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# Solving quadratic equations by factoring (old)

Video transcript

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.