ds - another difficult quadratic

[ccassel]

Hi,

How would you answer this question?

If s=(x+y)^2 and t=(x-y)^2, what is the value of 2^s/2^t?

(1) xy=12
(2) x/y=3

Cheers,

[vineeshp]

Are s and t equal?

Both have the same expression.

[ccassel]

It is corrected now . Thanks for the heads up.

Cheers,

[GMATGuruNY]

Hi,

How would you answer this question?

If s=(x+y)^2 and t=(x-y)^2, what is the value of 2^s/2^t?

(1) xy=12
(2) x/y=3

Cheers,

When the bases are the same and we're dividing, we subtract the exponents:
2^s/2^t = 2^(s-t).

Let's determine the value of s-t:
s = (x+y)² = x² + 2xy + y²
t = (x-y)² = x² - 2xy + y²
s-t = (x² + 2xy + y²) - (x² - 2xy + y²) = 4xy.

Thus, 2^(s-t) = 2^(4xy).

Question rephrased: What is the value of xy?

Statement 1: xy = 12.
Sufficient.

Statement 2: x/y = 3.
No way to determine the value of xy.
Insufficient.

The correct answer is A.

For those not comfortable with the algebra, an easy approach would be to plug in values.

Statement 1: xy = 12.
Let x=4, y=3.
s = (x+y)² = (4+3)² = 49.
t = (x-y)² = (4-3)² = 1.
2^s/ 2^t = 2��/2¹ = 2��.

Let x=6, y=2.
s = (x+y)² = (6+2)² = 64.
t = (x-y)² = (6-2)² = 16.
2^s/ 2^t = 2��/2¹� = 2��.

Since in each case the result is 2��, sufficient.

Statement 2: x/y = 3.
Let x=3, y=1.
s = (x+y)² = (3+1)² = 16.
t = (x-y)² = (3-1)² = 4.
2^s/ 2^t = 2¹�/2� = 2¹².

Let x=6, y=2.
s = (x+y)² = (6+2)² = 64.
t = (x-y)² = (6-2)² = 16.
2^s/ 2^t = 2��/2¹� = 2��.

Since in the first case the result is 2¹² and in the second case the result is 2��, insufficient.

The correct answer is A.

[ccassel]

Great explanation Mitch.

Thanks!

[nafiul9090]

s=(x+y)^2
t=(x-y)^2

2^s-t
=s-t
=(x+y)^2-(x-y)^2
=(x+y+x-y)(x+Y-X+y)
=2x*2y
=4xy

2^4xy


so statement 1 is sufficient