Lesson: FOIL Method for Expanding

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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

Shouldn't the answer be E???
gmat-admin's picture

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

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?
gmat-admin's picture

"...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,

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
gmat-admin's picture

Question link: https://gmatclub.com/forum/for-integers-a-and-x-which-of-the-following-v...

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

Check the answer choices....answer = A

Cheers,
Brent

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