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### Course: Digital SAT Math > Unit 4

Lesson 1: Factoring quadratic and polynomial expressions: foundations# Structure in expressions — Harder example

Watch Sal work through a harder Structure in expressions problem.

## Want to join the conversation?

- What is Sal talking about(81 votes)
- I'm going to presume you know your multiplication table. If you don't... good luck I guess? Also you need to memorise the formulae of factorisation, good luck with that

Anyways the first step is substitution. We know that a is x^2 + y^2 and b is xy so the given equation becomes 9(x^2 + y^2) - 18 xy, or expanded and rearranged in descenting order of x's exponentials, 9x^2 - 18xy + 9y^2

Now if you remember the formula for expansion/factorisation, you'll be able to see that this expression you just calculated looks like a^2 - 2ab + b^2 = (a-b)^2. Now 3*3 = 9 and 3*3*2 = 18. So the answer is (3x-3y)^2

Did I already say that you need to learn the multiplication table by heart? It's really,**really**important!(2 votes)

- ok so we just forget about the middle number(25 votes)
- we didnt really forget about it. we used it to determine that we were dealing with a 3x 3y perfect square. when you multiply it out its the same(11 votes)

- Isn't -1 squared, +1. In the video he changed the form of the original equation to the new one with -1 squared; even though we need -1 as the result. So is this appropriate?(11 votes)
- Short Version \/

To address your first statement, (-1) squared would indeed equal 1. However, you are confusing (-1) squared with -(1) squared.

Long winded, rambling version \/

This confusion is a result of a devious property of exponents, as when you look at, say -2^2; you would think that the answer would be 4. However, -2 can be factored in a not so obvious manner into -1*2. Then, the exponent is applied to the positive two, leading to 4, which is then multiplied by negative one to become negative 4. The reason the exponent is applied before the multiplication of the -1 is because of the order of operations, parenthesis, exponents, multiplication/division and finally addition/subtraction. As exponents are before multiplication, and since -2^2 is also -1*2^2, thus the answer of the example is -4. If, however, parenthesis are present around the -2, then you will get 4.

Don't worry about this confusion too much, as it has happened to me as well.(22 votes)

- hey i am getting confused what is the difference between 3rd choice and the 4th one? it seems both of them can work(9 votes)
- The third one has an a squared rather than an a(3 votes)

- can someone explain why he ignored the -2(3x)(3y)?(9 votes)
- the first one seems to work tho(10 votes)
- At0:12Sal says that it a difference of squares, which rules out option number one.(0 votes)

- bro just yAPPED

'(6 votes) - wait how is 9x^2 equal to (3x)^2 ? I’m so confused 😐(3 votes)
- When we have exponents, we have to distribute them to each factor that we're dealing with. This is similar to when you are multiplying things, you distribute to every term ( 2*(3+x) = 2*3 + 2*x ). If you were "factoring" out an exponent of 2 from 9x^2, you would have to take it out of every term:

9x^2 = ( sqrt(9) * sqrt(x^2) )^2

= (3 * x)^2(5 votes)

- I wish he gave another example to do on our own because right now I am not sure if I got it or not😭😭(5 votes)
- At0:57, how is the 1 squared? It's not even a difference of 2 squares!(2 votes)
- 1 is squared because no matter how many times you multiply 1 times itself, it will always equal 1. It's a very unique number.(7 votes)

## 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.