Hi everyone, I'm totally excited about making
this video because it's not every day that
you get to witness a big breakthrough in elementary
math.
It's a brand new way of solving the quadratic
equation. Yeah.
Right, so this new way is not only new it's
better from my point of view. It''s better,
it's easier to do, it's easier to understand
where it came from, it's, it to me is better
in every way.
And it's brand new. It was just posted online
by Professor Po-Shen Loh, who is a professor
of mathematics at Carnegie Mellon University
on October 16. So as of today, it's less than
two months ago. It's like a month and a half
ago.
Yeah. And, and, and the reason why it's so
exciting is because humans have been solving
quadratic equations for like several thousand
years since Babylonian time, and
using the modern notation, the kind of notation
in math that we're using now we've been solving
it with the quadratic equation for something
I want to say 500 maybe 400 but definitely
a lot of hundreds of years ago, and nobody
thought of this particular approach. So that's
why I'm so excited about it. And I'm so happy
I'm able to share it with you. So without
further ado, let's take a look at how we solve
a quadratic equation.
Right. So first of all, if you're given an
equation x^2 plus bx plus c equals zero, then
our first instinct, especially here in the
US is to try to factor. Okay. And when factoring
doesn't work,
then we move on to the quadratic formula.
Now the quadratic formula, on the one hand,
is your best friend because it can solve every
quadratic equation ever. You throw a quadratic
equation at it and it will solve it.
It's not like factoring where a lot of time,
you have seen it. Okay. You have seen situations
where you try to solve using factoring, you
can't
The quadratic formula doesn't have that problem,
but it has the opposite problem is it's so
powerful but it's hard. Okay.
People have to come up with like stories about
negative boy couldn't decide whether to go
to Radical Party, blah, blah, blah. Or they
have to come up with a song, and sing to it.
So you can get negative b plus or minus square
root b squared minus 4ac over 2a.
And the first few times you did it, and as
a math teacher I've seen this with my own
eyes multiple times, the first time people
carry out the quadratic equation. It was hard.
To keep track of what is a equal to what's
b equal to what's c equal to. You plug it
in and the order of operation whole PEMDAS
thing messes people up.
And the biggest problem is especially once
you've got really advanced and people try
to explain to you where the quadratic formula
came from, you, like, oh really completing
the square. And I can tell you a lot of my
students, even some of the good ones,
are afraid of completing the square. They're
like, I will give an exam, lots of different
problems and they will try them all.
But there will be students who will try them
all, except for the completing the square
problem. So I know it's kind of scary.
And that's the downside of the quadratic quadratic
formula, it's so powerful, but to carry it
out is hard and to understand where it came
from. You have to understand something even
harder.
So enter a math professor at Carnegie Mellon
University Professor Po-Shen Loh. He provided
this new method and he describes it in his
abstract. You can pause the video to read
it if you want but I agree with what he -- his
evaluation of his own method.
That is, very little computation. And it's
conceptually natural, uses the graph of
Graph of the quadratic and the fact that,
you know, average and all that. But you'll
see
So first, let's take a look at a simple example.
Okay.
Let's say you're given x squared plus six
x plus five. I said, simple, but yeah, it's
not like the most simple but it is simple
enough.
Okay, because it's something that you can
factor. Okay, you look at this equation, you
say, I'm gonna look for two numbers that add
to six and multiply to five.
Okay. And since you're here, I might as well
tell you when you're trying to factor a quadratic,
look for the two numbers of multiply to c
first.
Okay. Oh, and by the way, I said "c" because
this is where conveniently there's no "a"
terms here. So starting out with x squared
plus. That's the kind of thing we like, right,
we like to factor equations that don't have
an a in front.
Or has a equals one, right. So anyway, so
we look at the product first, concentrate
on that because there are fewer answers there.
What are two numbers that multiply to five?
Either plus one plus five positive one and
positive five, or negative one and negative
five.
Okay, so between those two pairs which one
has a sum of 6?
The positive 1 and positive 5. So the two
numbers are positive 1 and positive 1. Use
those two numbers in your factor x plus 1
times x plus 5 equals 0. You get the roots
of negative 1 and negative 5. Done. Okay.
Good, yeah? Now I'd like to point out something.
When you're trying to factor you look for
two numbers that add up to positive six. You
look for two numbers that add up to "b"
Okay, but by the time you solve
By the time you solve the two roots are negative
1 and negative 5. The two roots are the opposite
of the two numbers you use to factor.
Yeah?
Okay, and therefore their sum also be opposite,
ok. The roots -- they sum up to negative six,
not six. The two numbers you use to factor
add up to six, but the roots themselves add
to negative six.
So here's the rule. The rule is when you have
equation x squared plus bx plus c.
The roots will add to negative b. That will
be important in a minute.
Right. Now, let's make it harder.
Let's change that number five to seven.
Our first instinct. Let's think about factoring.
Two numbers that multiply to 7 and add to
6.
Either 1 and 7, that adds to 8 that doesn't
work.
Or negative 1, negative 7. Those add to negative
8, not 6. So don't work either.
Some of you may say, with, how about 7 and
negative 1. Well, nice try. Seven and negative
1 add to 6, but they multiply to negative
seven. And we're looking for positive seven
All right. So yeah, we have two choices, negative
one and negative seven positive one positive
seven and they don't work. So we can't do
it. We can't do it. We have to, guess what,
OMG pull out the quadratic formula.
Maybe not.
So here's Professor Loh's approach. Think
about the graph of the function x^2 plus 6x
plus 7.
Let's say that it has two solutions.
Here and there. Those are the roots. Right.
Then we know something about these roots.
We know that they add to negative six. And
we know they multiply to seven. So first we
know they add to negative six.
Okay, and the two roots are on the two sides
of the parabola like this. So they're symmetric.
they're symmetric across this number here.
And this value here is the average of the
two roots. Okay, yeah? With me so far?
So here's Professor Loh's first light bulb.
First Light bulb is this that there's an average
here. And what's the value of that average?
If the sum - if the two numbers add up to
negative 6 then the average is negative 3.
Yeah, okay.
That is his first light bulb.
So here's his first first big light bulb moment.
We don't know where the roots are. There's
one here. There's one there.
We don't know where they are exactly but we
do know that they add up to negative 6. If
the two numbers add up to negative six then
the average value of those two numbers has
to be negative 3. So that means this place
this location right here is negative 3.
And we don't know what this distance is between
the two.
Between the two points in the average. But
we know that this distance and that distance
are the same distance
Okay, so if the middle is negative three and
the first root is there, then we can call
this some letter Z plus some letter Z. So
from negative 3 you add z it goes to the first
root. And you subtract the Z, it goes to the
other root.
Okay, so
We don't know what the two routes are equal
to but we know that they must be negative
three plus some z for one of them and negative
three minus some Z for the other
So the 3 is in the middle. I'm sorry the negative
3 is in the middle.
The -3 is in the middle and this number here
has to be -
It has to be negative three plus z.
And this number back here must be negative
three minus z.
So that's the first light bulb.
So far, and you're like, Okay, how does that
help, okay. It's interesting, but how does
that help?
Well, so far we've used the fact that the
two roots have to add up to -6.
Well Professor Loh has another lightbulb moment.
His second light bulb moment is, well, the
product of the two roots has to be seven.
And so let's use that fact.
We know the two roots must be -3 plus z and
the other must be -3 minus z. So, the product
of them must be seven. So I have negative
three plus z on one side.
Negative three plus z is here and have negative
three minus z over here.
And they are the two roots, so they must multiply
to seven.
So two lightbulb moments. And we got a new
equation, the new equation is -3 plus z times
-3 minus z equals seven.
Hah, and you're like well how's that supposed
to help me right? Especially if you're familiar
with quadratic formula you know that you're
going to have to multiply this out. Move the
seven over and you're thinking like
I'm not sure if this helps. I'm just exchanging
one equation that's kind of bad with another
equation that also kind of bad
And this is Professor Lohs third and most
important lightbulb.
His new lightbulb moment is this
This is the formula for difference of squares.
You have a plus b.
Times a minus b
That's the formula for the difference of squares.
So if you multiply this out. Okay. A plus
B times A minus B is A squared minus B squared.
So if you multiply out the left side, you
get -3 squared minus z square, So basically
you got nine minus z squared equals to seven.
So, yeah.
It's a much easier equation to solve than
your original quadratic because there isn't
a z term, there's just z squared.
So you can just solve it.
By first solving for z squared.
Subtracting nine both side you have negative
z squared equals -2. z squared equals 2. So,
z is equal to plus or minus square root of
2.
Actually, since we defined z as a distance,
it's just the positive - it's just square
root of 2. So yeah, you know what you don't
even need to know about the plus or minus
square root, you just go square root of 2.
So remember this?
We now know what Z is so the solutions are,
one is -3 plus square root of 2 and the other
is -3 minus square root of 2.
So the roots are -3 plus square root of 2
and -3 minus square root of 2
Hah!
Done. Yeah.
Isn't that cool? No quadratic formula.
Okay, no quadratic formula and you don't need
to understand completing the square to know
where how I got my solution. I got my solution
from the fact that the sum of the two numbers
is equal to be sorry -b and therefore, the
average of the two numbers is -b [over 2]
Okay.
And what's also cool for me is that it reinforces
your knowledge of difference of squares. And
as an educator - as a math educator, I really
want you to understand the difference of squares.
I really want you to know how to work with
the special factors, the difference of squares,
the square of a sum, the square of a difference,
etc. Okay. So yeah, that's why I think it's
so cool.
So you can run through the video again to
see how it works. But I'll summarize it here.
In summary, first of all, given an equation
x squared plus Bx plus C equals zero then
the first thing we notice is that the sum
of the two roots is negative b.
So the average of the two roots is -b/2 okay
so it'll be that number in between those two
points on the parabola. Okay, so first you
compute that number to see what it's equal
to
Now you know that the product of the two roots
is "c". So you set up that equation. Okay,
the first part of the equation is the first
root, the average plus some number z.
And second number is the other root, the average
minus some number z. And this average you
got out of negative be over to so it's an
actual number. Okay, so that's an numerical
value here. There's a numerical value there
and there's no medical value where c is
And this product is the difference of squares.
So you multiply out the left side which is
a difference of squares. So you end up with
a bunch of numbers and a z squared, you don't
have a z term. So you don't need the quadratic
formula.
You solve for z squared and then you take
the square root you get z.
So from the first step you get the average
from the last step, you get the Z. OK, so
the roots are average plus z and average minus
z. Done.
Right. Now, those of you who are paying attention
are going to say, Wait, wait, wait. I have
a question. I see this. You can't fool me.
I see this
I see this part right here where you have
an equation with no a, or a is equal to 1.
Yeah.
I hear you.
So yeah, you
What if the quadratic equation has an "a"
that's not equal to 1? How do you make it
1? Divide it out.
Divide both sides by "a"
So yeah, if your given equation is 2x^2 minus
3x plus 2, just divide by 2.
And as a matter of fact, that's what I would
do in my next video and we'll do some more
examples where "a" is not equal to one and
also where the roots are complex numbers.
Professor Loh's method happens to also work
when the roots are complex numbers. So yeah,
just like the quadratic formula, it will work
all the time.
So thank you so much, Professor Po-Shen Loh.
This has been awesome. And thank you very
much for watching.
