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## Algebra (all content)

# Using deductive reasoning

Sal uses deductive reasoning to prove an algebraic identity. Created by Sal Khan and Monterey Institute for Technology and Education.

## Video transcript

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.