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Structure in expressions — Harder example

Watch Sal work through a harder Structure in expressions problem.

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Video transcript

- - [Narrator] We're asked if X squared plus Y squared is equal to A and X Y is equal to B, which of the following is equivalent to nine A minus 18 B. Pause this video and see if you can figure that out. All right, now let's work through this together. So we have nine A minus 18 B but we know that A is equal to X squared plus Y squared. And we know that B is equal to X Y. So we can rewrite this as nine times X squared plus Y squared minus 18 times X Y. X Y. And let's see, it looks like all of the choices We are squaring something, So this is going to be some type of a perfect square. And so let me distribute the nine out. So I'm going to get nine X squared plus nine Y squared. And then I'll write all of this like this minus 18 X Y. And let me put this in a form that at least my head likes to process when I'm factoring quadratics, is... Let me write the X term, the second degree X term first then I'll write the X Y term: minus 18 X Y, and then finally we have plus nine Y squared. Now let's see if we can see any patterns here, especially patterns that we associate with perfect square quadratics. So we can see that this over here, this is the same thing as three X squared. It actually could be plus or minus three X squared. The one that we see on the right. Let me do this in another color. This is plus or minus three Y squared, and let's see over here, three X times three Y would be nine X Y. This is negative two times that. Let me write that that way. So this is equal to negative two times three X times three Y. So it looks like we have a perfect square that's dealing with three X and three Y and it looks like we would have to subtract one of them. So we could write this as... if I were to factor it I could write this as three X minus three Y. One of them has to be negative. So minus three Y and then all of that squared or it could be the other way around. It could be three Y minus three X squared. So let's look at the choices and it looks like choice B is, for sure, exactly what we wrote down. Now, they could have had three Y minus three X squared and that actually would have been a legitimate choice as well. And if you don't believe me you can actually multiply this out and it will be equal to nine X squared minus 18 X Y plus nine Y squared.