SOLUTION: Could you please help me with an application question involving a quadratic equation? The question is: On a sales trip, Gail drives the 600 mi to Richmond at a certain speed. The

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Question 111421This question is from textbook
: Could you please help me with an application question involving a quadratic equation?
The question is: On a sales trip, Gail drives the 600 mi to Richmond at a certain speed. The return trip is made at a speed that is 10 mph slower. Total time for the round trip is 22 hr. How fast did Gail travel on each part of the trip?
This is what I've got so far:
t[1]=600/r
t[2]=600/(r-10)
(600/r)+(600/(r-10))=22
r(r-10)(600/r)+r(r-10)(600/(r-10))=r(r-10)22
600(r-10)+600r=22r(r-10)
600r-6000+600r=22r^2-220r
22r^2-1420r+6000=0
I know at this point to plug it into quadratic equation form... but when factoring this I get decimal numbers, so it seems wrong.
-(-64.545)+-(sqrt(-64.545)^2-4(1)(272.727))/2(1)
64.545+-(sqrt4166.057-1090.908)/2
64.545+-(sqrt3075.149)/2
(64.545+55.454)/2=60
or
(64.545-55.454)/2=4.5
Please help!
Thank you in advance for your help!
Brenda This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Everything is great until you reach the point where you factor out the leading coefficient (which you don't need to do). So I'm going to start off with the step that has 22r%5E2-1420r%2B6000=0


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for r:


Starting with the general quadratic


ar%5E2%2Bbr%2Bc=0


the general solution using the quadratic equation is:


r+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 22%2Ar%5E2-1420%2Ar%2B6000=0 ( notice a=22, b=-1420, and c=6000)





r+=+%28--1420+%2B-+sqrt%28+%28-1420%29%5E2-4%2A22%2A6000+%29%29%2F%282%2A22%29 Plug in a=22, b=-1420, and c=6000




r+=+%281420+%2B-+sqrt%28+%28-1420%29%5E2-4%2A22%2A6000+%29%29%2F%282%2A22%29 Negate -1420 to get 1420




r+=+%281420+%2B-+sqrt%28+2016400-4%2A22%2A6000+%29%29%2F%282%2A22%29 Square -1420 to get 2016400 (note: remember when you square -1420, you must square the negative as well. This is because %28-1420%29%5E2=-1420%2A-1420=2016400.)




r+=+%281420+%2B-+sqrt%28+2016400%2B-528000+%29%29%2F%282%2A22%29 Multiply -4%2A6000%2A22 to get -528000




r+=+%281420+%2B-+sqrt%28+1488400+%29%29%2F%282%2A22%29 Combine like terms in the radicand (everything under the square root)




r+=+%281420+%2B-+1220%29%2F%282%2A22%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




r+=+%281420+%2B-+1220%29%2F44 Multiply 2 and 22 to get 44


So now the expression breaks down into two parts


r+=+%281420+%2B+1220%29%2F44 or r+=+%281420+-+1220%29%2F44


Lets look at the first part:


x=%281420+%2B+1220%29%2F44


r=2640%2F44 Add the terms in the numerator

r=60 Divide


So one answer is

r=60




Now lets look at the second part:


x=%281420+-+1220%29%2F44


r=200%2F44 Subtract the terms in the numerator

r=50%2F11 Divide


So another answer is

r=50%2F11


So our solutions are:

r=60 or r=50%2F11



note: ignore the graph. It didn't turn out the way I wanted it to.

Now notice that we get two answers. However if we subtract 50%2F11-10 (or 5.45-10) we get 50%2F11-10=-50%2F11 which is a negative number. So we must discard the second solution.

So her original speed was 60 mph and her slower speed was 50 mph