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In the last video, I told you that if you had a quadratic

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equation of the form ax squared plus bx, plus c is

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equal to zero, you could use the quadratic formula to find

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the solutions to this equation.

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And the quadratic formula was x.

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The solutions would be equal to negative b plus or minus

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the square root of b squared minus 4ac, all

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of that over 2a.

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And we learned how to use it.

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You literally just substitute the numbers a for a, b for b,

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c for c, and then it gives you two answers, because you have

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a plus or a minus right there.

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What I want to do in this video is

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actually prove it to you.

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Prove that using, essentially completing the square, I can

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get from that to that right over there.

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So the first thing I want to do, so that I can start

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completing the square from this point right here, is--

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let me rewrite the equation right here-- so we have ax--

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let me do it in a different color-- I have ax squared plus

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bx, plus c is equal to 0.

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So the first I want to do is divide everything by a, so I

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just have a 1 out here as a coefficient.

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So you divide everything by a, you get x squared plus b over

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ax, plus c over a, is equal to 0 over a, which

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is still just 0.

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Now we want to-- well, let me get the c over a term on to

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the right-hand side, so let's subtract c

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over a from both sides.

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And we get x squared plus b over a x, plus-- well, I'll

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just leave it blank there, because this is gone now; we

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subtracted it from both sides-- is equal to negative c

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over a I left a space there so that we can

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complete the square.

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And you saw in the completing the square video, you

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literally just take 1/2 of this coefficient right here

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and you square it.

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So what is b over a divided by 2?

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Or what is 1/2 times b over a?

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Well, that is just b over 2a, and, of course, we are going

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to square it.

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You take 1/2 of this and you square it.

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That's what we do in completing a square, so that

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we can turn this into the perfect square of a binomial.

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Now, of course, we cannot just add the b over 2a squared to

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the left-hand side.

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We have to add it to both sides.

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So you have a plus b over 2a squared there as well.

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Now what happens?

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Well, this over here, this expression right over here,

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this is the exact same thing as x plus b over 2a squared.

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And if you don't believe me, I'm going to multiply it out.

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That x plus b over 2a squared is x plus b over 2a, times x

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plus b over 2a. x times x is x squared.

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x times b over 2a is plus b over 2ax.

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You have b over 2a times x, which is another b over 2ax,

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and then you have b over 2a times b over 2a, that is plus

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b over 2a squared.

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That and this are the same thing, because these two

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middle terms, b over 2a plus b over 2a, that's the same thing

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as 2b over 2ax, which is the same thing as b over ax.

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So this simplifies to x squared plus b over ax, plus b

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over 2a squared, which is exactly what we have written

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right there.

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That was the whole point of adding this term to both

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sides, so it becomes a perfect square.

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So the left-hand side simplifies to this.

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The right-hand side, maybe not quite as simple.

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Maybe we'll leave it the way it is right now.

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Actually, let's simplify it a little bit.

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So the right-hand side, we can rewrite it.

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This is going to be equal to-- well, this is

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going to be b squared.

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I'll write that term first. This is b-- let me do it in

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green so we can follow along.

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So that term right there can be written as b

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squared over 4a square.

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And what's this term?

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What would that become?

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This would become-- in order to have 4a squared as the

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denominator, we have to multiply the numerator and the

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denominator by 4a.

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So this term right here will become

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minus 4ac over 4a squared.

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And you can verify for yourself that that is the same

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thing as that.

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I just multiplied the numerator and the

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denominator by 4a.

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In fact, the 4's cancel out and then this a cancels out

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and you just have a c over a.

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So these, this and that are equivalent.

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I just switched which I write first. And you might already

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be seeing the beginnings of the quadratic formula here.

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So this I can rewrite.

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This I can rewrite.

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The right-hand side, right here, I can rewrite as b

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squared minus 4ac, all of that over 4a squared.

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This is looking very close.

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Notice, b squared minus 4ac, it's already appearing.

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We don't have a square root yet, but we haven't taken the

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square root of both sides of this

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equation, so let's do that.

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So if you take the square root of both sides, the left-hand

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side will just become x plus-- let me scroll down a little

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bit-- x plus b over 2a is going to be equal to the plus

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or minus square root of this thing.

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And the square root of this is the square root of the

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numerator over the square root of the denominator.

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So it's going to be the plus or minus the square root of b

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squared minus 4ac over the square root of 4a squared.

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Now, what is the square root of 4a squared?

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It is 2a, right?

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2a squared is 4a squared.

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The square root of this is that right here.

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So to go from here to here, I just took the square root of

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both sides of this equation.

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Now, this is looking very close to the quadratic.

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We have a b squared minus 4ac over 2a, now we just

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essentially have to subtract this b over 2a from both sides

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of the equation and we're done.

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So let's do that.

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So if you subtract the b over 2a from both sides of this

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equation, what do you get?

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You get x is equal to negative b over 2a, plus or minus the

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square root of b squared minus 4ac over 2a, common

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denominator.

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So this is equal to negative b.

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Let me do this in a new color.

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So it's orange.

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Negative b plus or minus the square root of b squared minus

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4ac, all of that over 2a.

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And we are done!

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By completing the square with just general coefficients in

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front of our a, b and c, we were able to derive the

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quadratic formula.

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Just like that.

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Hopefully you found that as entertaining as I did.