Answer
Sandy takes 9.1 hours.
Work Step by Step
Dave = $x$
Sandy = $y$
$1/x +1/y = 1/4$
$y-2=x$
$1/x +1/y =1/4$
$1/(y-2) + 1/y = 1/4$
$(y-2)*( 1/(y-2) + 1/y) = 1/4*(y-2)$
$1 + (y-2)/y = (y-2)/4$
$4y*(1 + (y-2)/y) =4y*(y-2)/4$
$4y + 4(y-2) = y*(y-2)$
$4y+4y-8 = y^2-2y$
$8y-8=y^2-2y$
$8y-8-8y+8=y^2-2y-8y+8$
$0 = y^2-10y+8$
$a=1$, $b=-10$, $c=8$
$y=(-b±\sqrt {b^2-4ac})/2a$
$y=(-(-10)±\sqrt {(-10)^2-4*1*8})/2*1$
$y=(10±\sqrt {100-32})/2$
$y=(10±\sqrt {68})/2$
$y=(10±8.246)/2$
$y=(10+8.246)/2$
$y=18.246/2$
$y=9.123$
$y=(10-8.246)/2$
$y=1.754/2$
$y=.8777$
This answer isn’t valid since we are looking at Sandy’s time, and we know Dave takes 2 fewer hours to paint a room alone. This would mean Dave takes $-1.1223$ hours to paint, which is impossible.
Sandy takes 9.1 hours.
