特殊積的因式分解 (英)

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特殊積的因式分解 (英) : 特殊積的因式分解

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Let's solve some quadratic equations by factoring.
So let's say I had x squared plus 4x is equal to 21.
Now your impulse might be to try to factor out an x and
somehow set that equal to 21.
And that will not lead you to good solutions.
You'll probably end up doing
something that's not justified.
What you need to do here is put the entire quadratic
expression on one side of the equation.
We'll do it onto the left-hand side.
So let's subtract 21 from both sides of this equation.
The left-hand side then becomes x squared
plus 4x minus 21.
And then the right-hand side will be equal to 0.
And the way you want to solve this, this
is a quadratic equation.
We have a quadratic expression being set equal to 0.
The way you want to solve this is you want to factor them,
and say, OK, each of those factors could
then be equal to 0.
So how do we factor this?
Well, we saw in the last video that we have to figure out two
numbers whose product is equal to negative 21, and whose sum
is equal to 4.
This would be a plus b would have to be equal to 4.
Since their product is negative, they have to be of
different signs.
And so let's see, the number that jumps out
at me is 7 and 3.
If I have negative 7 and positive 3, I would get
negative 4.
So let's do positive 7 and negative 3.
So the a and b are positive 7 and negative 3.
When I take the product, I get negative 21.
When I take their sum, I get positive 4.
So I can rewrite this equation here.
I could rewrite it as x plus 7, times x minus
3, is equal to 0.
And now I can solve this by saying, look, I have two
quantities.
Their product is equal to 0.
That means that one or both of them have to be equal to 0.
So that means that x plus 7 is equal to 0.
That's an x.
Or x minus 3 is equal to 0.
I could subtract 7 from both sides of this equation.
And I would get x is equal to negative 7.
And over here, I can add 3 to both sides of this equation.
And I'll get x is equal to 3.
So both of these numbers are solutions to this equation.
You could try it out.
If you do 7-- negative 7 squared is 49.
Negative 7 times 4 is minus 28, or negative 28.
And that does indeed equal 21.
And I'll let you try it out with the positive 3.
Actually, let's just do it.
3 squared is 9, plus 4 times 3 is 12.
9 plus 12 is, indeed, 21.
Let's do a bunch more examples.
Let's say I have x squared plus 49 is equal to 14x.
Once again, whenever you see anything like this, get all of
your terms on one side of the equation and get a 0 on the
other side.
That's the best way to solve a quadratic equation.
So let's subtract 14x from both sides.
We could write this as x squared minus 14x plus 49 is
equal to 0.
Obviously, 14x minus 14x is 0.
This quantity minus 14x is this quantity right there.
Now we just have to think about what two numbers, when I
take their product, I'm going to get 49, and when I take
their sum, I'm going to get negative 14.
So one, they have to be the same sign because this is a
positive number right here.
And they're both going to be negative because
their sum is negative.
And there's something interesting here.
49 is a perfect square.
Its factors are 1, 7, and 49.
So maybe 7 will work, or even better, maybe
negative 7 will work.
And it does!
Negative 7 times negative 7 is 49.
And negative 7 plus negative 7 is negative 14.
We have that pattern there, where we have 2 times a
number, and then we have the number squared.
This is a perfect square.
This is equal to x minus 7, times x minus
7, is equal to 0.
Don't want to forget that.
Or we could write this as x minus 7 squared is equal to 0.
So this was a perfect square of a binomial.
And if x minus 7 squared is equal to 0, take the square
root of both sides.
You'll get x minus 7 is equal to 0.
I mean, you could say x minus 7 is 0 or x minus 7 is 0.
But that'd be redundant.
So we just get x minus 7 is 0.
Add 7 to both sides, and you get x is equal to 7.
Only one solution there.
Let's do another one in pink.
Let's say we have x squared minus 64 is equal to 0.
Now this looks interesting right here.
A bell might be ringing in your head on
how to solve this.
This has no x term, but we could think of it
as having an x term.
We could rewrite this as x squared plus 0x minus 64.
So in this situation, we could say, OK, what two numbers,
when I multiply them, equal 64, and when I
add them equal 0?
And when I take their product, I'm getting a
negative number, right?
This is a times b.
It's a negative number.
So that must mean that they have opposite signs.
And when I add them, I get 0.
That must mean that a plus minus b is equal to 0, or that
a is equal to b, that we're dealing with the same number.
We're essentially dealing with the same number, the negatives
of each other.
So what can it be?
Well, if we're doing the same number and they're negatives
of each other, 64 is exactly 8 squared.
But it's negative 64, so maybe we're dealing with one
negative 8, and we're dealing with one positive 8.
And if we add those two together, we do
indeed get to 0.
So this will be x minus 8 times x plus 8.
Now you don't always have to go through this
process I did here.
You might already remember that if I have a plus b times
a minus b, that that's equal to a squared minus b squared.
So if you see something that fits the pattern, a squared
minus b squared, you could immediately say, oh, that's
going to be a plus b-- a is x, b is 8-- times a minus b.
Let's do a couple more of just general problems. I won't tell
you what type these are going to be.
Let me switch colors.
It's getting monotonous.
Let's say we have x squared minus 24x plus
144 is equal to 0.
Well, 144 is conspicuously 12 squared.
And this is conspicuously 2 times negative 12.
Or this is conspicuously negative 12 squared.
So this is negative 12 times negative 12.
This is negative 12 plus negative 12.
So this expression can be rewritten as x minus 12 times
x minus 12, or x minus 12 squared.
We're going to set that equal to 0.
This is going to be 0 when x minus 12 is equal to 0.
You can say either of these could be equal to 0, but
they're the same thing.
Add 12 to both sides of that equation and you get x is
equal to 12.
And I just realized, this problem up here, I factored
it, but I didn't actually solve the equation.
So this has to be equal to 0.
Let's take a step back to this equation up here.
And the only way that this thing over here will be 0 is
if either x minus 8 is equal to 0 or x plus
8 is equal to 0.
So add 8 to both sides of this.
You get x could be equal to 8.
Subtract 8 from both sides of this.
You get x could also be equal to negative 8.
Let's do one more.
Just to really, really get the point drilled in your head.
Let's do one more.
Let's say we have 4x squared minus 25 is equal to 0.
So you might already see the pattern.
This is an a squared.
This is a b squared.
We have the pattern of a squared minus b squared,
where, in this case, a would be equal to x, right?
This is 2x squared.
And b would be equal to 5.
So if you have a squared minus b squared, this is going to be
equal to a plus b times a minus b.
In this situation, that means that 4x squared minus 25 is
going to be 2x plus 5 times 2x minus 5.
And of course, that will be equal to 0.
And this will only be equal to 0 if either 2x plus 5 is equal
to 0 or 2x minus 5 is equal to 0.
And then we can solve each of these.
Subtract 5 from both sides.
You get 2x is equal to negative 5.
Divide both sides by 2.
You could get one solution is negative 5/2.
Over here, add 5 to both sides.
You get 2x is equal to positive 5.
Divide both sides by 2.
You get x could also be equal to positive 5/2.
So both of these satisfy that equation up there.