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I think this the explanation isn't clear enough, please elaborate. i dont understand why -2 is not a solution . for instance squrt of 4 has 2 soln i.e 2 and -2. please explain?

I think this the explanation isn't clear enough, please elaborate. i dont understand why -2 is not a solution . for instance squrt of 4 has 2 soln i.e 2 and -2. please explain?

That's not true.

When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT. _________________

Hi Bunuel, would the following be a valid approach to this question or just luck?

1.) Re-arrange and factor out \(x^2: x^2(x^2−x−6)=0\) --> So one solution is 0 -> \(x^2\) 2.) Use discriminant for the left equation \((x^2−x−6)=0\): \(b^2 - 4ac\) --> 1-(4*1*(-6)) -> 25 -> Since 25 is positive the left equation has two solutions 3.) 1.) + 2.) = 3 Solutions

Hi Bunuel, would the following be a valid approach to this question or just luck?

1.) Re-arrange and factor out \(x^2: x^2(x^2−x−6)=0\) --> So one solution is 0 -> \(x^2\) 2.) Use discriminant for the left equation \((x^2−x−6)=0\): \(b^2 - 4ac\) --> 1-(4*1*(-6)) -> 25 -> Since 25 is positive the left equation has two solutions 3.) 1.) + 2.) = 3 Solutions

You should find the actual roots and find out whether all of them are valid. Only 2 of them are valid for the equation at hand. Please refer to the discussion on previous 2 pages.
_________________

I am still not able to convince my self that -2 is not the root of the given equation. Because as per my knowledge this should have been the one solution. Now, the dilemma is what i will do in similar questions.

how did the following question become x*4-x*3-6x*2=0?

It does not.

Take \(x=\sqrt[4]{x^3+6x^2}\) to the 4th power to get \(x^4=x^3+6x^2\), which when re-arranged becomes \(x^4-x^3-6x^2=0\). It's x to the 4th power, minus x cubed, minus 6 times x squared, equal 0.
_________________

how did the following question become x*4-x*3-6x*2=0?

It does not.

Take \(x=\sqrt[4]{x^3+6x^2}\) to the 4th power to get \(x^4=x^3+6x^2\), which when re-arranged becomes \(x^4-x^3-6x^2=0\). It's x to the 4th power, minus x cubed, minus 6 times x squared, equal 0.

I think this should clarifies everybody's confusion regarding the answer, i hope everyone understands how we got 0,3,(-2)

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

Not sure I understand what you mean but GMAT doe not have its own math. That's simply not correct. The only thing is that GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers, so no imaginary numbers. That's why the even roots from negative numbers are not defined for the GMAT.

As for your example:

(-2)^4 = 16 but -2^4 = -16, this is true generally not only for the GMAT.
_________________

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

Not sure I understand what you mean but GMAT doe not have its own math. That's simply not correct. The only thing is that GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers, so no imaginary numbers. That's why the even roots from negative numbers are not defined for the GMAT.

As for your example:

(-2)^4 = 16 but -2^4 = -16, this is true generally not only for the GMAT.

Hey Bunuel,

Now i am totally confused. Yes it is obvious and universally understood that sqr.root of negative number is not a real number, basically it doesn't exist, why? Because no negative number raised to an even power will ever give us a negative result, yet according to your example -2^4 = -16, so we know that the number -2 raised to the power 4 gives us -16, isn't it a paradox? Now, according to the exponent rules: (-2)^4 = (-2^1)^4 = -2*(-2)*(-2)*(-2) right? Now according to your interpretation -2^4 is not the same, so it basically means -2*2*2*2, which doesn't make sense, because this goes against the whole idea behind exponents, which is: we take a number and multiply by ITSELF n times. So the correct way to write it would be -(2^4) = -16 or -(-2^4) = -16. Now back to the issue at hand: logically 2^2=4 as well as -2*(-2)=4, which is (-2)^2= 4, now i don't see why sqr.root of 4 can only be 2 and not -2 as well.

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

Not sure I understand what you mean but GMAT doe not have its own math. That's simply not correct. The only thing is that GMAT deals with only Real Numbers: Integers, Fractions and Irrational Numbers, so no imaginary numbers. That's why the even roots from negative numbers are not defined for the GMAT.

As for your example:

(-2)^4 = 16 but -2^4 = -16, this is true generally not only for the GMAT.

Hey Bunuel,

Now i am totally confused. Yes it is obvious and universally understood that sqr.root of negative number is not a real number, basically it doesn't exist, why? Because no negative number raised to an even power will ever give us a negative result, yet according to your example -2^4 = -16, so we know that the number -2 raised to the power 4 gives us -16, isn't it a paradox? Now, according to the exponent rules: (-2)^4 = (-2^1)^4 = -2*(-2)*(-2)*(-2) right? Now according to your interpretation -2^4 is not the same, so it basically means -2*2*2*2, which doesn't make sense, because this goes against the whole idea behind exponents, which is: we take a number and multiply by ITSELF n times. So the correct way to write it would be -(2^4) = -16 or -(-2^4) = -16. Now back to the issue at hand: logically 2^2=4 as well as -2*(-2)=4, which is (-2)^2= 4, now i don't see why sqr.root of 4 can only be 2 and not -2 as well.

-2^4 = -(2^4) = -16. It's order of operations thing.

As for the square roots: I tried to explain it several times on previous pages but I'll try this one last time:

When the GMAT provides the square root sign for an even root, such as \(\sqrt{x}\) or \(\sqrt[4]{x}\), then the only accepted answer is the positive root.

That is, \(\sqrt{25}=5\), NOT +5 or -5. Even roots have only a positive value on the GMAT.

In contrast, the equation \(x^2=25\) has TWO solutions, +5 and -5.

If it's still unclear or confusing then probably it's better to start from basics and brush-up fundamentals or enrol into a class.
_________________

Maybe i wasn't able to explain clearly what i was confused about, but i found the answer on the Magoosh website, boldfaced is the explanation i needed: The equation x^2 = 16 has two solutions, x = +4 and x = -4, because 4^2 = 16 and (-4)^2 = 16, and the GMAT will impale you for only remembering one of those two. At the same time, sqrt{16} has only one output: sqrt{16} = +4 only. When you yourself undo a square by taking a square root, that’s a process that results in two possibilities, but when you see this symbol as such, printed as part of the problem, it means find the positive square root only.

If \(x=\sqrt[4]{x^3+6x^2}\), then the sum of all possible solutions for x is:

A. \(-2\) B. \(0\) C. \(1\) D. \(3\) E. \(5\)

Take the given expression to the 4th power: \(x^4=x^3+6x^2\);

Re-arrange and factor out x^2: \(x^2(x^2-x-6)=0\);

Factorize: \(x^2(x-3)(x+2)=0\);

So, the roots are \(x=0\), \(x=3\) and \(x=-2\). But \(x\) cannot be negative as it equals to the even (4th) root of some expression (\(\sqrt{expression}\geq{0}\)), thus only two solution are valid \(x=0\) and \(x=3\).

The sum of all possible solutions for x is 0+3=3.

Answer: D

Thank you for this valuable info. Because in engineering we used to consider root(4) = + / - 2 . Now, I know that GMAT only considers positive values for even roots. . Ahh i screwed up one question in Main GMAT because of my ignorance in regards to this knowledge.

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

The gmat doesn't have its own version of math. The square root is a function, that's why it is always positive. As a function, it can't take a positive and a negative value (if it did, it wouldn't be a function, you can look at the graph to check that out).

When you solve x^2 = 4, applying the square root function and assuming two results for x is incorrect, because you would only get x = 2. The formal and correct method is: x^2 = 4 x^2 - 4 =0 (x + 2)(x - 2)=0 and from here get both results for x.

And here we go again. Sometimes i feel as if GMAT has it's own version of math that goes against universal one. I understand that on verbal there can be a whole lot of inerprteations of rules, simply because it is VERBAL, as in not a precise science which is basically a human imagination. But math has to be precise. And now, according to e-GMAT -b^m if m is even --> positive number and you are stating that −2^4=−16 and i made a whole lot of mistakes when practicing quant from both OG and scholaranium that involved sqr.root of a number, because i didn't include the negative value as well. Can comeone give a link or a screenshot of an OG question where when we take sqr root we don't consider the negative value?

The gmat doesn't have its own version of math. The square root is a function, that's why it is always positive. As a function, it can't take a positive and a negative value (if it did, it wouldn't be a function, you can look at the graph to check that out).

When you solve x^2 = 4, applying the square root function and assuming two results for x is incorrect, because you would only get x = 2. The formal and correct method is: x^2 = 4 x^2 - 4 =0 (x + 2)(x - 2)=0 and from here get both results for x.

That's correct. + 1.

Another way of solving x^2 = 4 would be:

\(x = \sqrt{4}=2\) or \(x = -\sqrt{4}=-2\).
_________________