Quadratic Equation word problem

Question

Using one engine of a ferry boat, it takes 6 hr. longer to cross a channel than it does using a second engine alone. With both engines operating, the ferry boat can make the crossing in 4 hr. How long would it take each engine, working alone to power the ferry boat across the channel?

Answer 1

it takes the 2nd engine x hours

it takes the 1st engine x+6 hours.

the work rate of the first engine is 1/(x+6) and the work rate of the second engine is 1/x

the combined work rate is 1/(x+6) + 1/x = 1/4

It takes 4 hours for both of them, so the combined work rate is 1/4

1/(x+6) + 1/x = 1/4

Multiplies by 4x(x+6)

4x + 4(x+6) = x(x+6)

4x + 4x + 24 = x^2 + 6x

8x + 24 = x^2 + 6x

0 = x^2 - 2x - 24

0 = ( x - 6 )( x + 4 )

x-6 = 0 or x+4 = 0

x+4=0 results in negative measures

x-6 = 0 ---> x = 6

it takes the 2nd engine 6 hours ---> work rate is 1/6

it takes the 1st engine 12 hours ---> work rate is 1/12

1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 is the combined work rate.

It takes 4 hours for both of them.