Question 1113179
The dimensions and area of the rug are:
{{{ ( 23 - 2x )*( 30 - 2x ) = 408 }}}
{{{ 690 - 60x - 46x + 4x^2 = 408 }}}
{{{ 4x^2 - 106x + 282 = 0 }}}
{{{ 2x^2 - 53x + 141 = 0 }}}
Use quadratic formula
{{{ x = ( -b +_sqrt( b^2 - 4*a*c )) / (2a) }}}
{{{ a = 2 }}}
{{{ b = -53 }}}
{{{ c = 141 }}}
{{{ x = ( -(-53) +-sqrt( (-53)^2 - 4*2*141 ))/4 }}}
{{{ x = ( 53 +-sqrt( 2809 - 1128 )) / 4 }}}
{{{ x = ( 53 + sqrt( 1681 )) / 4
{{{ x = ( 53 + 41 )/4 }}}
{{{ x = 94/4 }}}
{{{ x = 23.5 }}}  ( too large for the border )
Use the neg square root
{{{ x = ( 53 - 41 )/4 }}}
{{{ x = 12/4 }}}
{{{ x = 3 }}}
and
{{{ 23 - 2x = 23 - 6 }}}
{{{ 23 - 2x = 17 }}}
and
{{{ 30 - 2x = 30 - 6 }}}
{{{ 30 - 2x = 24 }}}
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The rug is 17 x 24
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check:
{{{ 17*24 = 408 }}}
OK