M03-18

Question

How many distinct roots does the equation  x^4 - 2x^2 + 1=0  have?

A. 0
B. 1
C. 2
D. 3
E. 4

Bunuel · Accepted Answer

Official Solution:

How many distinct roots does the equation  x^4 - 2x^2 + 1=0  have?

A. 0
B. 1
C. 2
D. 3
E. 4

We can apply the identity  a^2 - 2ab + b^2 = (a - b)^2  to the equation:
 
 x^4 - 2x^2 + 1 = 0 ;
 
 (x^2-1)^2=0 ;
 
 x^2-1 = 0 ;
 
 x^2=1 .
 
Thus, the roots are  x=1  and  x=-1 . Therefore, the given equation has two distinct roots.

Answer: C

Bunuel · Answer

7. Algebra 
     Theory   
 Math : Algebra 101 
 Algebra: Tips and hints 
 FOIL on the GMAT: Simplifying and Expanding 
 Factoring Quadratics 
 Solving Quadratic Equations  
 Using Algebra vs. Logic on GMAT Quant Questions 
     Questions   
 Tough Algebra Set 
 Algebra Ability Quiz by e-GMAT 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

Prashant10692 · Answer

Bunuel,

I rejected -1 because (x2−1)2=0(x2−1)2=0.

Could you please provide few links to get a firm grip on such questions.

I have seen one solution before in which one value was eliminated because it was raised to the power of 4.

willacethis · Answer

Bunuel  in this question, I took x = (x^2) and hence calculated the roots for (x^2)^2-2(x^2)+1 by the formula root(B^2-4ac) which turned out to be 0 and hence, I marked choice B. Am I missing something?

Bunuel · Answer

(x^2)^2-2(x^2)+1 =0

x^2=\frac{2+\sqrt{4-4}}{2}  -->  x^2=1  -->  x=1  or  x = -1 ;

x^2=\frac{2-\sqrt{4-4}}{2}  -->  x^2=1  -->  x=1  or  x = -1 .

JIAA · Answer

Hi  Bunuel ,

I was wondering how did you find the roots as illustrated above?? 
Is this some kind of formula??

Would appreciate your help!
THANKS

Bunuel · Answer

Solving and Factoring Quadratics: https://www.purplemath.com/modules/solvquad.htm https://www.purplemath.com/modules/factquad.htm Hope it helps.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

bhavik29 · Answer

What am I doing wrong?

x^4 - 2x^2 + 1 = 0
x^2(x^2-2) + 1 = 0
x^2-2=0 or x^2 + 1 = 0
x^2= 2 or x^2 = -1 
x= square root 2

Bunuel · Answer

Nothin there is correct. I think you'd benefit if you brush up algebra fundamentals.

7. Algebra 
     Theory   
 Math : Algebra 101 
 Algebra: Tips and hints 
 FOIL on the GMAT: Simplifying and Expanding 
 Factoring Quadratics 
 Solving Quadratic Equations  
 Using Algebra vs. Logic on GMAT Quant Questions 
     Questions   
 Tough Algebra Set 
 Algebra Ability Quiz by e-GMAT 
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

bhavik29 · Answer

x^4 - 2x^2 + 1 = 0

let u = x^2

u^2 - 2u + 1 
(u-1)(u-1)
Sub in x^2 in u 
(x^2-1)(x^2-1)
(x-1)(x+1)(x-1)(x+1)
so solutions are x=1 and x=-1

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