GMAT Club Olympics 2024 (Day 9): If n^4 + n^2 = 1, then which of the

Question

If  n^4 + n^2 = 1 , then which of the following must equal to  n^6 ?

A.  n^2 - 1

B.  n^2 + 1

C.  2n^2

D.  2n^2 - 1

E.  2n^2 + 1

Bunuel · Accepted Answer

GMAT Club Official Explanation:

n^6 = n^4*n^2

Since given that  n^4 = 1 - n^2 , then:

n^6 = n^4*n^2 =

= (1 - n^2)n^2 =

= n^2 - n^4 =

= n^2 - (1 - n^2) =

=2n^2 - 1

Answer: D.

jack5397 · Answer

IMO - D

n^4+n^2 = 1 ----- eq.1

multiply both sides by n^2

n^6+n^4 = n^2

Substitute the value of n^4 from eq.1

n^6 + (1-n^2) = n^2

n^6 = 2n^2-1

Option D

Milkyway_28 · Answer

A straightforward way to solve this is to equate  n^6  to the answer choices and see if they equal  n^4 + n^2 = 1

Intuitively, the lower options looked more suitable than the top ones so I started from the bottom and worked my way up.

2n^2 + 1=n^6

n^6 - 2n^2 = 1

n^2(n^4 - 2) = 1

Here I rearranged  n^4 + n^2 = 1  to  n^4 = 1 - n^2 ​​​​​​​ and plugged in  n^4

n^2(1 - n^2 - 2) = 1  
​​​​​​​
 - n^2 - n^4 = 1  or  n^2 + n^4 = - 1  Doesn't work

We try again with the next value

2n^2 - 1=n^6

n^6 - 2n^2 = - 1

n^2(n^4 - 2) = - 1

n^2(1 - n^2 - 2) = - 1

- n^2 - n^4 = - 1  or  n^2 + n^4 = 1  which works

The answer is D

pianogirl · Answer

n^4+n^2=1 => n^6 + n^4 = n^2 => n^6 = n^2 +n^2 - n^2 - n^4 = 2*n^2-1

The right answer is  D

bumpbot · Answer

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