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We need to factor 25x to the fourth minus 30x

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squared plus 9.

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And this looks really daunting because we have something to

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the fourth power here.

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And then the middle term is to the second power.

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But there's something about this that

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might pop out at you.

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And the thing that pops out at me at least is that 25 is a

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perfect square, x to the fourth is a perfect square, so

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25x to the fourth is a perfect square.

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And 9 is also perfect square, so maybe this is the square of

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some binomial.

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And to confirm it, this center term has to be two times the

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product of the terms that you're squaring on either end.

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Let me explain that a little bit better

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So, 25x to the fourth, that is the same thing as 5x squared

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squared, right?

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So it's a perfect square.

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9 is the exact same thing as, well, it could be plus or

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minus 3 squared.

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It could be either one.

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Now, what is 30x squared?

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What happens if we take 5 times plus or minus 3?

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So remember, this needs to be two times the product of

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what's inside the square, or the square root of this and

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the square root of that.

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Given that there's a negative sign here and 5 is positive,

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we want to take the negative 3, right?

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That's the only way we're going to get a negative over

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there, so let's just try it with negative 3.

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So what is what is 2 times 5x squared times negative 3?

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What is this?

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Well, 2 times 5x squared is 10x squared times negative 3.

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It is equal to negative 30x squared.

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We know that this is a perfect square.

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So we can just rewrite this as this is equal to 5x squared--

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let me do it in the same color.

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5x squared minus 3 times 5x squared minus 3.

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And we saw in the last video why this works.

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And if you want to verify it for yourself,

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multiply this out.

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You will get 25x to the fourth minus 30x squared plus 9.

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