What is the value of x?

1) \(x^2 = 9\). So x = +3 or -3. Insufficient

2) \(\sqrt{x^2} = -x\) x can be 0 or any negative number. Insufficient

1 & 2 together. x is negative and is either +3 or -3. So x is -3.

Answer C

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Squaring is not the solution for every problem. When you square both sides you sometimes lose valuable information. e.g.
x = -5
Square -> x^2 = 25

If you are given x^2 = 25, all you can say is that x is 5 or -5. You cannot say which one. So you lost information here.

As for this question, there is a concept that you need to use here \(\sqrt{x^2}= |x|\)
\(\sqrt{9} = 3\). It is not 3 or -3. Only the positive value is considered for square roots. Hence, the mod is used when dealing with a variable.

So from the second statement, you get |x| = -x
Now, we know that |x| = -x when x is negative. So the only thing that the second statement tells us is that x is negative.
Statement 1 tells you that x is 3 or -3. Statement 2 tells you that x is negative. SO using both statements, you can say that x = -3. Sufficient.
Answer (C)

HI Bunnel,

I am slightly confuse here. Isnt it true that when the GMAT provides the square root sign for an even root, then the only accepted answer is the positive root?

How is A and B different here? If x can be negative according to A then it could be negative according to B as well. Could you please help clarify this rule?

Thanks.

Please check again: where did we get negative result?

whatever value comes after square root ...put a modulus over it..and then you will not get confused....
as you said..square root gives positive value..hence modulus does the same thing..
example:
\sqrt{x^4}=modulus x^2==>since x^2 is always positive(or equal to zero) we can remove mod
hence it becomes==>x^2=9....now no more square root ...hence whatever value will satisfy ...it can be positive or negative.
hence x=+3..or -3========>insufficient.

in option 2
\sqrt{x^2}=-x
remove square root and put a mod
hence
mod x= -x===>this conditions is correct only when X IS NEGATIVE...==>NOT SUFFICIENT

NOW COMBINING WE CAN ANSWER X=3==>SINCE X IS NEGATIVE..HENCE C

I am referring to this statement - (1) \(\sqrt{x^4} = 9\) --> \(x^2=9\) --> \(x=3\) or \(x=-3\). Not sufficient.

As per your explanation in this statement, x could be 3 or -3. However, in the second statement, the explanation says

\sqrt{x^2}= |x|

My question is why in the first statement, \sqrt{x^4} not equal to |x^2|.

Thanks again.

That's because |x^2|=x^2.

Given : variable x
DS : value of x

Statement 1 : \(\sqrt{x^4} = 9\)
x^4 = 81
x = +/-3
putting in the same equation we find both +3,-3 are valid soln.
NOT SUFFICIENT

Statement 2 : \(\sqrt{x^2}=-x\)
|x| = -x
so x<0
NOT SUFFICIENT

Combined : x = -3

Answer C

Hi Experts,

I completely understand the above explanation but can you tell me why i cannot apply the below process,

statement ii.

x^(2x1/2)= x = -x
2x=0
x=0
Hence B.

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