What is the value of X?

Question

(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.

Bunuel · Answer

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.

Answer: C.

Hope it's clear.

stevennu · Answer

What is the rule regarding the handling of -X. When I square both sides, should the equation read (-X)^2 or -(X)^2?

Is squaring both sides the correct way to approach this problem or should I be looking at it as an absolute value equation?

Kconfused · Answer

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?

Thanks in advance
KC

EMPOWERgmatRichC · Answer

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,
Rich

susheelh · Answer

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!

Bunuel · Answer

Done. Thank you for the suggestion.

susheelh · Answer

Thank you  Bunuel !

shashankism · Answer

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 Answer C

oryahalom · Answer

Hi  Bunuel ,

Can you explain please your analysis to statement 2?

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.

Thanks in advance!

Bunuel · Answer

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.

oryahalom · Answer

Bunuel Thanks! Your explanations are the best

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?

JS1290 · Answer

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!

susheelh · Answer

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.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the value of X?

Prep Toolkit

Top 5 GMAT Debrief Videos

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

What is the value of X?

Prep Toolkit

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

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards