In this video I'm going to talk about
where, um, the quadratic formula came from.
It didn't just show up. You know, we have
negative b plus or minus the square root
of b squared minus 4ac all over 2a.
Um, it actually came from, uh, completing the square.
And that's why completing the square and
the quadratic formula are two methods you can
use to solve any quadratic equation. Um, and
they are, they're connected. Um, the quadratic,
the completing the square, um, if you
actually complete, um, ax squared plus bx
plus c, uh, equals zero, uh, you'll see that the
quadratic formula, um, happens. And I'm going
to show you how that, that looks today.
Um, so
let's jump into, um, the actual quadratic, um,
quadratic equation.
So, we're just going
to complete the square. I'm going to take
you through the same steps I did in the
other video. Um, now I just have variables a,
b, and c in there instead of numbers.
So the first thing we always did is we
subtracted c from both sides.
So we end
up with ax squared plus bx, equals
negative c. Now, at this point what we
need to do, is we need to, um,
get rid of that
a right there. Remember we said, um, we never
want the a there. We want it to be a one.
So we actually have to divide a by
everything. Alright? So I'm gonna do that.
And I'm gonna rewrite.
I get x
squared plus b over ax. our b is
actually now b divided by a. And then we
have equals negative c over a. So let's
go ahead and now, uh, make it a perfect
square trinomial. So we're gonna put a
plus here, and we're gonna take half of b
over a.
Now I could write b divided by a,
divided by two.
But instead of just doing that I like to
actually write it like this. I like to
write it multiplied by the reciprocal, one
over two. Because divided by two is the same
as multiplying by one-half. And I'm going to
square it. So I'm creating that perfect
square trinomia.l I'm gonna do that on
this side as well.
Alright.
Now what we can do from here, is we can
take the perfect square trinomial that
we have on the left side and write it as
a, um, um, a pair of a binomial squared.
So again,
we'll write the x here. The, um, the next term,
in it's going to be a plus here because
we have a plus sign right, you know right here.
Um, the next thing we're gonna look at
is this right here. This is going to be
that, that term after the x.
It's b times
one, which will just be b. All over a times two.
I like to write my number first. So
it's 2a.
I'm gonna put parentheses around it, and
I'm going to square it. At this point if
you wanted to check to make sure that
this binomial squared equals this
trinomial, you could. It does. Okay?
I'm gonna write equals. I'm gonna come over
here to this guy, and I'm gonna, I'm gonna
write this one a little different. I'm
gonna write my negative c over a to
start. I'm gonna write my plus sign, but
I'm going to square this. If you remember
we always took this number and we
squared it. Um, I'm not gonna really worry
about the one. But b squared will just be
in the numerator. The denominator will be,
I have to square both the a and the two.
So two squared is four. A squared is just a
squared. So I'm gonna write it like that.
I'm gonna put my number first. Okay?
I'm gonna stay on the left side here for a
second, and I want to combine these
together. Alright? Notice that we don't
have a common denominator right now. So
we have to, you know, find the l, the least
common denominator. Notice this a here.
What we need to do to this one right
here is we need to multiply it by, uh, let's
put multiplied by. A 4a in the bottom.
If you multiply by 4a in the bottom, 4a
times a would equal 4a squared.
But whatever you do to the top you must do
to the bottom. So I'm going to multiply
that by 4a. And I'm going to show you
what this looks like. And you're gonna
see maybe something happen here.
Uh, so 4a, uh,
this is times negative c. So it would end
up equaling a negative. This is a big
negative. One. So I could say negative 4ac
all over 4a squared.
We have plus b
squared all over 4a squared. Okay. Now, I
think at this point we should probably
combine those together. Alright?
So when we combine those, we end up with
this right here. Negative 4ac plus
b squared. Denominators are the same, so
I'm just gonna write it all over 4a squared.
I know that's kind of small down
there. Uh, I'm going to move it over to the
other page. Notice the numerator though.
I'm gonna, on the next page I'm gonna
rewrite it a little different. I'm gonna
write it as b squared minus 4ac. And if
you think about that, that is the
discriminant. Okay?
So I'm gonna go over to this page and
rewrite what we all have. Alright? So, we
still have an x, plus a b, over 2a squared.
Equals, I'm going to write my b squared
first now. So b squared minus 4ac, all
over 4a squared. Okay? So from here, I'm
going to now take the square root of
both sides. Okay? Square root.
And remember if I do the
square root, I have to put plus or minus
over there. So let's rewrite what we have.
Se have x plus b over 2a equals plus or
minus. I want you to take a look at this.
There's nothing you can do with the b
squared minus 4ac. Okay? We would have to
do that first. We're gonna leave that as
our discriminant inside the radical symbol
So I'm just gonna put this in the
numerator. B squared minus 4ac. But I want
you to take a look at the numerator. Do
you see the 4a squared right here? Four is
perfect. A squared is perfect. The square
root of four is two. And the square root of a
squared is a. This is probably starting
to look pretty similar to what, what you
know of the quadratic formula to look like. The last thing we need to do is we
need to subtract this b over 2a on both
sides.
So I'm gonna rewrite it. X equals.
Temember, and it's an x because we're
trying to find out what the x-intercepts are.
So remember, we add this to both sides.
We get negative b, over 2a, plus or minus
the square root of b
squared minus 4ac, all over 2a. At this
point there's just one last thing to do.
Because they both have a 2a in the
denominator. I would combine them
together. And this is where we get the
quadratic formula. We have negative b,
plus or minus the square root of b
squared minus 4ac all over 2a.
And that's
x equals that. And that right there is
the quadratic formula. So in this video
I've shared with you how the quadratic
formula actually came about, um, and it
wasn't just finally thrown out there. It
was by completing the square. and that's
why completing the square, and the
quadratic formula are two methods that
can solve any quadratic, um, equation for you.
Thanks for watching.
