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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 ?  [#permalink]

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New post 21 May 2015, 04:15
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What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post Updated on: 25 May 2015, 09:53
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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:

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Originally posted by EMPOWERgmatRichC on 22 May 2015, 10:54.
Last edited by EMPOWERgmatRichC on 25 May 2015, 09:53, edited 1 time in total.
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 21 May 2015, 05:55
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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
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 21 May 2015, 05:27
Bunuel wrote:
What is the range of all the roots of |x^2 - 2| = x ?

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



In short, we will be solving for 2 quadratics. The two are:

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

The roots for quadratic 1 are {-2,1} and the roots for quadratic 2 are {2,-1}.

Range is highest - lowest, i.e. 2-(-2) =4

answer is A... or so I hope
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 21 May 2015, 05:57
adityadon wrote:
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

Yeah you're right. Totally forgot to go back to the stem. -1 and -2 don't satisfy.
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 21 May 2015, 10:56
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Attached image.
there are only 2 roots.
We can solve it algebrically as well.

Answer: C
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 22 May 2015, 08:56
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adityadon wrote:
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


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.
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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New post 25 May 2015, 06:29
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Bunuel wrote:
What is the range of all the roots of |x^2 - 2| = x ?

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


OFFICIAL SOLUTION:

First of all notice that since x is equal to an absolute value of some number (|x^2 - 2|), then x cannot be negative.

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

First equation gives x = -1 or x = 2. Since x cannot be negative, we are left with only x = 2.
Second equation gives x = -2 or x = 1. Again, since x cannot be negative, we are left with only x = 1.

The range = {largest} - {smallest} = 2 - 1 = 1.

Answer: D.
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Re: What is the range of all the roots of |x^2 - 2| = x ?  [#permalink]

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Re: What is the range of all the roots of |x^2 - 2| = x ?   [#permalink] 30 Jan 2019, 22:23
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