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We're on problem 53.

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It says Toni is solving this equation by completing the

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square. ax squared plus bx plus c is equal to 0, where a

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is greater than 0.

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So this is just a traditional quadratic right here.

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And let's see what they did.

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First, he subtracted c from both sides and he got ax

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squared plus bx is equal to minus c.

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OK, that's fair enough.

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And then let's see.

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He divided both sides by a.

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Right, that's fair enough.

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He got minus c/a.

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Which step should be Step 3 in the solution?

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So he's completing the square.

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So essentially, he wants this to become a perfect square.

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So let's see how we can do that.

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So we have x squared plus b/a x-- and I'm going to leave a

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little space here-- is equal to minus c/a.

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So for this to be a perfect square we have to add

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something here, we have to add a number.

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And we learned from several videos in the past and we kind

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of pseudo-proved it.

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And actually, I have several videos I do solely on

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completing the square.

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You essentially have to add whatever number this is, add

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half of it squared.

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And if that doesn't make sense to you, watch the Khan Academy

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video on completing the square.

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But what is half of b/a?

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Well it's b over 2a.

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So 1/2 times b/a is equal to b over 2a.

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And then, we want to add this squared.

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So let's add that to both sides of this equation.

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So we're left with x squared plus b/a x.

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And we want to add this squared.

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Plus b over 2a squared is equal to minus c/a.

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Anything you add to one side of the equation, you have to

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add to the other.

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So we have to add that to both sides.

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Plus b over 2a squared.

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And let's see if we've solved the problem so

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far, what they want.

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X, b over 2-- right.

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This is exactly what we did. x squared plus b/a plus b over

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2a squared, and they add it to both sides of the equation.

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So D is the right answer.

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Now if you find that a little confusing or if it wasn't

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intuitive for you, I don't want you to

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memorize the steps.

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Watch the Khan Academy video on completing the square.

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Next problem, 56.

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No, 54.

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All right, this is another one that should be cut and pasted.

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All right, four steps to derive the quadratic formula

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are shown below.

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I said in previous videos that you can derive the quadratic

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formula by completing the square.

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And we actually do that in another video.

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I don't want to give too much of a plug for other videos,

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but let's see what they want to do.

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What is the correct order of these steps?

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So the first thing you want to start off with is just a

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quadratic equation.

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And this one is the first step.

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This is where we started off with in the last problem.

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Then what you want to do is add 1/2 of this squared to

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both sides.

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So b over 2a squared you want to add to both sides, and

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that's what they did here.

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So our order is I.

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And then you want to do IV.

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That's what we did in the last problem.

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We did IV.

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And then from here, you know that this expression right

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here is going to be equal to x plus b over 2a squared.

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And once again, watch soon. the completing the squared

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video if that didn't make sense.

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But the whole reason why you added this here is so that you

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know that, OK, what two numbers, when I multiply them

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equal b over 2a squared, and when I add them equal b/a?

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Well that's obviously, b over 2a.

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If you add it twice you're going to get b over a.

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If you square it, you're going to get this whole expression.

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So you say, oh, this is just x plus b over 2a squared and you

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get that there.

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And then, is equal to-- and then they just

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simplify this fraction.

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They found a common denominator and all the rest.

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And so the next step is Step II.

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And then all you have left is Step III.

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And you've pretty much derived the quadratic equation.

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So I, IV, II, III.

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That's choice A.

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Problem 55.

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Which of the solutions-- OK, I'll put all

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of the choices down.

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So which is one of the solutions to the equation?

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So immediately when you see all of the choices, they have

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these square roots and all that.

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This isn't something that you would factor.

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You would use a quadratic equation here.

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So let's do that.

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So the quadratic equation is, so if this is Ax squared plus

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Bx plus C is equal to 0.

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The quadratic equation is minus b.

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Well they do it lowercase.

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Plus or minus the square root of b squared minus 4ac, all of

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that over 2a.

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And this is just derived from completing the square with

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this, but we do that in another video.

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And so let's substitute it in.

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What is b?

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b is minus 1, right?

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So minus minus 1, that's a positive 1.

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Plus or minus the square root of b squared.

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Minus 1 squared is 1.

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Minus 4 times a.

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a is 2.

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Times 2.

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Times c.

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c is minus 4.

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So times minus 4.

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All of that over 2a.

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a is 2, so 2 times a is 4.

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So that becomes 1 plus or minus the square root.

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So we have a 1.

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So we have minus 4 times a 2 times a minus 4.

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That's the same thing as a plus 4 times 2 times a plus 4.

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Let's just take that minus out.

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So it's plus.

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There's no minus here.

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So let's see, 4 times 2 is 8.

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Times 4 is 32.

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Plus 1 is 33.

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All of that over 4.

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Let's see, we're not quite there yet.

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Well they say, which is one of the solutions to the equation?

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So let's see.

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If we wanted to simplify this out a-- well,

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this is right here.

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Because we have 1 plus or minus the square

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root of 33 over 4.

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Well they wrote just one of them.

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They wrote just the plus.

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So C is one of the solutions.

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The other one would have been if you had a minus sign here.

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Anyway, next problem.

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56.

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And this is another one I need to cut and paste.

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It says, which statement best explains why there's no real

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solution to the quadratic equation?

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OK, so I already have a guess of why this

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won't have a solution.

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But in general-- well, let's try the quadratic equation.

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Before even looking at this problem,

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let's get an intuition.

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It's negative b plus or minus you the square root of b

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squared minus 4ac, all of that over 2a.

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My question is to you, when does this not make any sense?

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Well you know, this'll work for any b, any 2a.

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But when does the square root sign really fall apart, at

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least when we're dealing with real numbers,

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and that's a clue?

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Well, it's when you have a negative number under here.

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If you end up with a negative number under the square root

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sign, at least if we haven't learned imaginary numbers yet,

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you don't know what to do.

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There's no real solution to the quadratic equation.

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So if b squared minus 4ac is less than

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0, you're in trouble.

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There's no real solution.

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You can't take a square root of a negative sign if you're

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doing with real numbers.

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So that's probably going to be the problem here.

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So let's see what b squared minus 4ac is.

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You have b is 1.

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So 1 minus 4 times a.

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a is 2.

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2 times c is 7.

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And sure enough, 1 times 4 times 2 times 7 is going to be

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less than 0.

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So let's just see what they have here.

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Right, the value of 1 squared-- oh, right.

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It's b squared.

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Well 1 squared, same thing as 1.

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1 squared minus 4 times 2 times 7,

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sure enough is negative.

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So that's why we don't have a real

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solution to this equation.

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Next problem.

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I'm actually out of space.

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OK, they want to know the solution set to

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this quadratic equation.

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I'll just copy and paste.

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So that's essentially the set of the x's that

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satisfy this equation.

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And obviously, for any x that you put in this, the left-hand

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side is going to be equal to 0.

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So what x's are valid?

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And they just want us to apply the quadratic equation.

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So we've written it a couple of times, but let's just do it

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straight up.

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So it's negative b.

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b is 2.

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So it's negative 2 plus or minus the

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square root of b squared.

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Well that's 2 squared.

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Minus 4 times a.

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a is 8.

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Times c, which is 1.

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All of that over 2 times a.

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So 2 times 8, which is equal to minus 2 plus or minus the

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square root of 4-- let's see.

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Did I write this down?

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

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4 times a times c.

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Right.

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So you get 4 minus 32.

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That's why I was double checking to see if I did this

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right because I'm going to get a negative number here.

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All of that over 16.

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And so we're going to end up with the same conundrum we had

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in the last. 4 minus 32, we're going to end with minus 2 plus

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or minus the square root of minus 28 over 16.

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And if we're dealing with real numbers, I mean there's no

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real solution here.

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And at first I was worried.

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I thought I made a careless mistake or there was an error

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in the problem.

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But then I look at the choices.

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They have choice D.

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And I'll copy and paste choice D here.

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Choice D.

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No real solution.

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So that's the answer, because you can't take a square root

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of a negative number and stay in the set of real numbers.

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Let's see, do I have time for another one?

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I'm over the 10 minutes.

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I'll wait for the next video.

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See

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