What is the range of all the roots of |x^2 - 2| = x ?

Question

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

Bunuel · Accepted Answer

Official Solution:

What is the range of all the roots of  |x^2 - 2| = x ?

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

First, note that since  x  is equal to the absolute value of some number ( |x^2 - 2| ),  x  cannot be negative.

Next,  |x^2 - 2| = x  implies that either  x^2 - 2 = x  or  -(x^2 - 2) = x .

The first equation,  x^2 - 2 = x , yields  x = -1  or  x = 2 . Since  x  cannot be negative, we only consider  x = 2 .

The second equation,  -(x^2 - 2) = x , yields  x = -2  or  x = 1 . Again, since  x  cannot be negative, we only consider  x = 1 .

The range is the difference between the largest and smallest values: range = 2 - 1 = 1.

Answer: D

adityadon · Answer

we get 2 quadratic equations here .. 
1) x^2-x-2=0 ....... roots 2 , -1
2) x^2+x-2=0 ........ roots -2, 1

Inserting each root in given equation , it can be seen that -1 and -2 do not satisfy the equations . 
So value of x for given equation .... x=2 or x=1
I guess range is 2-1 =1

Lucky2783 · Answer

Attached image.
there are only 2 roots. 
We can solve it algebrically as well.

Answer: C

Yogita25 · Answer

I guess all the four integers, -2,-1,1 & 2 satisfies the equation mod (x^2-2) = X. Please let me know if I am missing some point. Though I understood the explanation by graph method, I am not able to understand the way you have explained it.

EMPOWERgmatRichC · Answer

Hi All,

You have to be VERY careful with this question. It's actually built more to test your attention-to-detail than your "math skills"....

We're asked to find the RANGE of the roots of the following equation: |x^2 - 2| = x ?

Before you jump in and start doing calculations, there are a couple of points to note:

1) Absolute value calculations can NEVER equal a negative number. Here, we have an EQUATION set equal to X. As such, X CANNOT be NEGATIVE.
2) The answer choices are small integers, so the roots of the above equation are likely also integers that are relatively close to one another.
3) Since the question mentions ROOTS, there should be at least 2 solutions.

A bit of basic "brute force" is all that's really needed to find the roots of the equation....

Could X = 0?
|0-2| is NOT 0, so X cannot be 0

Could X = 1?
|1-2| does = 1, so X = 1 is a root

Could X = 2?
|4-2| does = 2, so X = 2 is a root

Could X = 3?
|9-2| is NOT 3, so X cannot be 3

As X gets bigger, the absolute value calculation gets even larger (and farther 'away') from X.

Thus, the only roots are 1 and 2. The range is 2-1 = 1

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

GMATGuruNY · Answer

Since the absolute value on the left side cannot be equal to a negative value, the right side of the equation must be NONNEGATIVE.
Thus, only nonnegative values for x are viable here.

Case 1: x²-2 = x
x² - x - 2 = 0
(x-2)(x+1) = 0.
x=2 or x=-1.
Since x must be nonnegative, only x=2 is viable.

Case 2: x²-2 = -x
x² + x - 2 = 0
(x+2)(x-1) = 0.
x=-2 or x=1.
Since x must be nonnegative, only x=1 is viable.

The range of the two roots = greater root - smaller root = 2-1 = 1.

D

pallavichsk · Answer

Thank you for your response.

But my question is, as they can't be non-negative, after finding the roots, I considered that x^2-2 has to be >=0 from which x^2 >= 2.

The roots that satisfy this condition are 2 and -2.

The roots 1 and -1 does not satisfy this. How to decide which one to consider?

Bunuel · Answer

|x^2 - 2| cannot be negative but x^2 - 2 (expression in the modulus) can be. For example, if x = 1, then x^2 - 2 = -1 but | x^2 - 2| = |-1| = 1.

KarishmaB · Answer

I would square both sides to get rid of the absolute values and then put the roots back to check which ones satisfy the original equation.
 
 |x^2 - 2|^2 = x^2

x^4 -5x^2 + 4 = 0

(x^2 - 4) (x^2 - 1) = 0

x = 2, -2, 1, -1

Only 1 and 2 satisfy so range = 2 - 1 = 1

Think about why it is critical in this case to put back the roots in original equation to check. Why do we get extraneous roots?

thekingslay · Answer

Solved this within a minute - 45 seconds without lifting a pen.

The mod is equal to +x. means the range lies in the positive side. 
With this, open the mod with no sign change. X^2-2=x

Now, simply solve the quadratic equation - x^2-2-x=0, i,e. x-2,x-1. Means either X=1, or X=2. 
Range of 2-1=1. So D is correct.

Rishm · Answer

Open the mod, x^2-x-2=0 bring x to the left hand side. Now You get range between -1 and 2, now substitute them and u get 1&2 as only option and the range would be 1 for them. 0 doesnt work and neither does -1.

Kinshook · Answer

What is the range of all the roots of |x^2 - 2| = x ?

|xˆ2 - 2| = x

Case 1: xˆ2 - 2 >=0; xˆ2 >=2
xˆ2 - 2 = x
xˆ2 - x - 2 = 0
(x-2)(x+1) = 0
x = 2; Feasible since xˆ2 = 4 >=2
x = -1; Not feasible since xˆ2 = 1<2

Case 2: xˆ2 - 2<0; xˆ2 < 2
2 - xˆ2 = x
xˆ2 + x - 2 = 0
(x+2)(x-1) = 0
x = 1; xˆ2 = 1 < 2; feasible
x = -2; x^2= 4 > 2; Not feasible

x = 2; x = 1

Range of all roots of x = |2-1| = 1

IMO D

What is the range of all the roots of |x^2 - 2| = x ?

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What is the range of all the roots of |x^2 - 2| = x ?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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