>> This is part 5 of Solving
Quadratic Equations Using the
Quadratic Formula, which
is shown here in purple.
And in this video we are going
to solve this problem using
the quadratic formula.
Now, before you can use
the quadratic formula,
you need to simplify
the problem and get it
in the form ax squared plus
bx plus c, and if possible --
which it is -- it's
always possible --
eliminate any fractions.
All right, so we need to
square this left-hand side.
Now if you want to use
the formula you can,
or remember this
is the same thing
as x minus 1/2 times
x minus 1/2.
Okay. So if I do
the FOIL method,
we have x squared minus 1/2x
minus 1/2x plus 1/4th equals --
now x over 2 is the
same thing as 1/2x.
So you could write all of
these as 1/2x or x over 2.
They're really the same thing.
Okay, so we could combine like
terms if we want right now,
or we could just
multiply everything
by the least common denominator,
which is gonna be 4.
I'm gonna go ahead and
just multiply everything
by the least common
denominator of 4.
So I have 4 times the x
squared minus 4 times 1/2 --
I wanted to write
that in green --
1/2x minus 4 times another 1/2x
plus 4 times 1/4th equals --
and don't forget to multiply
by 4 on the right-hand side
of the equation -- 4 times 1/2x.
Now keep in mind you could have
simplified both sides first,
or you could have set
it equal to zero first.
It doesn't matter
when you multiply
by the least common denominator.
So I've got 4x squared
and all right,
so 2 goes into 4
twice so there's 2x.
See what I did was I
eliminated a fraction there.
And 4 times 1/2,
that's another minus 2x,
4 times 1/4th is
gonna be plus 1.
And on the right-hand side
of the equation I have 2x.
All right?
So we're getting closer.
Now I need to subtract
this 2x from both sides.
So I have 4x squared and I've
got minus 2x minus another 2x
minus that other 2x is gonna
give you negative 6x plus 1
equals zero.
So I've taken that original
problem, and I've got it
into a form where it's easy
to state the values of a, b,
and c. So a is 4, b is
negative 6, and c is 1.
And now we could plug that
into the quadratic formula.
So put the video on pause
and try that on your own.
I like to figure out b
squared minus 4ac first,
that's just a personal
preference.
So b squared will be
negative 6 times negative 6,
that's 36, minus 4 times ac.
That means a times c,
4 times 1 which is 4.
So I've got 36 minus
16 which is 20.
So remember, that's the part
that goes underneath
the square root.
So I know I'm gonna end up
with the square root of 20.
And by the way, 20 is 4 times
5, so I know I'm gonna be able
to simplify that square root.
So now we're gonna plug it
in the quadratic formula.
So x equals negative b, ah-ha,
opposite of b is positive 6 plus
or minus the square root of
20, which I'm gonna write
as 4 times 5, all over 2a.
Well, I got 2 times 4, so you
can leave that as 2 times 4
or you can write it as 8.
Now I could simplify and
take a 2 out of here, right?
So that's gonna give me 6 plus
or minus 2 square
roots of 5, all over 8.
Now in order to reduce
a rational expression
like this, you need to factor.
So out of the numerator
I could factor out a 2.
So I have 2 times 3 plus
square root of 5 over 8,
which I'm gonna write as 2
times 4 so it's easy to see.
So I can cancel the 2's.
Oh I forgot to put the minus
sign there, plus or minus.
So I have two solutions for
x, 3 plus radical 5 over 4
and 3 minus radical 5 over 4.
So here was our original
problem,
x minus 1/2 squared
equals x over 2.
And believe it or not,
those are the two solutions.
I'm not gonna check it because
checking it takes quite a long
time, probably take
more than five minutes.
You have to be careful
that if you do plug in,
like 3 plus squared of 5 over 4
for x, and you're very careful
and square this binomial --
actually it'll be bigger
than a binomial, etcetera,
you will get the same
answer on both side.
All right?
So just keep practicing.
And the key here is if you
have a problem like this
with fractions, clear the
fractions first before you
decide to plug in your
values of a, b, and c.
