Squares, first question

[Hlabioa7]

If x^2 + 1/x^2 = 4 what is x^4 + 1/x^4

[Matt@VeritasPrep]

Nobody answered this? Come on, guys

x² + 1/x² = 4

(x² + 1/x²)² = 4² = 16

but we also know that (x² + 1/x²)² = x� + 2 + 1/x�

so x� + 2 + 1/x� = 16, and x� + 1/x� = 14

[Brent@GMATPrepNow]

If x² + 1/x² = 4, then x� + 1/x� = ?

A. 10
B. 11
C. 12
D. 14
E. 15

IMPORTANT: I notice that if we SQUARE x², we get x�, and if we SQUARE 1/x², we get 1/x�, so let's see what happens if we take the equation x² + 1/x² = 4 and SQUARE both sides:

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

Answer: D

Related Video:
Using the FOIL method to expand expressions: https://www.gmatprepnow.com/module/gmat ... /video/952

Cheers,
Brent

[Matt@VeritasPrep]

In addition to my algebraic method above, you can also always try to gut out an approximation. (This isn't ideal here, since the answers are packed tightly together, but it's a fallback plan on test day.)

If x² + 1/x² = 4, then we know x should be slightly less than 2. (2² + 1/2² = 4.25, which is close).

By that logic, our answer should be slightly less than 2� + 1/(2�). From there, we should have D or E (... and it's a coin flip).