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I. if x < 0 then value of sqrt ( |x| ) = sqrt(x) II. if x < 0 then value of sqrt ( x ) = i*sqrt(x)

I think the following should be the solution:
I. if x < 0 then value of sqrt ( |x| )
= sqrt(-x) II. if x < 0 then value of sqrt ( x )
= sqrt(x) - no change (- but when x is assigned some numeric value like say -3 then sqrt ( x )=3i but we do not need the 'i' as long as the expression is in terms of x)

I. if x < 0 then value of sqrt ( |x| ) = sqrt(x) II. if x < 0 then value of sqrt ( x ) = i*sqrt(x)

I think the following should be the solution: I. if x < 0 then value of sqrt ( |x| ) = sqrt(-x)

II. if x < 0 then value of sqrt ( x ) = sqrt(x) - no change (- but when x is assigned some numeric value like say -3 then sqrt ( x )=3i but we do not need the 'i' as long as the expression is in terms of x)

what do U think?

If you want to keep it in the "x" form then YES, but ultimately, what I wrote too is not wrong! So, let us take it this way, look at the choices & decide

If you want to keep it in the "x" form then YES, but ultimately, what I wrote too is not wrong! So, let us take it this way, look at the choices & decide

Agreed.
Also, your statement earlier "..We know that x<0, then -x > 0." is a very interesting way of looking [or - to look (..not sure if infinitive or -ing form is better here) ] at absolute values.
thanks

I saw this question earlier but could not find a logical explanation.
I still feel that sqrt(positive number) can be either positive or negative. In that case, the answer can be either x or -x.
For ex, sqrt(4) = 2 or -2.
Please explain why the OA is -x .... Thanks.

seems you have correct OA. but how? could you explain?

This is my logic---->

We know that x<0, then -x > 0. In other words, -x*|x| = +x^2

so, sqrt(-x*|x|) = sqrt(x^2) = +x or -x. but since it is GIVEN that x < 0, answer should be "-x".

(For any sqrt, answer is always "+" or "-" ; but if some condition is given, we can arrive at whether "+" OR "-")

but we cannot have -x as one of the value of sqrt(x^2) because sqrt(x^2) has only one value that is x. any square under radical sign has only +ve value.

seems you have correct OA. but how? could you explain?

This is my logic---->

We know that x<0, then -x > 0. In other words, -x*|x| = +x^2

so, sqrt(-x*|x|) = sqrt(x^2) = +x or -x. but since it is GIVEN that x < 0, answer should be "-x".

(For any sqrt, answer is always "+" or "-" ; but if some condition is given, we can arrive at whether "+" OR "-")

but we cannot have -x as one of the value of sqrt(x^2) because sqrt(x^2) has only one value that is x. any square under radical sign has only +ve value.

How do you say that? For example, "25" is a square of "5" & also of "-5".
sqrt(25) = +5 or -5. I didn't get your point.

seems you have correct OA. but how? could you explain?

This is my logic---->

We know that x<0, then -x > 0. In other words, -x*|x| = +x^2

so, sqrt(-x*|x|) = sqrt(x^2) = +x or -x. but since it is GIVEN that x < 0, answer should be "-x".

(For any sqrt, answer is always "+" or "-" ; but if some condition is given, we can arrive at whether "+" OR "-")

I am with you untill

sqrt(-x*|x|) = sqrt(x^2) = +x or -x.

so sqrt(-x*|x|) is -ve or +ve

and its absolute value is |x|

x or -x

how do you conclude that the value of the expression sqrt(-x*|x|) is -x?

OD,
Frankly speaking, I'm not very comfortable with this problem. Ideally, answer should be "+/-x", but since it was already mentioned that x<0, (and most important: answer choice has both +x & -x, WE HAVE TO SELECT ONE CHOICE) the value that I selected is "-x"

For example, if we are solving sqrt(a) for something like age of a person, -ve value wouldn't make sense, we have to select +ve value.