Question 768485
Let {{{ t }}} = Shane's time in hrs 
to complete job working alone
{{{ t + 2 }}} = Brian's time to complete
job working alone
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Shane's rate of working:
( 1 job done ) / ( t hrs )
Brian's rate of working:
( 1 job done ) / ( t + 2 hrs )
Their rate working together:
( 1 job done ) / ( 5 hrs )
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Add their rates to get their rate working together
{{{ 1/t + 1/( t + 2 ) = 1/5 }}}
Multiply both sides by {{{ t*( t + 2 )*5 }}}
{{{ 5*( t + 2 ) + 5t = t*( t + 2 ) }}}
{{{ 5t + 10 + 5t = t^2 + 2t }}}
{{{ -t^2 + 8t + 10 = 0 }}}
Using quadratic formula:
{{{ t = (-b +- sqrt( b^2 - 4*a*c )) / (2*a) }}} 
{{{ a = -1 }}}
{{{ b = 8 }}}
{{{ c = 10 }}}
{{{ t = (-8 +- sqrt( 8^2 - 4*(-1)*10 )) / (2*(-1)) }}} 
{{{ t = (-8 +- sqrt( 64 + 40 )) / ( -2 ) }}} 
{{{ t = (-8 +- sqrt(  104 )) / ( -2 ) }}} 
{{{ t = ( -8 - 10.2 ) / ( -2 ) }}}
{{{ t = 18.2 / 2 }}}
{{{ t = 9.1 }}}
{{{ t + 2 = 11.1 }}}
Note that using the positive square root gives me
a negative answer for time, so I used the negative
square root
Working alone, Shane takes 9.1 hrs
Brian takes 11.1 hrs
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check:
{{{ 1/t + 1/( t + 2 ) = 1/5 }}}
{{{ 1/9.1 + 1/11.1 = .2 }}}
{{{ .11 + .09 = .2 }}}
{{{ .2 = .2 }}}
OK