Factoring Quadratic Expressions

Factoring Quadratic Expressions

In this video I want to do a
bunch of examples of factoring

a second degree polynomial,
which is

often called a quadratic.

Sometimes a quadratic
polynomial, or just a

quadratic itself, or quadratic
expression, but all it means

is a second degree polynomial.

So something that's going to
have a variable raised to the

second power.

In this case, in all of the
examples we'll do, it'll be x.

So let's say I have the
quadratic expression, x

squared plus 10x, plus 9.

And I want to factor it into the
product of two binomials.

How do we do that?

Well, let's just think about
what happens if we were to

take x plus a, and multiply
that by x plus b.

If we were to multiply these
two things, what happens?

Well, we have a little bit
of experience doing this.

This will be x times x, which is
x squared, plus x times b,

which is bx, plus a times x,
plus a times b-- plus ab.

Or if we want to add these two
in the middle right here,

because they're both
coefficients of x.

We could right this as x squared
plus-- I can write it

as b plus a, or a plus
b, x, plus ab.

So in general, if we assume that
this is the product of

two binomials, we see that this
middle coefficient on the

x term, or you could say the
first degree coefficient

there, that's going to be
the sum of our a and b.

And then the constant term is
going to be the product

of our a and b.

Notice, this would map
to this, and this

would map to this.

And, of course, this is the
same thing as this.

So can we somehow pattern
match this to that?

Is there some a and b where
a plus b is equal to 10?

And a times b is equal to 9?

Well, let's just think about
it a little bit.

What are the factors of 9?

What are the things that a
and b could be equal to?

And we're assuming that
everything is an integer.

And normally when we're
factoring, especially when

we're beginning to factor,
we're dealing

with integer numbers.

So what are the factors of 9?

They're 1, 3, and 9.

So this could be a 3 and a 3,
or it could be a 1 and a 9.

Now, if it's a 3 and a 3, then
you'll have 3 plus 3-- that

doesn't equal 10.

But if it's a 1 and a
9, 1 times 9 is 9.

1 plus 9 is 10.

So it does work.

So a could be equal to 1, and
b could be equal to 9.

So we could factor this
as being x plus 1,

times x plus 9.

And if you multiply these two
out, using the skills we

developed in the last few
videos, you'll see that it is

indeed x squared plus
10x, plus 9.

So when you see something like
this, when the coefficient on

the x squared term, or the
leading coefficient on this

quadratic is a 1, you can just
say, all right, what two

numbers add up to this
coefficient right here?

And those same two numbers, when
you take their product,

have to be equal to 9.

And of course, this has to
be in standard form.

Or if it's not in standard form,
you should put it in

that form, so that you can
always say, OK, whatever's on

the first degree coefficient,
my a and b

have to add to that.

Whatever's my constant term, my
a times b, the product has

to be that.

Let's do several
more examples.

I think the more examples
we do the more

sense this'll make.

Let's say we had x squared
plus 10x, plus-- well, I

already did 10x, let's do a
different number-- x squared

plus 15x, plus 50.

And we want to factor this.

Well, same drill.

We have an x squared term.

We have a first degree term.

This right here should be
the sum of two numbers.

And then this term, the constant
term right here,

should be the product
of two numbers.

So we need to think of two
numbers that, when I multiply

them I get 50, and when
I add them, I get 15.

And this is going to be a bit of
an art that you're going to

develop, but the more practice
you do, you're going to see

that it'll start to
come naturally.

So what could a and b be?

Let's think about the
factors of 50.

It could be 1 times 50.

2 times 25.

Let's see, 4 doesn't
go into 50.

It could be 5 times 10.

I think that's all of them.

Let's try out these numbers,
and see if any of

these add up to 15.

So 1 plus 50 does not
add up to 15.

2 plus 25 does not
add up to 15.

But 5 plus 10 does
add up to 15.

So this could be 5 plus 10, and
this could be 5 times 10.

So if we were to factor this,
this would be equal to x plus

5, times x plus 10.

And multiply it out.

I encourage you to multiply this
out, and see that this is

indeed x squared plus
15x, plus 10.

In fact, let's do it. x
times x, x squared. x

times 10, plus 10x.

5 times x, plus 5x.

5 times 10, plus 50.

Notice, the 5 times
10 gave us the 50.

The 5x plus the 10x is giving
us the 15x in between.

So it's x squared plus
15x, plus 50.

Let's up the stakes a little
bit, introduce some negative

signs in here.

Let's say I had x squared
minus 11x, plus 24.

Now, it's the exact
same principle.

I need to think of two numbers,
that when I add them,

need to be equal
to negative 11.

a plus b need to be equal
to negative 11.

And a times b need to
be equal to 24.

Now, there's something for
you to think about.

When I multiply both of these
numbers, I'm getting a

positive number.

I'm getting a 24.

That means that both of these
need to be positive, or both

of these need to be negative.

That's the only way I'm going to
get a positive number here.

Now, if when I add them, I get
a negative number, if these

were positive, there's no way I
can add two positive numbers

and get a negative number, so
the fact that their sum is

negative, and the fact that
their product is positive,

tells me that both a
and b are negative.

a and b have to be negative.

Remember, one can't be negative
and the other one

can't be positive, because the
product would be negative.

And they both can't be positive,
because when you add

them it would get you
a positive number.

So let's just think about
what a and b can be.

So two negative numbers.

So let's think about
the factors of 24.

And we'll kind of have to think
of the negative factors.

But let me see, it could be 1
times 24, 2 times 11, 3 times

8, or 4 times 6.

Now, which of these when I
multiply these-- well,

obviously when I multiply
1 times 24, I get 24.

When I get 2 times 11-- sorry,
this is 2 times 12.

I get 24.

So we know that all these,
the products are 24.

But which two of these, which
two factors, when I add them,

should I get 11?

And then we could say,
let's take the

negative of both of those.

So when you look at these,
3 and 8 jump out.

3 times 8 is equal to 24.

3 plus 8 is equal to 11.

But that doesn't quite
work out, right?

Because we have a negative
11 here.

But what if we did negative
3 and negative 8?

Negative 3 times negative 8
is equal to positive 24.

Negative 3 plus negative 8
is equal to negative 11.

So negative 3 and
negative 8 work.

So if we factor this, x squared
minus 11x, plus 24 is

going to be equal to x minus
3, times x minus 8.

Let's do another
one like that.

Actually, let's mix it
up a little bit.

Let's say I had x squared
plus 5x, minus 14.

So here we have a different
situation.

The product of my two numbers is
negative, right? a times b

is equal to negative 14.

My product is negative.

That tells me that one of them
is positive, and one of them

is negative.

And when I add the two, a plus
b, it'd be equal to 5.

So let's think about
the factors of 14.

And what combinations of them,
when I add them, if one is

positive and one is negative,
or I'm really kind of taking

the difference of the
two, do I get 5?

So if I take 1 and 14-- I'm just
going to try out things--

1 and 14, negative 1 plus
14 is negative 13.

Negative 1 plus 14 is 13.

So let me write all of the
combinations that I could do.

And eventually your brain
will just zone in on it.

So you've got negative 1
plus 14 is equal to 13.

And 1 plus negative 14 is
equal to negative 13.

So those don't work.

That doesn't equal 5.

Now what about 2 and 7?

If I do negative 2-- let me do
this in a different color-- if

I do negative 2 plus 7,
that is equal to 5.

We're done!

That worked!

I mean, we could have tried 2
plus negative 7, but that'd be

equal to negative 5, so that
wouldn't have worked.

But negative 2 plus 7 works.

And negative 2 times
7 is negative 14.

So there we have it.

We know it's x minus
2, times x plus 7.

That's pretty neat.

Negative 2 times 7
is negative 14.

Negative 2 plus 7
is positive 5.

Let's do several more of these,
just to really get well

honed this skill.

So let's say we have x squared
minus x, minus 56.

So the product of the two
numbers have to be minus 56,

have to be negative 56.

And their difference, because
one is going to be positive,

and one is going to be
negative, right?

Their difference has
to be negative 1.

And the numbers that immediately
jump out in my

brain-- and I don't know if they
jump out in your brain,

we just learned this in
the times tables--

56 is 8 times 7.

I mean, there's other numbers.

It's also 28 times 2.

It's all sorts of things.

But 8 times 7 really jumped
out into my brain, because

they're very close
to each other.

And we need numbers that are
very close to each other.

And one of these has to be
positive, and one of these has

to be negative.

Now, the fact that when their
sum is negative, tells me that

the larger of these two should
probably be negative.

So if we take negative
8 times 7, that's

equal to negative 56.

And then if we take negative
8 plus 7, that is equal to

negative 1, which is exactly the
coefficient right there.

So when I factor this, this
is going to be x minus 8,

times x plus 7.

This is often one of the hardest
concepts people learn

in algebra, because it
is a bit of an art.

You have to look at all of the
factors here, play with the

positive and negative signs,
see which of those factors

when one is positive, one is
negative, add up to the

coefficient on the x term.

But as you do more and more
practice, you'll see that

it'll become a bit
of second nature.

Now let's step up the stakes
a little bit more.

Let's say we had negative x
squared-- everything we've

done so far had a positive
coefficient, a positive 1

coefficient on the
x squared term.

But let's say we had a
negative x squared

minus 5x, plus 24.

How do we do this?

Well, the easiest way I can
think of doing it is factor

out a negative 1, and then it
becomes just like the problems

we've been doing before.

So this is the same thing as
negative 1 times positive x

squared, plus 5x, minus 24.

Right?

I just factored a
negative 1 out.

You can multiply negative 1
times all of these, and you'll

see it becomes this.

Or you could factor the negative
1 out and divide all

of these by negative 1.

And you get that right there.

Now, same game as before.

I need two numbers, that when
I take their product I get

negative 24.

So one will be positive,
one will be negative.

When I take their sum,
it's going to be 5.

So let's think about
24 is 1 and 24.

Let's see, if this is negative 1
and 24, it'd be positive 23,

if it was the other way around,
it'd be negative 23.

Doesn't work.

What about 2 and 12?

Well, if this is negative--
remember, one of these has to

be negative.

If the 2 is negative, their
sum would be 10.

If the 12 is negative, their
sum would be negative 10.

Still doesn't work.

3 and 8.

If the 3 is negative,
their sum will be 5.

So it works!

So if we pick negative 3 and
8, negative 3 and 8 work.

Because negative
3 plus 8 is 5.

Negative 3 times 8
is negative 24.

So this is going to be equal
to-- can't forget that

negative 1 out front, and then
we factor the inside.

Negative 1 times x minus
3, times x plus 8.

And if you really wanted to,
you could multiply the

negative 1 times this,
you would get 3

minus x if you did.

Or you don't have to.

Let's do one more of these.

The more practice, the
better, I think.

All right, let's say I had
negative x squared

plus 18x, minus 72.

So once again, I like to factor
out the negative 1.

So this is equal to negative
1 times x squared,

minus 18x, plus 72.

Now we just have to think of
two numbers, that when I

multiply them I get
positive 72.

So they have to be
the same sign.

And that makes it easier in our
head, at least in my head.

When I multiply them,
I get positive 72.

When I add them, I
get negative 18.

So they're the same sign, and
their sum is a negative

number, they both must
be negative.

And we could go through all
of the factors of 72.

But the one that springs up,
maybe you think of 8 times 9,

but 8 times 9, or negative 8
minus 9, or negative 8 plus

negative 9, doesn't work.

That turns into 17.

That was close.

Let me show you that.

Negative 9 plus negative 8, that
is equal to negative 17.

Close, but no cigar.

So what other ones are there?

We have 6 and 12.

That actually seems
pretty good.

If we have negative 6 plus
negative 12, that is equal to

negative 18.

Notice, it's a bit of an art.

You have to try the different
factors here.

So this will become negative
1-- don't want to forget

that-- times x minus 6,
times x minus 12.