Today we'll be looking at a quadratic
formula word problem and figuring out
how to figure out the information and
the problem and apply that to what it
wants us to know. So we're building a
rectangular dog pen that must have an
area of 400 square feet. The length must be
10 feet longer than the width and we're
supposed to find the dimensions of the
pen. So find the length and width of the pen.
Let's start out by just writing down the
important stuff, the stuff that we
already know. We know that the area is
enough that is a is equal to 400 square
feet which I'll right just like that.
We also have a length and a width which
I'll write as a small cursive L and
small W. So we don't really know what the
width is. It doesn't give us any
information in the problem about what
that could be. But we do know that the
length is 10 feet longer than the width
or in other words the width plus 10. So W
plus 10. We also know that the dog pen is
rectangular. So I'll draw a little
rectangle here and I'll write an L on
the longest sides and to denote the
length and a small W on the other side
to denote the width. And I'll also write, one
more time, that the length is equal to W
plus 10. And then, of course, everything on
the inside here will be the area which
we know is equal to 400. So the problem
here is, we don't have the width so we
don't have the length either because the
length is dependent on how long the
width is. But we do have area and we do
know how to find the area of a rectangle.
It's just a formula. It's area equals
length times width. And we know that L is
equal to W plus 10, so we could transform
this into a equals W plus 10 times W and
if should all be equal to 400 because a
is also equal to 400. So this equation
here this is what we're gonna be trying
to solve for. We're gonna be trying to
find W in this equation and once we have
W or we can also solve for L. So let's
work on this cell. I'll write this
equation down again. We'll have a equals...
I guess I'll write our variables first. a
equals W plus 10 all times W and that
equals 400. All right.
Well let's go ahead and simplify this by
multiplying everything together. So W
times W is w squared and then 10 times W
is of course 10 W and this will all be
equal to 400. And now this is starting to
look pretty familiar. We see now that we
have the beginnings of the standard form
of a quadratic equation and, in fact, this
will eventually become a standard form
quadratic equation. All we need to do to
accomplish that is subtract 400 from
both sides so we can get the right side
to equal 0. Once we do that we do have
our standard form and we can just use
our quadratic formulas of W equals
negative b plus or minus square root b
we're minus 4ac divided by 2a. We also
know that these numbers in front of our
variables here are going to be our A B
and C. So a is equal to one. B is equal to
10 and C is equal to negative 400. All
right. Now that we have our A B and C we
can go ahead and plug this into the
quadratic formula which will be W equals
negative B. So negative 10 plus or minus
the square root of 10 squared minus 4
times a times C which is a negative.
Don't forget about that and then this is
all over 2a. It's just 1 so I'll simplify
this a little bit. I'll go ahead and tell
you that that big number under the
square root there is gonna simplify to
1700 and then that will be over 2. And
then the square root of 1700 is going to
be about 41 but for the purposes of this
problem we're just gonna go ahead and round
to 41. It'll just make everything simpler.
But you do already know how to simplify
square roots so we don't really need to
go into that.
So now that we have this this w equals
negative 10 plus or minus 41 over 2 we
can use this. We know that W now equals
a 15.5 and also a negative 25.6, but what
is w? Well W is width, which is going to
be in feet and we can't have negative
feet. If we try to measure something
that's three feet long it's never going
to equal a negative number because feet
is a skill or value. It's never going to
be negative. So we can go ahead and get
rid of this value right here. So we have
this W, which is awesome that's our width
we found it that's all you need to do
for width. We're not quite done. We also
need our length. If our W is equal to
fifteen point five feet and we know that
our length L is equal to width plus ten,
then our length must equal twenty five
point six feet. And these are, in fact, our
dimensions. These are, in fact, the numbers
that we were trying to solve for. And if
we look back to our rectangle here, we
can see that that makes sense that L is
greater than W. And so we have our
answers here. W is equal to fifteen point
five feet and our length L is equal to
twenty five point five feet. Sorry I
wrote 0.6 that should be point five! And
that's how you apply the quadratic
formula to one simple word problem.
