Question 945590
Let {{{ t }}} = Michelle's time to paint a house in hours
{{{ t + 9 }}} = Shawn's time to paint a house in hours
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( 1 house ) / ( t hours ) = Michelle's rate of painting
( 1 house ) / ( t + 9 hours ) = Shawn's rate of painting
( 1 house ) / ( 10.8 hours ) = their rate painting together
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Add their rates of painting to get their rate painting together
{{{ 1/t + 1 / ( t + 9 ) = 1 / 10.8 }}}
Multiply both sides by {{{ 10.8*t*( t+ 9 ) }}}
{{{ 10.8*( t + 9 ) + 10.8t = t*( t + 9 ) }}}
{{{ 10.8t + 97.2 + 10.8t = t^2 + 9t }}}
{{{ t^2 + 9t - 21.6t - 97.2 = 0 }}}
{{{ t^2 - 12.6t - 97.2 = 0 }}}
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Use the quadratic formula
{{{ t = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}}
{{{ a = 1 }}}
{{{ b  -12.6 }}}
{{{ c = -97.2 }}}
{{{ t = (-(-12.6) +- sqrt( (-12.6)^2 - 4*1*(-97.2) )) / (2*1) }}}
{{{ t = ( 12.6 +- sqrt( 158.76 + 388.8 )) / 2 }}}
{{{ t = ( 12.6 +- sqrt( 547.56 )) / 2 }}}
{{{ t = ( 12.6 + 23.4 ) / 2 }}}
{{{ t = 36/2 }}} ( can't use the negative square root )
{{{ t = 18 }}}
{{{ t + 9 = 27 }}}
Shawn takes 27 hrs to paint the house by himself
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check:
{{{ 1/t + 1 / ( t + 9 ) = 1 / 10.8 }}}
{{{ 1/18 + 1 / 27 = 1 / 10.8 }}}
Divide both sides by {{{ 9 }}}
{{{ 1/2 + 1/3 = 1/1.2 }}}
Multiply both sides by {{{ 1/10 }}}
{{{ 1/20 + 1/30 = 1/12 }}}
Multiply both sides by {{{ 60 }}}
{{{ 3 + 2 = 5 }}}
OK