In this example we're going to solve an
application using the quadratic equation.
Naomi bikes the 36 miles to Hillsboro
averaging a certain speed. The return
trip is made at a speed
that is 3 miles per hour slower. The
total time for the round trip
7 hours. Find Naomi's average
speed on
each part of the trip. First let's set up a
table
that lists out the information in our
in our problem.
 
 
The distance to and from her destination
is the same 36 miles. The rate
going is a certain speed
the return trip is 3 miles per hour
slower
the total time is 7 hours
so we can list the time to as T
at the time from as 7 minus
T now we're going to set up a system of
equations here
so that we can solve. Notice that
we have a formula that says
time equals the distance divided by the
rate
so in our first
in our first line we see
that T equals
36 over R. From our second one
we would see that
7 minus T equals
36 over
R-3. Now we have two equations. Now we
can use these and use substitution
and solve for R. Now the
thing I wanted  to touch with
you is that we could have used the fact
that R
equals the Distance over Time but I want
to solve
for R. I noticed that doing it this
way
I can replace T in my second equation
with 36 over R and therefore we only
have Rs
dealing with it with solve directly for
R. Had we used
this one we would have been solving for
T and then we have to it
do some work to find what R is.
So we have 7
minus 36 over R equals 36
over R-3
Now we're going to use the principle
that we we recently showed
where we multiply everything by that LCM
so we can eliminate
Rs from the denominator. The LCM, or the
least common denominator,
is R times R-3 so multiply
everything by that common denominator
Here I have shown that
step. So let's see what happens. 
Here we have 7R times R-3
minus, notice here that the Rs cancel out,
leaving me with just 36
times R-3. On the right hand side
R-3s cancel out leaving me with 36R
lets distribute
7R squared
minus 21R minus 36R
plus 108.
Let's combine our like terms:
 
 
subtract 36 for both sides
 
 
now we can state that
A is 7, B is -93
C is 108. Let's plug this into our
quadratic formula and solve
Let's plug in our values and
simplify
 
I'm going to plug this entire expression
into my calculator all at once
and find that discriminate.
square root of 5625
is 75 and now we need to simplify.
We have 2 possibilities
one that's positive and one that's
negative.
 
For the positive one,
we find 12. For the negative, we find
1.286
The logical answer would be twelve miles
per hour.
Now let's take this back to our original
problem.
The original problem wants us to find the
speed for both parts.
Since R is 12
and we stated that R was going to be
the rate To, that means the rate From
would be would 9. So let's state
this
This is my final answer.
