Answer
9.12 hours for Sandy
Work Step by Step
Dave and Sandy paint a room in 4 hours combined. Dave takes 2 hours less than Sandy if painting alone.
Dave and Sandy are represented by $D$ and $S$, respectively.
$1/D +1/S = 1/4$
$D=S-2$
$1/D +1/S = 1/4$
$1/(S-2) +1/S = 1/4$
$(S-2)*S[1/(S-2) +1/S] =(S-2)*S* 1/4$
$(S-2)*S[1/(S-2) +1/S] =(S-2)*S* 1/4$
$S +(S-2) = (S^2-2S)/4$
$2S -2 = (S^2-2S)/4$
$(2S -2)*4 = 4*(S^2-2S)/4$
$8S-8=S^2-2S$
$8S-8-8S+8=S^2-2S-8S+8$
$0=S^2-10S+8$
$a=1$, $b=-10$, $c=8$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(-10)±\sqrt {(-10)^2-4*1*8})/2*1$
$x=(10±\sqrt {100-32})/2$
$x=(10±\sqrt {68})/2$
$x=(10±\sqrt {4*17})/2$
$x=(10±2\sqrt {17})/2$
$x=5±\sqrt {17}$
$x=5±4.123$
$x=5+4.123$
$x=9.123$
$x=5-4.123$
$x=.877$
We ignore this answer for $x$ since we solved for Sandy's time, and Dave takes 2 fewer hours than Sandy. This would result in negative time for Dave, which doesn't exist.
$9.123-2=7.123$
