Quadratic Formula (proof) Deriving the quadratic formula by completing the square.
Quadratic Formula (proof)
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- In the completing the square video I kept saying that all
- the quadratic equation is completing the square
- as kind of a short cut of completing square.
- And I was under the impression that I had done this
- proof already but now I realize that I haven't.
- So let me prove the quadratic equation to you, by
- completing the square.
- So let's say I have a quadratic equation.
- I guess a quadratic equation is actually what you're trying to
- solve, and what a lot of people call the quadratic equation is
- actually the quadratic formula.
- But anyway I don't want to get caught up in terminology.
- But let's say that I have a quadratic equation that
- says ax squared plus bx plus c is equal to 0.
- And let's just complete the square here.
- So how do we do that?
- Well let's subtract c from both sides so we get ax squared plus
- the bx is equal to minus c.
- And just like I said in the completing the square video
- I don't like having this a coefficient here.
- I like just having one coefficient on my x squared
- term so let me divide everything by a.
- So I get x squared plus b/a x is equal to-- you have
- to divide both sides by a --minus c/a.
- Now we are ready to complete the square.
- What was completing the square?
- Well it's somehow adding something to this expression so
- it has the form of something that is the square
- of an expression.
- What do i mean by that?
- Well, I'll do a little aside here.
- if I told you that x plus a squared, that equals
- x squared plus two ax plus a squared, right?
- So if I can add something here so that this left hand side
- this expression looks like this, then I could
- go the other way.
- I can say this is going to be x plus something squared.
- So what do I have to add on both sides?
- If you watched the completing the square video this should be
- hopefully intuitive for you.
- What you do is you say well this b/a, this corresponds to
- the 2a term, so a is going to be half of this, is going to
- be half of this coefficient.
- That would be the a.
- And then what I need to add is a squared.
- So I need to take half of this and then square it and
- then add it to both sides.
- Let me do that in a different color.
- Do it in this magenta.
- So I'm going to take half of this-- I'm just completing
- square, that's all I'm doing, no magic here --so
- plus half of this.
- Well half of that is b/2a right?
- You just multiply by 1/2.
- And I have to square it.
- Well if I did it to the left hand side of the equation, I
- have to do it to the right hand side.
- So plus b/2a squared.
- And now I have this left hand side of the equation in the
- form that it is the square of an expression that is
- x plus something.
- And what is it?
- Well that's equal to-- let me switch colors again --what's
- the left hand side of this equation equal to?
- And you can just use this pattern and go to the left.
- It's x plus what?
- Well we said a, you can do one of two ways. a is 1/2 of this
- coefficient or a is the square root of this coefficient or
- since we didn't even square it we know that this
- is a. b/2a is a.
- So this is the same thing as x plus b over 2a everything
- squared, and then that equals-- let's see if we can simplify
- this or make this a little bit cleaner --that equals--
- See, if I were to have a common denominator-- I'm just doing a
- little bit of algebra here --see, when I square this it's
- going to be 4a squared-- let me let me write this.
- This is equal to b squared over 4a squared.
- And so if I have to add these two fractions, let me make
- this equal to 4a squared.
- And if the denominator is 4a squared, what does
- the minus c/a become?
- I See if I multiply the denominator by 4a, I have to
- multiply the numerator by 4a.
- So this becomes minus 4ac, right?
- And then b squared over 4a squared, well that's
- just still b squared.
- I'm just doing a little bit of algrebra.
- Hopefully I'm not confusing you.
- I just expanded this.
- I just took the square of this, b squared over 4a squared.
- And then I added this to this, I got a common denominator.
- And minus c/a is the same thing as minus 4ac over 4a squared.
- And now we can take the square root of both
- sides of this equation.
- And this should hopefully start to look a little
- bit familiar to you now.
- So let's see, so we get x.
- So if we take the square root of both sides of this equation
- we get x plus b/2a is equal to the square root of this-- let's
- take the square root of the numerator and the demoninator.
- So the numerator is-- I'm going to put the b squared first, I'm
- just going to switch this order, it doesn't matter --the
- square root of b squared minus 4ac, right?
- That's just the numerator.
- I just the square root of it, and we have to get the square
- root of the denominator too.
- What's the square roof of 4a squared?
- Well it's just 2a, right?
- And now what do we do?
- Oh, it's very important!
- When we're taking the square root, it's not just the
- positive square root.
- It's the positive or minus square root.
- We saw that couple of times when we did the-- and you could
- say it's a plus or minus here too, but if you look plus or
- minus on the top and a plus or minus on the bottom, you can
- just write it once on the top.
- I'll let you think about why you only have to write it once.
- If you had a negative an a plus, or negative and a plus
- sometimes cancel out, or a negative and a negative,
- that's the same thing as just having a plus on top.
- Anyway, I think you get that.
- And now we just have to subtract b/2a from both sides.
- and we get, we get-- and this is the exciting part --we get x
- is equal to minus be over to 2a plus or minus this thing, so
- minus b squared minus 4ac, all of that over 2a.
- And we already have a common denominator, so we can
- just add the fractions.
- So we got --and I'm going to do this in a vibrant bold-- I
- don't know maybe not so much bold, well green color --so we
- get x is equal to, numerator, negative b plus or minus square
- root of b squared minus 4ac, all of that over 2a.
- And that is the famous quadratic formula.
- So, there we go we proved it.
- And we proved it just from completing the square.
- I hope you found that vaguely interesting.
- See in the next Video.
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