Algebra (all content)
Sal uses deductive reasoning to prove an algebraic identity. Created by Sal Khan and Monterey Institute for Technology and Education.
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.