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# What is the value of X?

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Re: What is the value of X? [#permalink]
Bunuel wrote:
What is the value of X?

(1) $$\sqrt{x^4} = 9$$ --> $$x^2=9$$ --> $$x=-3$$ or $$x=3$$. Not sufficient.
(2) $$\sqrt{x^2} = -x$$ --> $$|x|=-x$$. This implies that $$x\leq{0}$$. Not sufficient.

(1)+(2) Since $$x\leq{0}$$ then $$x=-3$$. Sufficient.

Hope it's clear.

Bunuel, can you please expand the explanation for the second statement a little more?
Say x= -2

Sqrt ( (-2)^2) = 2 Yes? It gets a little confusing here.

By definition, sqrt of any number x yields both positive and negative values. Sqrt9 = +or- 3
So are there cases when sqrt yields just a positive value or a negative value.

Can you point me to resources where I can sharpen my understanding on this topic?

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Re: What is the value of X? [#permalink]
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Hi ALL,

The concepts of squares and square-roots appear on the GMAT every time you take it, so you have to be clear on the rules (and there are several that you need to know. I'm going to focus on the ones that apply to this question though.

You have focus on the specific information that a prompt gives you...

If a prompt tells you that X^2 = 4, then X can be 2 or -2

If a prompt tells you that √X = 4, then X can ONLY be 2

When the above math concepts "overlap" though, you have to use both rules:

If a prompt tells you that √(X^2) = 4, then X can be 4 or -4, since PEMDAS rules tell us that X^2 has to be dealt with FIRST and X^2 gives us more than one solution.

GMAT assassins aren't born, they're made,
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Re: What is the value of X? [#permalink]
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Hello Moderators,

Can you please make the question stem math friendly - if possible? Its a god questions with some nice learnings. But, Its a bit confusing when giving a quick look at the stem. Thanks in advance!
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Re: What is the value of X? [#permalink]
susheelh wrote:
Hello Moderators,

Can you please make the question stem math friendly - if possible? Its a god questions with some nice learnings. But, Its a bit confusing when giving a quick look at the stem. Thanks in advance!

Done. Thank you for the suggestion.
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Re: What is the value of X? [#permalink]
Thank you Bunuel!

Bunuel wrote:
susheelh wrote:
Hello Moderators,

Can you please make the question stem math friendly - if possible? Its a god questions with some nice learnings. But, Its a bit confusing when giving a quick look at the stem. Thanks in advance!

Done. Thank you for the suggestion.
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What is the value of X? [#permalink]
stevennu wrote:
What is the value of X?

(1) $$\sqrt{x^4} = 9$$
(2) $$\sqrt{x^2} = -x$$

What I did was first square both sides of the equation given in the first promt. This gives us X^4 = 81. From this we see that X is either equal to 3 or -3.

On the second promt, I again squared both sides of the equation and arrived at X^2 = - (X)^2. I concluded that X can only equal 0 and selected (B) as my answer.

The OA is (C).

My question is, was I correct in squaring both sides of the equation in statement 2? Additionally, what is the rule regarding the handling of -X. When I square both sides, should the equation read (-X)^2 or -(X)^2?

Thanks for the help.

Given thing
DS : Value of X

Option 1 : $$\sqrt{x^4} = 9$$
|x^2| = 9
x^2 = 9
x = +/- 3
NOT SUFFICIENT

Option 2: $$\sqrt{x^2} = -x$$
|x| = -x
So, x<=0
NOT SUFFICIENT..

Combined : x = -3
SUFFICIENT

Originally posted by shashankism on 27 Jun 2017, 04:13.
Last edited by shashankism on 27 Jun 2017, 05:42, edited 1 time in total.
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Re: What is the value of X? [#permalink]
Hi Bunuel,

i thought that the answer should be B because radical of X in a square suppose to be equal to X (after dividing the exponent by the radical - 2/2=1). According to this analysis it comes that X=-X and thus the only option is that x=0.

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Re: What is the value of X? [#permalink]
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oryahalom wrote:
Hi Bunuel,

i thought that the answer should be B because radical of X in a square suppose to be equal to X (after dividing the exponent by the radical - 2/2=1). According to this analysis it comes that X=-X and thus the only option is that x=0.

Notice that 0 as well as negative number satisfy the second statement. For example, test x=0, x=-1, x=-2, ...

The reason is given below.

MUST KNOW: $$\sqrt{x^2}=|x|$$:

The point here is that since square root function cannot give negative result then $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function: $$|x|=x$$, if $$x\geq{0}$$ and $$|x|=-x$$, if $$x<0$$. That is why $$\sqrt{x^2}=|x|$$.

Hope it's clear.
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What is the value of X? [#permalink]
Bunuel

Thanks!

According to your analysis if radical of x^2 or absolute value of x are equal to -x, x can also be equal to 0 and not just negative number as mentioned in your comment. Right?
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Re: What is the value of X? [#permalink]
I have a question regarding S2. Can we square both of the sides? If yes, would it be x^2=x^2 or x^2=-x^2?

Thanks!
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What is the value of X? [#permalink]
Hello csaluja,

Let me attempt to answer this. Do let me know in case of any questions, It will help me clear my understanding as well

Lets look at S2. Of course we can square both sides. But thats a trap laid by the test maker. It wont lead us anywhere. To avoid this trap we will have to approach it using Algebra.

S2 says :

$$\sqrt{x^2} = -x$$. To understand this better lets take a parallel example - square root of a perfect square - $$\sqrt{9}$$ = $$\sqrt{3^2} = ±3$$. Mathematically ±3 is also written as |3|.

Generalizing we get : $$\sqrt{x^2} = |x|$$

Essentially what S2 is saying is that $$\sqrt{x^2}$$ can only take the negative value. I hope this clears your understanding of S2.

Bunuel has (as usual) excellently explained the remaining part above.

csaluja wrote:
I have a question regarding S2. Can we square both of the sides? If yes, would it be x^2=x^2 or x^2=-x^2?

Thanks!
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Re: What is the value of X? [#permalink]
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Re: What is the value of X? [#permalink]
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