# Lesson: FOIL Method for Expanding

## Comment on FOIL Method for Expanding

### For the last ques from Math

For the last ques from Math Rev...

Is x^2 > y^2?

(1) x + y = 2
(2) x > y

How is the OA C? my explanation:

-Assume 1 and 2 are not suff on their own
-1+2)
x= 2, y= 0 (satisfying 1 and 2) --> Yes, 4 > 0
x= 2, y= -4 (satisfying 1 and 2) --> No, 4 is not greater than 16 ### x= 2, y= -4 does not satisfy

x= 2, y= -4 does not satisfy statement 1

### Brent, since statement 2

Brent, since statement 2 clearly states that, X>Y, it should be true for x^2>y^2 or even x^3>y^3, Right? Going by this logic, clearly, Statement two alone is sufficient? ### "...since statement 2 clearly

"...since statement 2 clearly states that x > y, it should be true for x^2 > y^2..."

That's not true. Here's a counter-example:

x = -3 and y = 1
Here, x < y, but it is NOT the case that x² < y².
If x = -3, then x² = (-3)² = 9
If y = 1, then y² = (1)² = 1
So, x² is greater than y²

Cheers,
Brent

### Hi Brent,

Hi Brent,

Im having trouble understanding the base concept behind this question, do you have another method of explanation other than the one helpfully provided in the comments.

https://gmatclub.com/forum/for-integers-a-and-x-which-of-the-following-values-of-a-guarantees-229421.html We're told that 4x² + ax + 16 is a perfect square.
Since this expression (4x² + ax + 16) resembles a quadratic expression, we should recall the two special products (which happen to be perfect squares):

x² + 2xy + y² = (x + y)²
x² - 2xy + y² = (x - y)²

So, as we can see, x² + 2xy + y² and x² - 2xy + y² are both perfect squares, because we can rewrite both of them as (x + y)² and (x - y)²

So, 4x² + ax + 16 = (something + something else)²
OR
4x² + ax + 16 = (something - something else)²

Let's examine each case

CASE A: 4x² + ax + 16 = (something + something else)²
Notice that 4x² = (2x)² and 16 = 4²
So, we can write: 4x² + ax + 16 = (2x + 4)²
If we EXPAND and simplify the right side, we get: 4x² + ax + 16 = 4x² + 16x + 16
So, in this case, a = 16

CASE B: 4x² + ax + 16 = (something - something else)²
Notice that 4x² = (2x)² and 16 = 4²
So, we can write: 4x² + ax + 16 = (2x - 4)²
If we EXPAND and simplify the right side, we get: 4x² + ax + 16 = 4x² - 16x + 16
So, in this case, a = -16

So, if a = 16, then 4x² + ax + 16 is a perfect square
And, if a = -16, then 4x² + ax + 16 is a perfect square

So, there are TWO values that make 4x² + ax + 16 a perfect square

Cheers,
Brent

### https://gmatclub.com/forum

https://gmatclub.com/forum/m19-184198.html
i did this question like this a = 0
then b = 4
simplifying 2x^2 -8x
now by using options x=-1,0,2,3,5 at x=2 expression is smallest hence c
sir in this type of questions can we assume value? For this question, the best option is to test values, as you have done.

Cheers,
Brent

Is x² > y²?
(1) x + y = 2
(2) x > y

I first simplified the target question by taking roots both sides, so is x>y?

(1) x + y = 2, this condition can be satisfied when,
x=1,y=1.....x=y
x=0,y=2.....x<y
x=3,y=-1.....x>y
we cant determine if x>y...Not sufficient

(2) x > y....this is what we need, so sufficient, answer is B

I don't understand why the above approach in the forum is complicated and the answer comes to C??!! ### Hi Gerrard,

Hi Gerrard,

The issue with your solution lies at the very beginning, when you rephrased the target question as "Is x > y?"

Asking "Is x> y?" is not the same as asking "Is x² > y²?".

For example, if x = 1 and y = -2, then it is true that x > y?
So, the answer to your REPHRASED target question is "YES, x is greater than y"

However, if x = 1 and y = -2, then it is NOT true that x² > y²?
So, the answer to the ORIGINAL target question is "NO, x² is NOT greater than y²"

In general, we cannot say that √(x²) = x
For example, if x = -3, we get: √(-3²) = -3
Simplify to get: √9 = -3 (not true)

Does that help?

Cheers,
Brent

### Understood.Thats great Brent!

Understood.Thats great Brent!.never thought you could again prove me wrong here! Thanks.

### Hi Brent.

Hi Brent.

On this question:
https://gmatclub.com/forum/if-x-2-1-x-2-4-what-is-the-value-of-x-4-1-x-175113.html

"(x² + 1/x²)² = 4²
So, (x² + 1/x²)(x² + 1/x²) = 16
Expand to get: x⁴ + 1 + 1 + 1/x⁴ = 16
Simplify: x⁴ + 1/x⁴ = 14"

How would I know to subtract the two ones in x⁴ + 1 + 1 + 1/x⁴ = 16 from the 16? How did you know to bring the two ones over but to leave the x⁴ + 1/x⁴ ? Why not bring the 1 over from 1/x⁴? It's important to remember that our goal is to determine the value of x⁴ + 1/x⁴

So, once we have x⁴ + 1 + 1 + 1/x⁴ = 16, we should recognize that, except for the two 1's, the left side of the equation has the x⁴ + 1/x⁴ we need.

So, once we subtract 2 from both sides, we have x⁴ + 1/x⁴, which is what we're trying to determine the value of.

Does that help?

Cheers,
Brent

### Hi Brent,

Hi Brent,

Is x² > y²?

(1) x + y = 2
(2) x > y

Simplify the question to x² - y² > 0 or (x+y)(x-y) > 0

Stmt 1. 2(x-y)>0 or divide both sides by 2 to get x-y>0 or x>y, which is not sufficient.

Stmt 2. x>y, not sufficient.

Together, the statements provide the same info so E.

Where have I gone wrong?
Thanks ### You did a great job

You did a great job rephrasing the target question to get: "Is (x+y)(x-y) > 0?"

However, statement 1 (x + y = 2) does not translate into 2(x-y) > 0

Here's my full solution: https://www.beatthegmat.com/is-x-2-y-2-t296831.html