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# D01-13

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Intern
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19 Apr 2016, 12:07
I think this is a high-quality question and I agree with explanation.
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07 Aug 2016, 01:49
I think this is a high-quality question and I agree with explanation.
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21 Sep 2016, 09:38
how did the following question become x*4-x*3-6x*2=0?
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21 Sep 2016, 23:59
manchitkapoor wrote:
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.
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25 Nov 2016, 07:17
This is a nice question. I forgot that 0 is a multiple of 9.
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01 Apr 2017, 08:38
Bunuel wrote:
manchitkapoor wrote:
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)

$$x=\sqrt[4]{x^3+6x^2}$$
x=0
$$0=\sqrt[4]{0^3+6*0^2}$$
$$0=\sqrt[4]{0}$$
$$0=0$$

$$x=\sqrt[4]{x^3+6x^2}$$
x=3
$$3=\sqrt[4]{3^3+6*3^2}$$
$$3=\sqrt[4]{27+6*9}$$
$$3=\sqrt[4]{27+54}$$
$$3=\sqrt[4]{81}$$
$$3=3$$

$$x=\sqrt[4]{x^3+6x^2}$$
x=(-2)
$$(-2)=\sqrt[4]{(-2)^3+6*(-2)^2}$$
$$(-2)=\sqrt[4]{-8+6*4}$$
$$(-2)=\sqrt[4]{-8+24}$$
$$(-2)=\sqrt[4]{16}$$
$$(-2) =| 2$$
Thus solution x = (-2) isn't applicable.
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12 Apr 2017, 03:49
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?
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12 Apr 2017, 03:58
stan3544 wrote:
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.

(-2)^4 = 16 but -2^4 = -16, this is true generally not only for the GMAT.
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12 Apr 2017, 04:58
Bunuel wrote:
stan3544 wrote:
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.

(-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.
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12 Apr 2017, 05:05
stan3544 wrote:
Bunuel wrote:
stan3544 wrote:
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.

(-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.
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12 Apr 2017, 06:37
1
Thanks for the explanation

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.

Hope this clears all the doubts for everyone.
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01 May 2017, 08:38
Bunuel wrote:
Official Solution:

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.

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.
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30 May 2017, 22:18
1
stan3544 wrote:
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.
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Joined: 02 Sep 2009
Posts: 52141

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30 May 2017, 22:21
CristianJuarez wrote:
stan3544 wrote:
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 $$x = -\sqrt{4}=-2$$.
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31 May 2017, 08:13
Can someone please provide the logic behind not taking the negative root for '√9=3, NOT +3 or -3;'
Is this mentioned anywhere int the OG or any other official material?
How do we know the GMAT doesn't consider the negative root ?
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31 May 2017, 09:20
AllenF wrote:
Can someone please provide the logic behind not taking the negative root for '√9=3, NOT +3 or -3;'
Is this mentioned anywhere int the OG or any other official material?
How do we know the GMAT doesn't consider the negative root ?

This is explained many times. $$\sqrt{}$$ denotes a function. Mathematically the square root function cannot give negative result.

OFFICIAL GUIDE:
$$\sqrt{n}$$ denotes the positive number whose square is n.

Piece of advice, people here, experts/tutors, are here to help students, not to deceive them. So, if you see the same thing repeated many times by experts/tutors you should now that it's true.
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31 May 2017, 18:33
Bunuel wrote:
CristianJuarez wrote:
stan3544 wrote:
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 $$x = -\sqrt{4}=-2$$.

Yes, actually I'd like to correct myself, I hope it isn't more confusing.

Bunuel's solution applying square root function comes from the fact that the function f(x)=x^2 allows x to take positive or negative values (or 0):

I don't have 5 posts so I can't include the image , please search for it on google.

Meanwhile, the function $$f(x)= \sqrt{x}$$ doesn't (this is why square root is always positive):

So, if people want to solve this excercise by applying square root function, you must consider that this is incorrect:

$$x^2=4$$ Apply $$\sqrt{}$$
$$\sqrt{x^2}=\sqrt{4}$$
$$x = \sqrt{4}$$
$$x=2$$ or $$x= -2$$

It is incorrect because $$\sqrt{4}$$ is always 2 and you lost a solution in the process.

So, you have to consider that mathemathically, $$\sqrt{x^2} = |x|$$. This is only because $$f(x)=x^2$$ can take a positive or a negative value for x (or 0 of course). Indeed, this doesn't mean the square root takes a positive or a negative solution, its always positive (or zero)!! (remember, $$|x|$$ is always $$>= 0$$ for any value of $$x$$).

You can check it out like this:
$$\sqrt{4} = \sqrt{2^2}= |2|=2, not -2!!$$ or:
$$\sqrt{4} = \sqrt{(-2)^2}= |-2|=2, not -2!!$$ It's impossible to get to $$-2$$ like that

So the correct process by applying square root function is:
$$x^2=4$$ Apply $$\sqrt{}$$
$$\sqrt{x^2}=\sqrt{4}$$
$$|x| = 2$$
$$x=2$$ or $$x= -2$$
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18 Aug 2017, 12:43
I think this is a high-quality question and I agree with explanation.
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31 Dec 2017, 09:14
very good question. Definitely eye opener.
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03 Feb 2018, 13:24
I love this question.

Did anyone come across any similar ones in the forum question banks? I'm not really able to drill down enough on filters to find similar ones with any consistency.
Re: D01-13 &nbs [#permalink] 03 Feb 2018, 13:24

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# D01-13

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