Question 171739
Call the width of the uniform space around the
rug {{{x}}}
The dimensions of the rug will then be
{{{(9 - 2x)*(12 - 2x)}}}
Half of the area of the 9 x 12 room is
{{{(9*12)/2 = 54}}}
{{{54 = (9 - 2x)*(12 - 2x)}}}
{{{54 = 108 - 24x - 18x + 4x^2}}}
{{{54 = 108 - 42x + 4x^2}}}
{{{4x^2 - 42x + 54 = 0}}}
Solve using quadratic formula
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
{{{a = 4}}}
{{{b = -42}}}
{{{c = 54}}}
{{{x = (-(-42) +- sqrt( (-42)^2-4*4*54 ))/(2*4) }}}
{{{x = (42 +- sqrt( 1764 - 864 )) / 8 }}}
{{{x = (42 +- sqrt( 900 )) / 8 }}}
{{{x = (42 + 30) / 8 }}}
{{{x = (42 - 30) / 8 }}}
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The answers are:
{{{x = 9}}}
{{{x = 1.5}}}
The 1st answer is impossible, so the width of
the border around the rug is 1.5 m
The dimensions of the rug are
{{{(9 - 2*1.5)*(12 - 2*1.5)}}} 
{{{(9 - 3)*(12 - 3)}}}
{{{6*9 = 54}}}
The dimensions of the rug are 6m x 9m