Before we move on entirely from the
quadratic formula, let's work on one more
application problem that we'll be able
to use to solve other problems like this.
And hopefully give you a better grasp on
quadratic formula word problems in
general. So the problem states, the
hypotenuse of a right triangle is 75
meters long. The length of one leg is 51
meters less than the other, and it wants
us to find the lengths of these legs. So
let's draw our picture first. If we make
a triangle, a right triangle, and I've drawn
some square in the corner to
denote that it's a right triangle. We
know that our hypotenuse, let's shorten
that to H, is equal to 75 meters. 75 M in short hand. Now we need to find
the lengths of these legs here. I'll call
one leg A and one leg B. That's just
arbitrary the problem doesn't
really care what we call the legs as
long as we find out which one is which
and how long they are. So the problem
also tells us that one of the legs is 51
meters less than the other. So I'll take
the leg B, and I'll say that B is equal
to 51 less than A or A minus 51. And
remember, this is gonna end up in meters.
We need to remember that. So we have a
hypotenuse. We have a right triangle. We
have two legs of unknown length, one of
which is dependent on the other for its
length. So what are some things we know
how to do with right triangles? Oh! Of
course, we know the
the Pythagorean theorem. Which is just A
squared plus B squared equals C squared.
We all know the Pythagorean theorem
helps us find the lengths of the sides
of our right triangle. Now C, of course,
right here is just our hypotenuse. Which
we already have as 75. And A and B
appropriately are our legs. So if we plug
in everything, we have here we would
write down an A squared. That's all we
can write down here because A isn't
really dependent on B. All we know is,
that it is a leg of some length. Then we
could write B squared. We also know that
B is equal to a minus 51. So we'll
substitute that in for B, minus 51, and
of course that will all be squared. And
then C is our hypotenuse, which is 75.
I'll write that down here, and again we
need to square that. Now looking at this
this is Pythagorean theorem, but we'll need to
turn it into a quadratic formula to
solve for our leg lengths. So I'll write
this equation down one more time. Again,
we found this using the Pythagorean
theorem. And now we'll have to find some
way to use the quadratic formula. So the
first thing we need to do is get rid of
this squared term. This a minus 51
squared. The way we need to do that is
FOIL. If you don't remember what foiling
is, it stands for first outside inside
last. And it refers to the pattern that
you need to take when you're multiplying
two term expressions
like(A minus 51) times (A minus 51). So, to foil
first outside inside and last. I'll write
this down here. We'd have A squared still
out front. This is all part of one
equation and then our first term for our
foil is A times A. So A squared. Our
second term is negative 51 times A. Our
third term is also negative 51 times A.
And then our last term is negative 51
times negative 51 which results in a
positive 2601.
And this is all equal to 5625, which is
our 75 squared. Now that we have this, we
can add and simplify everything together
as much as we want. You know A squared
plus A squared is of course 2A squared.
Those two negative 51 A's will simplify
into a negative 102A. And then we'd have
+ 2601 equals 5625. We can just bring
this 5,625 back over into the
left side of our equation. So we get 2y
squared and that's 102A minus
3024 all equal to zero. Now we have this
equation in our standard form and we can
use the quadratic formula to solve. I'll
write this down one more time.
2A squared minus 102A - 300 - 4
equals 0. And then I'll go ahead and
write down the quadratic formula. Now
usually our quadratic formula is x
equals and then some a B and C terms. I'm
going to be using capital A's B's and C's
so that we don't confuse it with our
already lowercase A that we're trying to
solve for. So our numbers here are going
to be A B and C. So our formula is A equals
negative B plus or minus square root B
squared minus 4ac sorry AC all over 2a.
If we plug all our values into this
equation we'll get our A equals negative
negative 1 or 2 which would be a
positive 1 or 2 and then plus or minus 2
times negative 3. So we're working with
really big numbers here but the
quadratic formula still holds even with
these big numbers. There's no need to be
scared of them. You know I'll go ahead
and simplify this a little bit without
doing all this math inside the square
root. Those big numbers will turn into an
even bigger number we'll get 13,788
over 4 which is gonna mean that A is
about 102 plus or minus 117 over 4.
This 117 comes from me just rounding the
square root term right here. So it's not
exactly a hundred and seventeen it's
like on its 117.4 and then it goes on and
on but I did just round it to 117 for
simplification purposes. So I'll tell you
that this number right right here it's
gonna be a equals 54 0.85 and negative
3.8 but we have to think about what A is.
Can A have a negative value?
Well A is gonna be in meters. You can't
really measure a negative
number on a meter stick. So we're just
gonna forget about this for now. So now
we have A. That's great! I would draw
another triangle here so we know that
our hypotenuse is 75 meters and we know
that A is about 55 meters and then what
was B? well B was equal to a minus 51 and
54.8 minus 51
makes B equal to a positive three
point eight five. And again
these are in meters. Don't forget that. So
our B is equal to three point eight five
meters. Our A is equal to 54 point eight
five meters and if you put those into
the Pythagorean theorem you'll see that
we will get a hypotenuse equal to 75
meters. So here we've solved our problem!
