Question 988568
Let u be the size of the border's width at all four sides.
The carpet area would be  {{{(12-2u)(20-2u)}}};
The room area is  {{{12*20}}};
The border area is  {{{12*20-(12-2u)(20-2u)}}}.


Border area must be one-third the area of the carpet:
{{{highlight_green(12*20-(12-2u)(20-2u)=(1/3)(12-2u)(20-2u))}}}
Let me skip a couple of steps...
{{{3*64u-12u^2=240-64u+4u^2}}}
{{{4u^2-64u+240+12u^2-3*64u=0}}}
{{{16u^2-64u-3*64u+240=0}}}
{{{16u^2-64*4u+240=0}}}
{{{highlight_green(u^2-16u+15=0)}}}
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Use formula for general solution of quadratic equation.  Can you finish this?


Discriminant is...   {{{14^2}}}.
The form you want is  {{{u=(16-14)/2=2/2=1}}}.