SOLUTION: Rachelle has taken (4x-6)hours to travel 102 km at a speed of (25x+1) kmh^-1. How fast was she traveling?

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Question 772331: Rachelle has taken (4x-6)hours to travel 102 km at a speed of (25x+1) kmh^-1. How fast was she traveling?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
speed%2Atimes=distance

THE ALGEBRA WAY:
Since the units agree,
%2825x%2B1%29%284x-6%29=102 --> 100x%5E2-150x%2B4x-6=102 --> 100x%5E2-146x-6=102 --> 100x%5E2-146x-6-102=0 --> 100x%5E2-146x-108=0
Now we solve that quadratic equation.
Using the quadratic formula, x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ,
x+=+%28-%28-146%29+%2B-+sqrt%28%28-146%29%5E2-4%2A100%2A%28-108%29%29%29%2F%282%2A100%29+-->x+=+%28146+%2B-+sqrt%2821316%2B43200%29%29%2F200+-->x+=+%28146+%2B-+sqrt%2864516%29%29%2F200+-->x+=+%28146+%2B-+254%29%2F200+
x+=+%28146+-+254%29%2F200=-108%2F200=-0.54 would give a negative 4x-6=-8.16 number for the hours driving.
That does not make sense, so we can only use the other solution.
x+=+%28146+%2B+254%29%2F200+-->x+=+400%2F200+-->x+=+2
So the speed (in km/h) is 25x%2B1=25%2A2%2B1=50%2B1=51
The speed is highlight%2851km%2Ah%5E%28-1%29%29

THE FIFTH GRADER WAY:
If 25x%2B1 is the speed, x must be a small number, maybe 0, or 1, or 2, or 3.
The numbers 0 and 1, give believable speeds, but the time driving turns out negative.
For x=0, 4x-6=-6, and for x=1, 4x-6=4-6=-2
Let's try x=2
That makes the speed 25x%2B1=25%2A2%2B1=50%2B1=51 km/h.
It makes the time 4x-6=4%2A2-6=8-6=2 hours.
That makes the distance (2 hours) X (51 km/h)= 102 km, so highlight%7Bx=2%29 and the speed is highlight%2851km%2Ah%5E%28-1%29%29