GMATPrep Problem

[yankee.musk]

If x<0, then Sqrt( -x|x| ) is

(A) -x
(B) -1
(C) 1
(D) x
(A) sqrt(x)

OA [spoiler]A[/spoiler]

[vinay89]

Isnt it D?

[yankee.musk]

Nope. My first response was D, but I was wrong

[vinay89]

k..i got it. During my calculations since x is a negative number i converted the -x to a +x...did not to that to my final result. Do you think that's right?

[indiantiger]

Sqrt( -x|x| )
=> sqrt(-(-x)|-x|) ( as x has negative sign)
=> sqrt(x^2)
=>-x ( x is less then 0)

[uwhusky]

Is this a legitimate GMAT question? I thought GMAT doesn't test questions like sqrt of a negative number.

I guess to approach this question, I would convert sqrt to exponent: (-x|x|)^1/2 and then (-x^2)^(1/2) is equal to -x^(2/2), which is -x^1 = -x.

Due to its ugliness, I don't think it is a GMAT question.

[Testluv]

When we apply the square root function, the result must be positive. So, we can eliminate negative answers. Choice B is clearly negative.

But because x itself is negative, we can also automatically eliminate choice D (square root can't result in negative).

Similarly, because we can't take the square root of a negative number, we can eliminate choice E.

We don't know x's value, so we can eliminate choice C.

The answer must be choice A!

Because x is negative, "-x" must be positive. The "-" sign doesn't mean we were taking the square root of a negative or that the square root operation yielded a negative result. The "-" doesn't denote the sign of "x". It just tells you to multiply x by -1.

If x<0, then square root of (x^2) is (-1)*x or -x.

[this_time_i_will]

If x<0, then square root of (x^2) is (-1)*x or -x.

Can you please show, in detail ,how this works.
i was working on following lines:
for x<0 ,sq rt(x^2)=>sqrt (-1*x*x)=>x*sqrt(-1)...am stuck here..

[uwhusky]

Ya I completely ignored the x < 0...

[selango]

If x<0, then square root of (x^2) is (-1)*x or -x.

Can you please show, in detail ,how this works.
i was working on following lines:
for x<0 ,sq rt(x^2)=>sqrt (-1*x*x)=>x*sqrt(-1)...am stuck here..

Sqrt(x^2)=x if x>0 or -x if x<0

if x<0,sqrt(x^2)=sqrt(-x*-x)=-x

Hope this clarify

[uwhusky]

Think of this way, what is sqrt of (x * x)?

This question is essentially asking the same thing, except that x is a negative number, so -x is a positive number.

Sqrt( -x|x| ) is essentially a positive x multiplies an absolute value of x, which is the same as x * x.

[brijesh]

When we apply the square root function, the result must be positive. So, we can eliminate negative answers. Choice B is clearly negative.

But because x itself is negative, we can also automatically eliminate choice D (square root can't result in negative).

Similarly, because we can't take the square root of a negative number, we can eliminate choice E.

We don't know x's value, so we can eliminate choice C.

The answer must be choice A!

Because x is negative, "-x" must be positive. The "-" sign doesn't mean we were taking the square root of a negative or that the square root operation yielded a negative result. The "-" doesn't denote the sign of "x". It just tells you to multiply x by -1.

If x<0, then square root of (x^2) is (-1)*x or -x.

Hi!

From your answer can we conclude :

sqrt(x2)= IxI (for all real x)

but i have a query here,

for all x<0 or x>0

x2 is always positive (i.e. x2>0)

so sqrt(x2) = x or -x (as a square of +ve or -ve number is always +ve, so sqrt (x2) in its result show both the number, as it )
x2

[uwhusky]

x square will always be a positive number is correct, but we're not squaring x in the question. Read the question carefully, it is Sqrt( -x|x| ), which is -x * |x|.

-x * |x| is not the same as (-x)^2 or x^2.

[Testluv]

If x<0, then square root of (x^2) is (-1)*x or -x.

Can you please show, in detail ,how this works.
i was working on following lines:
for x<0 ,sq rt(x^2)=>sqrt (-1*x*x)=>x*sqrt(-1)...am stuck here..

|x| = sqrt(x^2) (this is how absolute value is defined)

Both the absolute value and the square root functions must be positive. Therefore:

If x>0, then both sqrt(x^2) and |x| are x;

and

if x<0, then both sqrt(x^2) and |x| are -x.

(if x<0, then because absolute value and square root operations must result in positive, |x| and sqrt(x^2) have to be -x; they can't yield x because x is negative)
________________

In this question, confusion may arise from wanting/having to diagram "-x" when x itself is negative. (We can't let "-x" be x since "-x" is positive and x is negative).

x is negative. But suppose there is a number, z. If z is positive and equal in absolute value to x, then z = -x or x = -z.

The question asks for sqrt(-x*|x|) or sqrt[(-1)*(x)*|x|]

Subbing "-z" into "x", the question becomes:

what is sqrt[(-1)*(-z)*|-z|]?

Because z is positive, clearly (-1)*(-z) is just z. Likewise |-z| is just z. Thus, we have:

sqrt(z*z)

which is just z

which is equal to -x.

[brijesh]

If x<0, then square root of (x^2) is (-1)*x or -x.

Can you please show, in detail ,how this works.
i was working on following lines:
for x<0 ,sq rt(x^2)=>sqrt (-1*x*x)=>x*sqrt(-1)...am stuck here..

|x| = sqrt(x^2) (this is how absolute value is defined)

Both the absolute value and the square root functions must be positive. Therefore:

If x>0, then both sqrt(x^2) and |x| are x;

and

if x<0, then both sqrt(x^2) and |x| are -x.

(if x<0, then because absolute value and square root operations must result in positive, |x| and sqrt(x^2) have to be -x; they can't yield x because x is negative)
________________

In this question, confusion may arise from wanting/having to diagram "-x" when x itself is negative. (We can't let "-x" be x since "-x" is positive and x is negative).

x is negative. But suppose there is a number, z. If z is positive and equal in absolute value to x, then z = -x or x = -z.

The question asks for sqrt(-x*|x|) or sqrt[(-1)*(x)*|x|]

Subbing "-z" into "x", the question becomes:

what is sqrt[(-1)*(-z)*|-z|]?

Because z is positive, clearly (-1)*(-z) is just z. Likewise |-z| is just z. Thus, we have:

sqrt(z*z)

which is just z

which is equal to -x.

thank you