Deductive and inductive reasoning
Deductive Reasoning 3 Deductive Reasoning 3
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- Use deductive reasoning and the distributive property to
- justify x plus y squared is equal to x squared plus 2xy
- plus y squared.
- Provide the reasoning for each step.
- Now when they say use deductive reasoning and all
- this stuff, it might seem like something daunting and new,
- but this is no different than what we've done in the past.
- In fact, we've done this very exact problem.
- So let's just do it step by step, show our logic, and
- that's essentially deductive reasoning.
- We're starting with a statement and we're going to
- deduce that this is going to be equal to something else.
- So let's start with x plus y squared.
- We know an exponent means to multiply something by itself
- that many times.
- So we know that this is the same thing, or we can deduce
- that it is the same thing, as x plus y times-- and I'll do
- this next x plus y in a different color--
- times x plus y.
- That's what x plus y squared is.
- Now they say to use the distributive property.
- And this is a bit of review here.
- And we've seen it many, many, many times before.
- The distributive property just tells us that if we have a
- times b plus c-- I want to do as many colors as possible--
- that this is equal to a times each of these terms. a times b
- plus a times c.
- It's called distributive property cause you're
- distributing the a in all of the terms in the expression
- that your multiplying a by.
- Now we can do the exact same thing here.
- Instead of an a, you could imagine this is an x plus y.
- And we can take this entire x plus y and we can distribute
- it on to both of the terms on this expression that it is
- multiplying.
- If this was an a, it'd be ax plus ay.
- Now that it's an x plus y, we multiply the x plus y times
- each of those terms. And that's just by the
- distributive property.
- So by the distributive property that's going to be
- equal to-- we'll distribute this on to each of them-- x
- plus y times x.
- And actually I don't even have to write the x after it.
- I could just write it there.
- It doesn't matter whether you multiply x times x plus y, or
- x plus y times x.
- Order doesn't matter.
- So that's that times that.
- And then it's going to be plus y times x plus y.
- And now we can apply the distributive property again.
- We have x being multiplied by x plus y, then we have a y
- being multiplied by x plus y.
- So let's just do that again.
- So then we get this is equal to x times x plus x times y.
- I'm going through great pains to keep the colors consistent.
- Plus y times x.
- Plus y times this x over here.
- Plus y times that y over there.
- I'm doing this a lot slower, and I'm not
- skipping any steps here.
- Now what do each of these things equal? x times x.
- That is the same thing as x squared.
- So this is equal to x squared.
- This right here, xy, we have one xy.
- But then we have yx is also the same thing as xy.
- It doesn't matter what order you multiply it in.
- So xy plus xy is 2xy.
- Plus 2xy.
- And then this last term right here, y times y, that's the
- same thing as y squared.
- So we're done.
- We've used deductive reasoning.
- We've just used logical steps to start with a statement, to
- start with an expression really.
- And we essentially just logically manipulated it.
- We started with.
- I guess you could call this a statement-- I guess that's the
- best thing to call it-- and we logically manipulated to come
- up with another statement, another fact.
- We know that this is equal to this using logical properties
- and distributive property and things like that and
- properties of exponents.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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