uniform motion math with solution please

Question

if a train had traveled 5 miles an hour faster. it would have needed 1 1/2 hours less time to travel 150 miles. find the rate of the train.

Answer 1

5 mph faster would have the train arrive 1 1/2 hours earlier in a 150 mile distance

d= rt

distance = rate of speed times time

150 = rt

150 = (r+5)(t-1.5) = rt

rt +5t -1.5r - 7.5 = rt

5t -1.5r = 7.5

5t = 7.5+1.5r

t = 1.5 +.3r

rt = 150

t = 150/r

150/r = 1.5 + .3r

150 = 1.5r +.3r^2

.3r^2 + 1.5r -150 = 0

r^2 + 5r - 500 = 0

r^2 +5r + (5/2)^2 = 500 + (5/2)^2

(r+5/2)^2 = 500 +25/4 = 500 + 6.25= 506.25

r +5/2 = sqr506.25

r = -5/2 + sqr506.25

r = -2.5 + 22.5 = 20 mph = train's speed

it would take 150/20 = 15/2 = 7.5 hours to go 150 miles

at 25 mph, it would take 150/25 = 6 hours to go 150 miles which is 1 1/2 hours less time than at 20 mph