Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 14 (Transcript)

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Now let me show you ...

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So what if I do ...

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... a x squared plus b x plus c.

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Is that scared?

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Can we complete square there?

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Oh, yes, we can. We factor a, right?

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We get x squared plus b over a x plus c over a.

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And then I have to take ...

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... this part inside.

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And I have to look at the middle term.

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And I have to divide that number by two.

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And that will make a square for me.

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x plus b over two a ...

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... squared ...

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... is what the complete square should be.

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Well making that complete square ...

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... I add it ...

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... a square of this number ...

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... that I have to subtract immediately.

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So that it doesn't change the total value.

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b over two a squared.

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So this difference is equal to these two terms together.

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And I still have that extra c over a.

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All right.

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And then what? And then I would ...

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Well I'm done with completing the square.

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So let me try to simplify a little.

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It is a times x plus b over two a squared ...

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If I multiply by a it's going to be ...

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... minus b squared over ...

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... four ...

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... a ...

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... plus c.

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And I'd also like to bring that c ...

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... with this term to the common denominator.

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So c ...

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I have just plus c.

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So it will come to four a times c.

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Is that right?

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No. That's right.

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So it just may be common denominator for all the remaining terms.

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Isn't it reminding you of something?

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Minus b squared ...

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So er ...

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Isn't it reminding you of the quadratic equation?

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Right? If you want to solve that ...

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This is what I want to solve. And this is what you want to solve.

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And that is what you want to solve. Can you solve this?

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Well you have your ... You have to solve for this x. It appears only once.

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So all you have to do is you have to undo all the remaining stuff.

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So ...

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Let me do that.

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a times x plus b over two a squared.

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I will take this to the right.

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So that should equal to ...

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... b squared minus four a c divided by four a.

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And then of course I have to divide by a.

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We divide by a ...

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... making this four a squared.

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And then I will have to take a square root on both sides.

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Can I always take the square root?

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Of any number?

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What if that number is negative?

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That's a problem, right?

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So what can possibly make this number negative?

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Can four a squared make it negative?

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Never, right?

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So it is only that number.

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If b squared minus four a c is less than zero ...

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... then what?

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Then I cannot take a square root.

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And then I cannot solve the equation.

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Is that of some importance?

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From what you learned in the school, right?

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So this is how you decide whether there are roots or not.

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And if this number is positive then what?

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Well then you solve fot x by taking the square root.

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So x equals negative that number.

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b over two a.

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And then of course we will take a square root. It should be plus-minus ...

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Because there are two possible way of taking the square root.

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b squared minus four a c over ...

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... four a squared.

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And you notice that the square root of four a squared if two a.

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So the whole thing is common denominator two a ...

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... and the numerator is minus b plus-minus square root of that ...

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... our favorite number b squared minus four a c.

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Now my point is in doing this ...

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... is ... Well, remember I did the Product Rule for some reason, right?

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I did this for the reason of showing you what is essential in the whole thing.

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Because probably some of you may thought that the most essential thing about it is the formula.

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And you memorized that formula and probably forgot it already several times.

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And no. The formula is not the most important thing.

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So what is it that makes the whole thing work?

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What is the most essential thing?

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Student: Completing the square.
NB: Completing the square.

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One step that makes the whole thing work.

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So that's why that operation of completing square is so important.

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And unfortunately ...

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... it is probably not emphasized in school ...

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... that solving quadratic equation amounts to learning how to complete a square.

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Well at some point you have to learn it.

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So you've just learned it today.

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Congratulations.

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