .
The solutions to that equation will be {{{x}}} and {{{y}}} ,
and the diagonals of the rhombus are {{{2x}}} and {{{2y}}} .
 
For example, if {{{side=65cm}}} and {{{area=2016cm^2}}} respectively,
{{{b=-sqrt(area+side^2)=-sqrt(2016+65^2)=sqrt(2016+4225)=sqrt(6241)=79}}} and
{{{c=2016/2=1008}}} .
Solving the equation
{{{z^2-79z+1008=0}}}
either by factoring it into
{{{(z-16)(z-63)=0}}} or by applying the quadratic formula
{{{z=(79 +- sqrt(79^2-4*1*1008 ))/2=(79 +- sqrt(6241-4032))/2=(79 +- sqrt(2209))/2=(79 +- 47)/2}} ,
gives you the solutions {{{z=16}}} and {{{z=63}}} ,
for a rhombus with diagonals measuring {{{2*16=32}}}cm and {{{2*63=126}}}cm