MathRevolution wrote:

\(x^2 + \frac{1}{x^2}\) = ?

1) \(x - \frac{1}{x} = 4\)

2) \(x + \frac{1}{x} = 2√5\)

This question appears to be very simple if you know the properties of foiling; however, this is a question designed to bait the test taker into picking the classic "C" both are sufficient trap. While it knowing (x-y) and (x+y) would be sufficient to solve the question, it is actually not necessary and a skill the GMAT is testing you on... that the GMAT is just testing high school math and algebra is a myth because this clearly this question shows how arithmetic and manual become derelict when calculators are used unnecessarily - well in Singapore it really is but that's another story. Anyways,

Statement 1\(x - \frac{1}{x} = 4\) square both sides

\(x - \frac{1}{x} [m]x - \frac{1}{x} = 16

x^2- x(1/x)-x(1/x) + (1/x^2) = 16

x^2-2x(1/x) + (1/x^2) =16 ( pay attention to the reciprocal property)

x^2- 2 +(1/x^2) = 16

x^2 + (1/x^2) = 18

Statement 2[m]x + \frac{1}{x} = 2√5\)

[m]x + \frac{1}{x} [m]x + \frac{1}{x} = 20

x^2 + x(1/x) + x(1/x) + 1/x^2 = 20

x^2 + 2x (1/x) + (1/x^2) = 20 (pay close attention to the reciprocal property- x of any number times its reciprocal is 1 so 2x times the reciprocal of just x is always 2)

x^2 + 2 + (1/x^2) = 20

x^2 + (1/x^2) = 18

Lastly- it is important to note that even though statement 1 and 2 both reduced to a sum of 18- they don't necessarily have to have the same sum. For example, if statement 1 allows you to find x^2 + (1/x^2) and statement 2 also allows you to find x^2 + (1/x^2) but happens to have a different result, say 16 instead of 18, it would still be D.

Thus "D" is the correct answer