Justin recently drove to visit his parents who live 240 miles away. On his way there his average speed was 11 miles per hour faster than on his way home (he ran into some bad weather). If Justin spent a total of 8 hours driving, find the two rates.

rate there = x + 11
rate back = x

240/(x) + 240 / (x + 11) = 8
240(1/x + 1/(x + 11)) = 8
Divide by 8
30 (1/x   +  1/(x + 11) ) = 1
30 ((x + 11) + x ) / (x + 11)*x = 1
30 (2x + 11) =  (x + 11) * x
60x + 330 = x^2 + 11x
x^2 - 49x - 330 = 0

I think you have to use the quadratic on this.
a = 1
b = - 49
c = 330
When you do this you get
x = 55 or 
x = - 6. This is an extraneous speed. The car can't be going at - 6 miles/hour

One speed = 55
The other is  55 + 11 = 66

Check
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240/55  = 4.36  hours.
240/66 =  3.63 hours
Total = 7.99 hours which rounds to 8. These hours check.