How to factor quadratic functions step by step
Factoring Quadratics... How? (NancyPi)
Timecodes:
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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