Is sqrt((x-3)^2) = 3 - x?

[katejohn7]

Just so i am clear, the sqrt(x^2) is always positive as you can not have a negative number as square root, correct? I just want to make sure I am not confusing myself, which tends to happen after staring at these questions for too long!

Thanks for your help!

[mschwrtz]

imanemekouar, which sentence is unclear? If Ron's account still isn't clear, could you quote the sentence you find confusing?

[RonPurewal]

Just so i am clear, the sqrt(x^2) is always positive as you can not have a negative number as square root, correct? I just want to make sure I am not confusing myself, which tends to happen after staring at these questions for too long!

Thanks for your help!

well, it's not "always positive"; if x is zero, then that quantity will likewise be zero (which is not positive).
however, you're correct that the quantity can't be negative.

[csvvenkat]

ron, pls. help me understand...is not for example, sqrt (16)=+/- 4? or is it just +4..?

[mschwrtz]

The radical sign indicates the primary (non-negative) square root. Part of the confusion here--on the forum, not the GMAT--stems from using the term "sqrt" in place of the radical sign. The GMAT uses the sign, which is unambiguously non-negative.

People much more knowledgeable about math than I am use the terms "square root" and "radical" interchangeably, but once upon a time there was a convention that "radical" was the narrower term.

Anyway, to repeat, if by "sqrt 16" you mean "radical sign 16," then yes, it means only +4.

[selimsulos]

Folks, I was reading this past thread and I am getting confused here.. The square root of a number is NOT always positive - For example 4^(1/2) has two different roots - one is 2 and the other is (-2).. For that reason, if I plug in 5 fore X, the equation gives us -2 for the answer which is a root of 4... Can you please explain further why 5 doesn't work for this example??

[jnelson0612]

Folks, I was reading this past thread and I am getting confused here.. The square root of a number is NOT always positive - For example 4^(1/2) has two different roots - one is 2 and the other is (-2).. For that reason, if I plug in 5 fore X, the equation gives us -2 for the answer which is a root of 4... Can you please explain further why 5 doesn't work for this example??

In GMATland, the square root of a number is ALWAYS the positive root only (or zero if the number is zero). This is confusing, so let me illustrate:

1) if x^2=4, then x can be 2 or -2.

2) However, if I am asked the square root of 4, it is only positive 2. There are no negative square roots on the GMAT.

[me.parashar]

Ron,
It's stated in the thread that square root of a number is always positive! how come?
sqrt((x-3)^2) = |x-3|

square root can be negative as well right?
sqrt(4) = +- 2 right?

[RonPurewal]

Ron,
It's stated in the thread that square root of a number is always positive! how come?
sqrt((x-3)^2) = |x-3|

square root can be negative as well right?
sqrt(4) = +- 2 right?

no, the "√" sign must refer to a non-negative number.

think about it -- the "√" notation would be completely useless if it could refer to either sign.
for instance, the diagonal of a square with side 1 is √2 (which is a positive number). if "√2" could be either positive or negative -- which, thankfully, it can't -- then it would actually be impossible to write the length of the diagonal without using weird things like absolute value signs.

the more general principle at work here is that symbols should have one meaning. in the case of the "√" symbol, that meaning is the nonnegative one.

[laxmsun]

Isn't the variable under the square root sign positive? If so,
can't we not just write it as x-3? Or, does it have to be explicitly
stated as the positive root under the radical?

Is x-3=3-x? i.e Is x=3?

Thanks

[me.parashar]

Ron,
It's stated in the thread that square root of a number is always positive! how come?
sqrt((x-3)^2) = |x-3|

square root can be negative as well right?
sqrt(4) = +- 2 right?

no, the "√" sign must refer to a non-negative number.

think about it -- the "√" notation would be completely useless if it could refer to either sign.
for instance, the diagonal of a square with side 1 is √2 (which is a positive number). if "√2" could be either positive or negative -- which, thankfully, it can't -- then it would actually be impossible to write the length of the diagonal without using weird things like absolute value signs.

the more general principle at work here is that symbols should have one meaning. in the case of the "√" symbol, that meaning is the nonnegative one.

Thanks again Ron. Wonderfully explained :)

[RonPurewal]

Isn't the variable under the square root sign positive? If so,
can't we not just write it as x-3? Or, does it have to be explicitly
stated as the positive root under the radical?

Is x-3=3-x? i.e Is x=3?

Thanks

i'm not quite sure what you're asking, but it has probably been answered already on the first page of the thread. did you read the thread?

if you haven't already, please check out the following post, as well as the one below it:
post27226.html#p27226

if there is anything you don't understand those two posts, then please reply to them explicitly (with a quote). thanks.

[rahul.popat]

Hi Ron,

I am amazed to find GMAT considers sqrt as positive only. But in that case.

why sqrt((x-3)^2) = |x-3|

my question is, if LHS is positive, why mod is required in RHS.
i mean why it cant be just:

sqrt((x-3)^2) = x-3 (without mod)

[shubham_sagijain]

Rahul, as mentioned by Ron, the square root of a number is always positive in GMATLand.

So if you are asked sqrt (16), the answer is 4 and not (+ or -)4.

Thanks,
Shubh

[shubham_sagijain]

Hi Ron,

I am amazed to find GMAT considers sqrt as positive only. But in that case.

why sqrt((x-3)^2) = |x-3|

my question is, if LHS is positive, why mod is required in RHS.
i mean why it cant be just:

sqrt((x-3)^2) = x-3 (without mod)

Rahul,

Try substituing x= 2 in your example...