Question 1006415
{{{x}}}= length of the side of the painting (in feet).
{{{x-2}}}= length of the side of the mirror (in feet).
{{{x^2}}}= area of the painting (in ft squared).
{{{(x-2)^2}}}= area of the mirror (in ft squared).
The problem says that
{{{(x-2)^2=x^2-32}}} ,
because the difference of their areas is 32 ft squared,
and the mirror, with shorter sides, must have a smaller area.
 
Solving {{{(x-2)^2=x^2-32}}} :
{{{(x-2)^2=x^2-32}}}
{{{x^2-2*x*2+2^2=x^2-32}}}
{{{x^2-4x+4=x^2-32}}}
{{{-4x+4=-32}}}
{{{-4x=-32-4}}}
{{{-4x=-36}}}
{{{x=(-36)/(-4)}}}
{{{highlight(x=9)}}}
{{{x-2=9-2}}} ---> {{{highlight(x-2=7)}}}
The length of the sides of the mirror is {{{highlight(7ft)}}} , and
the length of the sides of the painting is {{{highlight(9ft)}}} .