Translating a word problem into a quadratic expression

Question

I need help translating this problem into a quadratic form.

A train travels a distance of 480 km at a uniform speed. If the speed had been
8 km/h less, then it would have taken 3 hours more to cover the same distance.

How would I set up a quadratic expression to find the speed of the train?

Answer 1

speed = distance/time, so distance = speed * time

s = speed, t = time

First train 480 = st or s = 480/t

Second train 480 = (s - 8)(t + 3)

Plug the first equation into the second for s (I'm also going to switch sides of the second equation)

(s - 8)(t + 3) = 480

(480/t -  8)(t + 3) = 480

t*480/t + 3*480/t - 8t - 8*3 = 480

480 +1440/t - 8t - 24 = 480

The 480's cancel on each side, so we have

1440/t - 8t - 24 = 0

Multiply the whole equation by t (I'm going to move stuff around on the left also)

-8t² - 24t + 1440 = 0

All 3 terms are divisible by 8, so you can divide that out.  Dividing by -8 gets rid of negative in front of your t² term.

t² + 3t - 180 = 0

To factor, what two numbers have a difference of 3 & a product of 180?  12 & 15, so

(t + 15)(t - 12) = 0

For that to be true, t = -15 or 12.

You can't have negative time, so the answer is 12 hours.

speed = distance/time = 480 km/12 hr = 40 km/hr

Answer 2

Start with a system of equations:

v * t = 480 km

( v - 8 kmph) * ( t - 3 h) = 480 km

Where v=speed and t=time

Solve for t and simplify

t = 480 / v  and  t = 480 / ( v - 8 )    + 3

 480 / v  = 480 / ( v - 8 )    + 3

for 

v^2  - 8v  +  1280 = 0