What is the value of x? (1) √x^4 = 9 (2) √x^2 = -x

Question

What is the value of x?

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

(2)  \sqrt{x^2} = -x

PKN · Answer

St1:-  \sqrt{x^4} = 9

Squaring both sides,
 (\sqrt{x^4})^2 = 9^2 
Or,  ((x^4(^{\frac{1}{2}}))^2=9^2 
Or,  x^(4*\frac{1}{2}*2)=9^2 
Or,  x^4=81 
Or, x=3 ,-3
Insufficient.

St2:-  \sqrt{x^2} = -x 
Since square root is always positive, hence 'x' in RHS must be less than 0.
Squaring both sides,
  (\sqrt{x^2})^2 = (-x)^2 
Or,  x^2=x^2 
Or,  x\le \:0 
Insufficient.

Combined, We have x=-3
Sufficient.

Ans. (C)

PKN · Answer

Hi  askhere ,
You have to always handle roots operations carefully.
You know , In GMAT even root operation always yield +value.
For example,  \sqrt{4} = +2 NOT -2 since  \sqrt{-2}  is imaginary and GMAT stays in real world. Only real values are accepted.
However,  x^2=4  has two solutions x=+2 or -2.

Now, as per above concepts,  \sqrt{x^2}  is always positive.
But RHS is negative of x. Negative of x will be positive when x<0 (or x is negative).--------------(1)

1) For example if x=-4, then  LHS= \sqrt{x^2}=\sqrt{(-4)^2}=\sqrt{16}= +4 
 RHS=-x=-(-4)=4 
So LHS=RHS when x<0

2) LHS will be equal to RHS when x=0

So, x can never be +ve.

From (1) and (2),  x\leq{0}  (More than one value of x, so insufficient)

Your method:- The moment we simplified the statement, we lost the essence of the statement ,i.e, what GMAT wanted to test us. Simplify when it is necessary or when the original statement has no worth significance. if the original statement conveys various possibilities, then we mustn't simplify.

Hope it helps.

P.S:- Never simplify root operations , first think of the logic behind the root operation, then do the needful.

CounterSniper · Answer

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

\sqrt{x^2}  = |x| = +x , -x
so  \sqrt{x^4}  = +3,-3
insufficient

(2)  \sqrt{x^2} = -x 
X is -ve

insufficient

using both x= -3 
C

sejm · Answer

Hi  PKN  , I still do not understand why statement 2 is insufficient. In your example, why is -2 not a possible option? Because when I square -2 it becomes +4 and sqrt of +4 is = or - 2. Is there more material to understand this concept in detail? I am not clear on it. Thank you!

chipsy · Answer

Bunuel  : please post official explanation or explain it once.

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 means that x is not a positive number, so x is 0 or a negative number. Not sufficient.

(1)+(2) Since from (2) x cannot be positive, then from (1) x can only be -3. Sufficient.

Answer: C.

Kinshook · Answer

Asked: What is the value of x?

(1)  \sqrt{x^4} = 9 
x = {-3,3}
NOT SUFFICIENT

(2)  \sqrt{x^2} = -x 
x = - ve 
NOT SUFFICIENT

(1) + (2) 
(1)  \sqrt{x^4} = 9 
(2)  \sqrt{x^2} = -x 
x =-3
SUFFICIENT

IMO C

MathRevolution · Answer

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 1 variable: Let the original condition in a DS question contain 1 variable. Now, we know that each condition (1) and (2) would usually give us an equation each, however, since we need 1 equation to match the numbers of variables and equations in the original condition, the unequal number of equations and variables should logically give us an answer D.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find value 'x' .

Second and the third step of Variable Approach: From the original condition, we have 1 variable (x).To match the number of variables with the number of equations, we need 1 equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Let’s take a look at each conditions .

Condition(1) tells us that \(\sqrt[2]{x^4}\) = 9.

=> \(x^2\) = 3 and therefore, x = 3, -3

Since the answer is not unique, condition(1) is not sufficient by CMT 2.

Condition(2) tells us that \(\sqrt[2]{x^2}\) = -x.

=> This means x is less than equal to zero.

Since the answer is not unique, condition(2) is not sufficient by CMT 2.

Let's take both conditions together.

From condition(1), x = -3 and 3 and from the condition(2) x should be less than equal to zero. hence, x = -3

Since the answer is unique, both conditions together are sufficient by CMT 2.

So, C is the correct answer.

Answer: C

jabhatta2 · Answer

VeritasKarishma  - Can S2 actually be true ? I know in DS questions, we have to assume S1 and S2 are accurate but S2 does not seem to follow GMAT theory.

If I read the Quarter Wit, Quarter Wisdom blog post below

""" x^2 = 4 has two roots, 2 and -2, while √4 has only one value, 2.""

If that is the case, how can S2 say  \sqrt{x^2} = -x to begin with ?

Please let us know your thoughts if S2 can be true to begin with ?

Link -  https://www.veritasprep.com/blog/2016/05/squares-square-roots-gmat/

KarishmaB · Answer

The same post also tells you that  \sqrt{x^2}= |x|

Now |x| can be equal to -x, right? When x is negative or 0.

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What is the value of x? (1) √x^4 = 9 (2) √x^2 = -x

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