factor
## Factoring Quadratics... How? (NancyPi)

Hi guys! I'm Nancy and today I'm going to show you

how to factor any quadratic expression.

So factoring can be a nightmare to some people because they feel like they're

just doing trial and error, stabbing in the dark

without any direction. Don't worry I have a way that doesn't involve

any guessing and will work for any quadratic expression.

First I'm going to show you a simple case and then I'm going to show you a trick

called "The Magic X"

for factoring any tougher quadratic.

OK. Say you have a quadratic expression like this

X squared plus 4X - 12 and you need to factor it.

What you need to find are two numbers that multiply to give you

this last number, -12, and which add to give you

the second number, positive 4.

So again, you need to find two numbers

which multiply

to -12

and which also add

to positive 4.

OK.

So first think all the numbers, all the pairs of numbers that would multiply to

-12.

And list them

in a column over here. List all your options and you can rule them out later.

So what pairs of numbers multiply -12?

We could have 1 and -12. That would give you a product of -12.

You can flip the signs to -1 and 12.

You could have 2 and -6.

-2 and 6. It's a little tedious.

You're writing all your options. 3 and -4. -3 and 4.

And those are all your possible pairs of numbers that multiply to -12.

So you've taken care of that requirement.

Now, you need to figure out which of these pairs

would also add to positive 4.

So check all of them.

1 plus -12 would give you some big negative number like -11.

Rule it out. -1 plus 12 would give you positive 11.

No. 2 plus -6 would give you -4.

Close, but not positive 4. -2 plus 6 will give you positive 4.

So those are your answers. Your numbers

for factoring and you can ignore the others. You don't need to check them at

that point.

All need to do is rewrite your quadratic

as two sets of parentheses multiplied together.

Each of them starting with X. And fill in those two numbers that you found.

-2 and 6.

Fill in -2 and 6.

Now of course you can simplify that. And just write it as

X - 2 times X + 6.

So that's your answer

for how to factor this quadratic.

Now if you want to you, you can always check

your factoring answer, by multiplying this out.

Foiling it out. And checking to make sure it's the same as your original

quadratic.

OK.

Say you're given a quadratic that doesn't start with X squared,

that actually has a term like 3X squared or 2X squared in the beginning.

First thing to do is check to see if

an overall number will factor out front. In this case

for instance, you have 3 that can go into

every one of the three terms. You can pull out an

overall 3 constant. When you do

you're left with

just X squared

plus 4X minus 12.

Which you'll remember is the same as

the last problem we just did. So this is actually not tougher factoring problem. This is

the same as the last problem, just disguised by this

overall 3 constant. And this would factor

the same as before

X - 2 times X + 6.

OK. Next we are going to look at a truly tougher example.

And I'll show you the "magic X trick", that will work for

any factoring problem.

OK. Say that you have a quadratic

and it doesn't start with just X squared, and it has a term like

3X squared in the beginning. You can use trial and error to factor this if you want,

but that may take a long time.

And I have faster, quicker method called "magic X"

That's a sure-fire way to factor.

For the "magic X" method you do

literally draw an X off to the side.

Now, at the top of your X

you're going to put the number you get from multiplying

your first coefficient, 3,

by your last constant, -8,

which is -24. You put that

in the top of your X. In the bottom you're going to put

your middle number, 10. Now what you need to do for the trick

is find two numbers that multiply

to give you -24, and add to give you 10.

So we can write that.. Find two numbers

that multiply

to -24

and

add

to 10. Then you can list pairs and you'll find that

12 and -2 are your two numbers.

Because 12 and -2 multiply to -24 and 12

plus -2 gives you 10.

OK. Next step

in the "magic X" method is

for each these numbers divide them

by your leading coefficient. In this problem its 3.

3 is the first coefficient on your X squared.

So you divide this number by 3,

and you divide this number you found by 3.

Those fractions simplify, so I'm going to write

a simplified X down here.

12 over 3 simplifies to 4 over 1.

-2 over 3 stays the same. It's already in simplest form.

OK. You're almost done with factoring.

You're going to use these fractions to write your final factoring.

The bottom number

in this faction gives you the coefficient of X.

So we have 1X.

The top number gives you your constant.

So just plus 4. Same for the other term. The other factor.

Your bottom number here, 3, gives you

your coefficient of X.

And the top number, -2, gives you

your constant.

And you're done.

This is your factorization, your factoring

of your quadratic.

And again, if you want, you can always check your answer

by multiplying this out. Foiling

all the terms. And checking the you get back your original quadratic.

And you will.

I hope this helped you figure out factoring. I know factoring

is super fun. It's okay you don't have to like math,

but you can like my video. So if you did, please click like below!

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