Gmat Prep problem

[lateraldeath]

Can someone explain to me why this problem is A, I really don't understand how one can get a negative after the square root. Unless I'm wrong in thinking -x and x being a negative number are two different things.

Thanks in advance!!

[Stuart@KaplanGMAT]

A square root is always positive.

So, if x is negative, then sqrt(x^2) = -x

For example:

If x = -3, then sqrt(-3^2) = sqrt(9) = 3 = -x
If x = -4, then sqrt(-4^2) = sqrt(16) = 4 = -x

In this question, we want to solve for:

sqrt(-x * |x|) = sqrt(x^2) which is always going to be positive. So, since x is negative, the answer is -x.

[lateraldeath]

ahh got it, dang it so that was the trick. I thought that problem looked way too easy when i did it so i knew i had to have missed something but couldn't figure out what for the life of me. Thanks

[Ian Stewart]

A square root is always positive.

Ah, here we need to be careful. Square roots can certainly be zero, so they are not always positive (the square root of zero is zero).

The symbol used in the question above, the 'square root symbol', means the non-negative square root. It can never give you a negative result. I'm sure that's what Stuart means above. But every positive number has two square roots, one positive, one negative. 4 has two square roots, 2 and -2, because 2^2 = 4 and (-2)^2 = 4. Square roots can certainly be negative as well.

But if you see something under the square root sign, the result will be positive or zero, because of the definition of that root sign.

[beeparoo]

Since we're being specific, let's not discount those queer imaginary numbers, which gives hope to all negative numbers living under the square root sign.

Thankfully, the good people at GMAC have relegated such numbers as to be figments of the imagination (pun intended!). We can all breathe a little easier now... High fives all around.

Sandra

[greatchap]

A square root is always positive.

So, if x is negative, then sqrt(x^2) = -x

For example:

If x = -3, then sqrt(-3^2) = sqrt(9) = 3 = -x
If x = -4, then sqrt(-4^2) = sqrt(16) = 4 = -x

In this question, we want to solve for:

sqrt(-x * |x|) = sqrt(x^2) which is always going to be positive. So, since x is negative, the answer is -x.

Thanks for explanation, but I still have a doubt. sqrt(-x * |x|) means sqrt(-x^2) so if x = -1 then does this mean- sqrt(- (-1)^2) so sqrt(-1) . but we cant have neg inside square root can we???
OR if I plug in value -3 then sqrt(- * -3 |-3|) or sqrt(3*3) sqrt(9) sqrt = 3 so thats positive so how is answer -x.

[Stuart@KaplanGMAT]

A square root is always positive.

So, if x is negative, then sqrt(x^2) = -x

For example:

If x = -3, then sqrt(-3^2) = sqrt(9) = 3 = -x
If x = -4, then sqrt(-4^2) = sqrt(16) = 4 = -x

In this question, we want to solve for:

sqrt(-x * |x|) = sqrt(x^2) which is always going to be positive. So, since x is negative, the answer is -x.

Thanks for explanation, but I still have a doubt. sqrt(-x * |x|) means sqrt(-x^2) so if x = -1 then does this mean- sqrt(- (-1)^2) so sqrt(-1) . but we cant have neg inside square root can we???
OR if I plug in value -3 then sqrt(- * -3 |-3|) or sqrt(3*3) sqrt(9) sqrt = 3 so thats positive so how is answer -x.

The answer is -x because x itself is negative.

If x = -3, and the answer we want is +3, then we have to multiple x by (-1) to get a postitive result.

-(-3) = +3

[durgesh79]

A square root is always positive.

Ah, here we need to be careful. Square roots can certainly be zero, so they are not always positive (the square root of zero is zero).

The symbol used in the question above, the 'square root symbol', means the non-negative square root. It can never give you a negative result. I'm sure that's what Stuart means above. But every positive number has two square roots, one positive, one negative. 4 has two square roots, 2 and -2, because 2^2 = 4 and (-2)^2 = 4. Square roots can certainly be negative as well.

But if you see something under the square root sign, the result will be positive or zero, because of the definition of that root sign.

exactly !!!!

Actually if you solve the quardetic equation x^2 = 4.
the +/- sign comes before the 'sqrt root symbol' So if there is no sign before the symbol given, we'll assume its +ve ... like we do with all other numbers.

[greatchap]

I understand guys, that sqrt(4) = +/- 2

Right, but what is puzzling me is that ques says -

X < 0 and what will be sqrt(-x |x|)...right

Isnt it asking the outcome of the above, so if x is neg the outcome will be positive. So then why are we choosing -x as answer & not x.

[gabriel]

exactly !!!!

Actually if you solve the quardetic equation x^2 = 4.
the +/- sign comes before the 'sqrt root symbol' So if there is no sign before the symbol given, we'll assume its +ve ... like we do with all other numbers.

Actually Durgesh, when we evaluate x^2=4, we are not looking for the sqroot of 4 but the roots of the function x^2-4=0, that is we are looking for values of "x" that would make the above equation 0. There is a small difference between this and finding the sqroot of 4.

The rule of thumb is that unless it is a question in inequalities or functions, one should only consider the principal sqroot of a number (positive sqroot). This is not true for only GMAT (an argument I have heard more than once) but for maths in general. One of the reasons for this convention is the use of sqroots in geometry, where only positive values can be used.

Regards

[Ian Stewart]

Actually Durgesh, when we evaluate x^2=4, we are not looking for the sqroot of 4 but the roots of the function x^2-4=0, that is we are looking for values of "x" that would make the above equation 0. There is a small difference between this and finding the sqroot of 4.

The rule of thumb is that unless it is a question in inequalities or functions, one should only consider the principal sqroot of a number (positive sqroot). This is not true for only GMAT (an argument I have heard more than once) but for maths in general. One of the reasons for this convention is the use of sqroots in geometry, where only positive values can be used.

Regards

I'm not sure what this means. Every positive number has two square roots; that is mathematical fact. When you are solving the equation x^2 = 4, you are, by definition, solving for both square roots of 4: the square roots of 4, are, by definition, the numbers whose square is 4. The square root symbol, which I take it is what you mean by sqroot, is defined as a function; it produces the non-negative square root of whatever is underneath the symbol. This is sometimes called the 'principal square root'. In day-to-day speech, when we talk about 'the square root' of a number, we mean the 'principal square root', and often we encounter geometry or word problems involving distances, or numbers of things, where the negative answer can be discarded. But if a question says 'x is a square root of 9', there is no 'rule of thumb' that lets us ignore the possibility that x is negative. And since the square root symbol is a function (known as 'the principal square root function'), I don't understand the exceptions you provide to your 'rule of thumb'.

Mind you, the GMAT is never ambiguous on this point, so this discussion is merely of theoretical interest. If you see a square root symbol, the result can never be negative. If you see an equation like x^2 = 4, there are two solutions for x.

[AleksandrM]

https://www.beatthegmat.com/gmat-prep-pr ... 12840.html

[greatchap]

my question still remains unanswered guys.

[AleksandrM]

If x < 0 then sqroot(-x * |x|) is:

sqroot (-(-x) * x) = x * x = sqrootx^2, which is is positive x. However since x is negative, you need to put a negative sign infront of it, to make it positive.