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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.