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## SAT (Fall 2023)

### Course: SAT (Fall 2023) > Unit 10

Lesson 2: Passport to advanced mathematics- Solving quadratic equations — Basic example
- Solving quadratic equations — Harder example
- Interpreting nonlinear expressions — Basic example
- Interpreting nonlinear expressions — Harder example
- Quadratic and exponential word problems — Basic example
- Quadratic and exponential word problems — Harder example
- Manipulating quadratic and exponential expressions — Basic example
- Manipulating quadratic and exponential expressions — Harder example
- Radicals and rational exponents — Basic example
- Radicals and rational exponents — Harder example
- Radical and rational equations — Basic example
- Radical and rational equations — Harder example
- Operations with rational expressions — Basic example
- Operations with rational expressions — Harder example
- Operations with polynomials — Basic example
- Operations with polynomials — Harder example
- Polynomial factors and graphs — Basic example
- Polynomial factors and graphs — Harder example
- Nonlinear equation graphs — Basic example
- Nonlinear equation graphs — Harder example
- Linear and quadratic systems — Basic example
- Linear and quadratic systems — Harder example
- Structure in expressions — Basic example
- Structure in expressions — Harder example
- Isolating quantities — Basic example
- Isolating quantities — Harder example
- Function notation — Basic example
- Function notation — Harder example

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# Structure in expressions — Harder example

Watch Sal work through a harder Structure in expressions problem.

## Want to join the conversation?

- 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?(9 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.(14 votes)

- hey i am getting confused what is the difference between 3rd choice and the 4th one? it seems both of them can work(7 votes)
- The third one has an a squared rather than an a(2 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)

- What is Sal talking about(9 votes)
- ok so we just forget about the middle number(5 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(2 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.(6 votes)

- can someone explain why he ignored the -2(3x)(3y)?(4 votes)
- wait how is 9x^2 equal to (3x)^2 ? I’m so confused 😐(2 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(4 votes)

- why not the first choice is seems right ?(1 vote)
- "A" does not work because the coefficient before the parenthesis is a^2 so in reality it is a^2/1. When you multiply it out, you would only multiply the top and not the bottom so the bottom half would still remain as 2x-y.(2 votes)

- Why is the first choice not correct?(3 votes)
- Because (2x-y)^2 is not 4x^2-y^2.

Remember, (a+b)^2 equals a^2 + 2ab + b^2

while (a+b)(a-b) equals a^2 - b^2

Hope this helped!(0 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.