Algebra -rational equation and factoring

Question

You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. If the round trip takes 2 hours, use a rational equation and factoring to find your average rate on the outgoing trip to work?

Answer 1

distance=rate*time

d=rt

40=rt

40=(r+30)(2-t)

4=rt implies t=40/r

substitute into the second equation

40=(r+30)(2-[40/r])

multiply both sides by r

40r=(r+30)(2-[40/r])(r)

40r=(r+30)(2r-40)

40r=2r^2+60r-40r-1200

2r^2+60r-40r-40r-1200=0

2r^2-20r-1200=0

divide both sides by 2

r^2-10r-600=0

factor

600=2*2*2*3*5*5

rearrange the factors and group

(2*2*5)(2*3*5)=20*30 and 20-30=-10, the coefficient of the middle term

(r-30)(r+20)=0

set r-30 equal to 0 and disregard r+20 as it will yield a negative answer

r-30=0

r=30 mph for going

r+30=30+30=60 mph for outgoing

also, the time for going is 4/3 hours and the time for outgoing is 2/3 hours