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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).Stuart Kovinsky wrote:A square root is always positive.
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???Stuart Kovinsky wrote: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.
The answer is -x because x itself is negative.greatchap wrote: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???Stuart Kovinsky wrote: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.
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.
exactly !!!!Ian Stewart wrote: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).Stuart Kovinsky wrote:A square root is always positive.
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.
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.durgesh79 wrote:
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.
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'.gabriel wrote: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.
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