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- In this video, I'm going to show you a technique called
- completing the square.
- And what's neat about this is that this will work for any
- quadratic equation, and it's actually the basis for the
- quadratic formula.
- And in the next video or the video after that I'll prove
- the quadratic formula using completing the square.
- But before we do that, we need to understand even
- what it's all about.
- And it really just builds off of what we did in the last
- video, where we solved quadratics
- using perfect squares.
- So let's say I have the quadratic equation x squared
- minus 4x is equal to 5.
- And I put this big space here for a reason.
- In the last video, we saw that these can be pretty
- straightforward to solve if the left-hand side is a
- perfect square.
- You see, completing the square is all about making the
- quadratic equation into a perfect square, engineering
- it, adding and subtracting from both sides so it becomes
- a perfect square.
- So how can we do that?
- Well, in order for this left-hand side to be a perfect
- square, there has to be some number here.
- There has to be some number here that if I have my number
- squared I get that number, and then if I have two times my
- number I get negative 4.
- Remember that, and I think it'll become
- clear with a few examples.
- I want x squared minus 4x plus something to be equal to x
- minus a squared.
- We don't know what a is just yet, but we
- know a couple of things.
- When I square things-- so this is going to be x squared minus
- 2a plus a squared.
- So if you look at this pattern right here, that has to be--
- sorry, x squared minus 2ax-- this right here has to be 2ax.
- And this right here would have to be a squared.
- So this number, a is going to be half of negative 4, a has
- to be negative 2, right?
- Because 2 times a is going to be negative 4.
- a is negative 2, and if a is negative 2, what is a squared?
- Well, then a squared is going to be positive 4.
- And this might look all complicated to you right now,
- but I'm showing you the rationale.
- You literally just look at this coefficient right here,
- and you say, OK, well what's half of that coefficient?
- Well, half of that coefficient is negative 2.
- So we could say a is equal to negative 2-- same idea there--
- and then you square it.
- You square a, you get positive 4.
- So we add positive 4 here.
- Add a 4.
- Now, from the very first equation we ever did, you
- should know that you can never do something to just one side
- of the equation.
- You can't add 4 to just one side of the equation.
- If x squared minus 4x was equal to 5, then when I add 4
- it's not going to be equal to 5 anymore.
- It's going to be equal to 5 plus 4.
- We added 4 on the left-hand side because we wanted this to
- be a perfect square.
- But if you add something to the left-hand side, you've got
- to add it to the right-hand side.
- And now, we've gotten ourselves to a problem that's
- just like the problems we did in the last video.
- What is this left-hand side?
- Let me rewrite the whole thing.
- We have x squared minus 4x plus 4 is equal to 9 now.
- All we did is add 4 to both sides of the equation.
- But we added 4 on purpose so that this left-hand side
- becomes a perfect square.
- Now what is this?
- What number when I multiply it by itself is equal to 4 and
- when I add it to itself I'm equal to negative 2?
- Well, we already answered that question.
- It's negative 2.
- So we get x minus 2 times x minus 2 is equal to 9.
- Or we could have skipped this step and written x minus 2
- squared is equal to 9.
- And then you take the square root of both sides, you get x
- minus 2 is equal to plus or minus 3.
- Add 2 to both sides, you get x is equal to 2 plus or minus 3.
- That tells us that x could be equal to 2 plus 3, which is 5.
- Or x could be equal to 2 minus 3, which is negative 1.
- And we are done.
- Now I want to be very clear.
- You could have done this without completing the square.
- We could've started off with x squared minus
- 4x is equal to 5.
- We could have subtracted 5 from both sides and gotten x
- squared minus 4x minus 5 is equal to 0.
- And you could say, hey, if I have a negative 5 times a
- positive 1, then their product is negative 5 and their sum is
- negative 4.
- So I could say this is x minus 5 times x plus
- 1 is equal to 0.
- And then we would say that x is equal to 5 or x is equal to
- negative 1.
- And in this case, this actually probably would have
- been a faster way to do the problem.
- But the neat thing about the completing the square is it
- will always work.
- It'll always work no matter what the coefficients are or
- no matter how crazy the problem is.
- And let me prove it to you.
- Let's do one that traditionally would have been
- a pretty painful problem if we just tried to do it by
- factoring, especially if we did it using grouping or
- something like that.
- Let's say we had 10x squared minus 30x minus
- 8 is equal to 0.
- Now, right from the get-go, you could say, hey look, we
- could maybe divide both sides by 2.
- That does simplify a little bit.
- Let's divide both sides by 2.
- So if you divide everything by 2, what do you get?
- We get 5x squared minus 15x minus 4 is equal to 0.
- But once again, now we have this crazy 5 in front of this
- coefficent and we would have to solve it by grouping which
- is a reasonably painful process.
- But we can now go straight to completing the square, and to
- do that I'm now going to divide by 5 to get a 1 leading
- coefficient here.
- And you're going to see why this is different than what
- we've traditionally done.
- So if I divide this whole thing by 5, I could have just
- divided by 10 from the get-go but I wanted to go to this the
- step first just to show you that this really
- didn't give us much.
- Let's divide everything by 5.
- So if you divide everything by 5, you get x squared minus 3x
- minus 4/5 is equal to 0.
- So, you might say, hey, why did we ever do that factoring
- by grouping?
- If we can just always divide by this leading coefficient,
- we can get rid of that.
- We can always turn this into a 1 or a negative 1 if we divide
- by the right number.
- But notice, by doing that we got this crazy 4/5 here.
- So this is super hard to do just using factoring.
- You'd have to say, what two numbers when I take the
- product is equal to negative 4/5?
- It's a fraction and when I take their sum, is equal to
- negative 3?
- This is a hard problem with factoring.
- This is hard using factoring.
- So, the best thing to do is to use completing the square.
- So let's think a little bit about how we can turn this
- into a perfect square.
- What I like to do-- and you'll see this done some ways and
- I'll show you both ways because you'll see teachers do
- it both ways-- I like to get the 4/5 on the other side.
- So let's add 4/5 to both sides of this equation.
- You don't have to do it this way, but I like to get the 4/5
- out of the way.
- And then what do we get if we add 4/5 to both
- sides of this equation?
- The left-hand hand side of the equation just becomes x
- squared minus 3x, no 4/5 there.
- I'm going to leave a little bit of space.
- And that's going to be equal to 4/5.
- Now, just like the last problem, we want to turn this
- left-hand side into the perfect square of a binomial.
- How do we do that?
- Well, we say, well, what number times 2 is equal to
- negative 3?
- So some number times 2 is negative 3.
- Or we essentially just take negative 3 and divide it by 2,
- which is negative 3/2.
- And then we square negative 3/2.
- So in the example, we'll say a is negative 3/2.
- And if we square negative 3/2, what do we get?
- We get positive 9/4.
- I just took half of this coefficient, squared it, got
- positive 9/4.
- The whole purpose of doing that is to turn this left-hand
- side into a perfect square.
- Now, anything you do to one side of the equation, you've
- got to do to the other side.
- So we added a 9/4 here, let's add a 9/4 over there.
- And what does our equation become?
- We get x squared minus 3x plus 9/4 is equal to-- let's see if
- we can get a common denominator.
- So, 4/5 is the same thing as 16/20.
- Just multiply the numerator and denominator by 4.
- Plus over 20.
- 9/4 is the same thing if you multiply the
- numerator by 5 as 45/20.
- And so what is 16 plus 45?
- You see, this is kind of getting kind of hairy, but
- that's the fun, I guess, of
- completing the square sometimes.
- 16 plus 45.
- See that's 55, 61.
- So this is equal to 61/20.
- So let me just rewrite it.
- x squared minus 3x plus 9/4 is equal to 61/20.
- Crazy number.
- Now this, at least on the left hand side,
- is a perfect square.
- This is the same thing as x minus 3/2 squared.
- And it was by design.
- Negative 3/2 times negative 3/2 is positive 9/4.
- Negative 3/2 plus negative 3/2 is equal to negative 3.
- So this squared is equal to 61/20.
- We can take the square root of both sides and we get x minus
- 3/2 is equal to the positive or the negative
- square root of 61/20.
- And now, we can add 3/2 to both sides of this equation
- and you get x is equal to positive 3/2 plus or minus the
- square root of 61/20.
- And this is a crazy number and it's hopefully obvious you
- would not have been able to-- at least I would not have been
- able to-- get to this number just by factoring.
- And if you want their actual values, you can get your
- calculator out.
- And then let me clear all of this.
- And 3/2-- let's do the plus version first. So we want to
- do 3 divided by 2 plus the second square root.
- We want to pick that little yellow square root.
- So the square root of 61 divided by 20, which is 3.24.
- This crazy 3.2464, I'll just write 3.246.
- So this is approximately equal to 3.246, and that was just
- the positive version.
- Let's do the subtraction version.
- So we can actually put our entry-- if you do second and
- then entry, that we want that little yellow entry, that's
- why I pressed the second button.
- So I press enter, it puts in what we just put, we can just
- change the positive or the addition to a subtraction and
- you get negative 0.246.
- So you get negative 0.246.
- And you can actually verify that these satisfy our
- original equation.
- Our original equation was up here.
- Let me just verify for one of them.
- So the second answer on your graphing calculator is the
- last answer you use.
- So if you use a variable answer, that's this number
- right here.
- So if I have my answer squared-- I'm using answer
- represents negative 0.24.
- Answer squared minus 3 times answer minus 4/5-- 4 divided
- by 5-- it equals--.
- And this just a little bit of explanation.
- This doesn't store the entire number, it goes up to some
- level of precision.
- It stores some number of digits.
- So when it calculated it using this stored number right here,
- it got 1 times 10 to the negative 14.
- So that is 0.0000.
- So that's 13 zeroes and then a 1.
- A decimal, then 13 zeroes and a 1.
- So this is pretty much 0.
- Or actually, if you got the exact answer right here, if
- you went through an infinite level of precision here, or
- maybe if you kept it in this radical form, you would get
- that it is indeed equal to 0.
- So hopefully you found that helpful, this whole notion of
- completing the square.
- Now we're going to extend it to the actual quadratic
- formula that we can use, we can essentially just plug
- things into to solve any quadratic equation.