hello everyone well we know by now that
completing the square can be a real pain
if you have to complete the square every
time to solve a quadratic equation and
you know I've always thought that
probably centuries ago there was an old
mathematician some summer evening and he
was setting out on his porch just
enjoying the evening and thinking about
how his students also did not enjoy
completing the square every time they
wanted to solve the quadratic formula
and so is he set out on the side of his
porch he had his rocking chair and off
to the side of the porch there was some
sand and he took a stick and he started
to think a little bit and he etched into
the sand x squared plus bx plus c equals
0 we know that that's the general form
of a quadratic equation and he thought
you know there's got to be a better way
than having to complete the square every
time so as he SAT there and he rocked
and he thought he thought well you know
I know that I'm going to complete the
square i need that leading coefficient
to be a one and maybe it is maybe it's
not all we know is that it's some real
number a but he knew that he could
guarantee that it would be a one if he
divided everywhere by a and so he did
that x squared plus B over ax plus C
over a he set that equal to zero and as
he said there any wat rock his wife
Ethel came outside and she said I'm
gonna bake some cookies would you like
some and he said honey I'm working on
something right now and she says all I
want to know is do you want some
chocolate chip cookies and he said yes
that would be great she said all right
you go back to your math and so he SAT
there for a little bit and he knew that
to complete the square he needed a
perfect square trinomial on the
left-hand side on one side of the
equation and he already had the x
squared here and he realized that this
constant was just in his way so he moved
it to the other side of the equal sign
to get it out of the way they thought
okay how do I go about in general making
a perfect square trinomial I thought
well I need to take one half of the
number in front of the x whatever that
happens to be in this case it's be over
a and I need to square that and I need
to add that to both sides because we
need to keep our equation balanced so he
took half of the be over a squared it
and add it to both sides and he set
there and he took he stick in the dust
in the sand and he continued to write x
squared plus B over a X plus he squared
the B and that gave me b squared c
squared the two to get a 4 d squared the
a to get a squared and so he had that
over here and he wrote it on the right
side of the equation as well and he was
happy and he rocked a little faster
because he realized that he had just
created a perfect square trinomial on
the left side and he knows that that'll
factor the square root of the first guy
he takes this sign right here and then
he takes the square root of what he has
here which is B over 2a and he squares
that and he notices over here that his
students probably wouldn't be very happy
right now because he's going to have to
add these two fractions and even back
centuries ago students did not enjoy
finding common denominators to add
fractions either so he kind of chuckled
at that because he knew his students
would have to do that and so he
recognized that the common denominator
would be 4a squared and so I thought
well i have to multiply here by for an a
and i have to multiply here by four
a and so he had negative for AC plus B
squared on top and he'd always told
these students that it's kind of nice to
have the positive term first so we used
the community of property of addition to
reorder those b squared minus 4ac all
over 4a squared on the Left he hadn't
done anything this was still this
factored form of a perfect square
trinomial and then he started to rock a
little faster because he realized over
here he had his perfect square trinomial
this was a nice expression over here he
could he could use the extraction of
roots idea that they'd went over in
class taking the square root of both
sides of the equation taking the square
root of the top and the bottom and as he
had told these students anytime you use
extraction of roots you need to put a
plus minus in front of this and so he
had an X plus B over 2a over here now I
didn't know what the square root of b
squared minus 4ac was so he just left
that under the square root but he did
know that the square root of 4a squared
was 2a and he smiled and he sniffed a
little bit and he knew that his wife had
just about finished the chocolate chip
cookies he could smell them coming out
of the oven as he subtracted B over 2a
from both sides and what he found was x
equal to the opposite of b plus or minus
the square root of b squared minus 4ac
all over 2a now his wife comes out with
a plate of chocolate chip cookies and a
cold glass of lemonade for each of them
and they set there and he realizes he
has just come up with an idea that will
save students everywhere for centuries
to come from having to complete the
square every time they need to solve a
quadratic equation that can't be
factored easily now as I tell my
students in the classroom there probably
was no old guy on a porch anywhere with
the stick and sand and dust but as I
tell them explaining how the quadratic
formula comes about is a lot more
entertaining for me if I put an old guy
and his wife and some chocolate chip
cookies in the story I think that's more
entertaining than standing hearing st.
okay look folks we want to solve this
general equation ax squared plus bx plus
c equals 0 if we complete the square on
it we need the leading coefficient to be
a 1 so we divide everywhere by eight we
need the constant to be out of the way
we need it on the other side of the
equal sign so we subtract C over a from
both sides we know we want to create a
perfect square trinomial on the left and
so we take half of the be over a square
add it to both sides thus creating a
perfect square trinomial right here that
we can factor once we've got that
factored we do our work over here to
clean it up as I mentioned we've got to
get that common denominator and it's
nice to have the positive term first so
we as i mentioned use the community
property of multiplication to turn those
two around and then we can take the
square root of both sides and we
subtract the B over 2a here's our
quadratic formula and just to give you
one quick example of how this works if
we wanted to solve the equation for x
squared minus 4x minus 1 equals 0 we
write that down and we want to identify
our characters here we want to identify
the fact that a is for
b is negative 4 and c is negative 1 so
we remember our formula X equal the
opposite of b plus or minus the square
root of b squared minus 4ac all over 2a
and if you've already listed your A or B
in your seat then it's very easy to plug
in the opposite of B which is negative 4
plus or minus the square root of b
squared B again is negative 4 we're
going to square that minus 4 times a
which is four times c which is negative
1 all over 2 times a which we said was 4
and so this gives me 4 plus or minus the
square root of 16 plus 16 4 times 4 16
times one is 16 and a negative times a
negative gives me a positive all over 8
and then we can simplify this we've got
four plus or minus the square root of 32
all over 8 I need to break down the
square root of 32 4 plus or minus a
factor of 32 that's a perfect square is
16 all over eight and then rewrite the
square root of sixteen is four root to
all over eight notice how we can factor
a four out of my top these guys will
reduce and so we are left with X equal 1
plus or minus the square root 2 all over
2 but you can use the quadratic formula
to solve any quadratic equation you
would like
