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Math Expert V
Joined: 02 Sep 2009
Posts: 57995
Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 41% (02:04) correct 59% (02:21) wrong based on 128 sessions

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Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

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Math Expert V
Joined: 02 Aug 2009
Posts: 7810
Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

TIP- whenever you see a SQUARE ROOT or a MODULUS on one side, and the choices/statements give you no specific values but type of integers they are, such as zero, negative etc, CHECK if the other side is NEGATIVE

So here whatever be in the equation, the LHS is a square root and a square root can never be NEGATIVE, so check RHS ..
RHS is a+x...
Combined we know x is non Positive, so max value will be 0 and a is NEGATIVE so a+x will be NEGATIVE..
Thus RHS is NOT equal to LHS..
Sufficient

C
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GMAT 1: 200 Q1 V1 Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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1
$$\sqrt{6a + x^{2}} = a + x$$
Simplifies to...
$$6a + x^2 = a^2 + x^2 + 2ax$$
$$6a = a^2 + 2ax$$
$$6 = a + 2x$$

Statement 1 tells us that $$x\leq{0}$$. It doesn't say anything about $$a$$. So we can have $$x = -1$$ and $$a = 8$$, in which case $$a + 2x = 6$$. Or we can have $$x = -1$$ and $$a = -1$$, in which case $$a + 2x = -3$$. Insufficient.

Statement 2 tells us that $$a<0$$. It doesn't say anything about $$x$$. Like for Statement 1, for any negative value of $$a$$ we can use a value of $$x$$ which may or may not fit the equation $$6 = a + 2x$$. Insufficient.

Combining both statements, since $$a$$ is negative and $$x$$ is at most 0, $$a + 2x$$ is negative and thus can never equal 6. Sufficient.
##### General Discussion
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Joined: 25 Feb 2013
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Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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1
Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

Statement 1: Nothing mentioned about $$a$$, hence the value cannot be calculated. $$x$$ is either $$0$$ or a negative no. Insufficient

Statement 2: nothing mentioned about x, hence value cannot be calculated. Insufficient

Combining 1 & 2, we know that x=0 or x<0 and a<0 so a+x<0. So RHS is always negative.
But LHS of the question stem is a square root which is always positive. Hence LHS is not equal to RHS. Sufficient

Option C
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Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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afa13 wrote:
chetan2u wrote:
Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

TIP- whenever you see a SQUARE ROOT or a MODULUS on one side, and the choices/statements give you no specific values but type of integers they are, such as zero, negative etc, CHECK if the other side is NEGATIVE

So here whatever be in the equation, the LHS is a square root and a square root can never be NEGATIVE, so check RHS ..
RHS is a+x...
Combined we know x is non Positive, so max value will be 0 and a is NEGATIVE so a+x will be NEGATIVE..
Thus RHS is NOT equal to LHS..
Sufficient

C

Chetan2u,
when the statement says "not a positive integer", is it safe to assume that it is either zero or a negative integer. I mean why do we rule out the fact that x is either a positive fraction or a negative fraction?
Do u understand what my concern is?
thank u in advance.

Hi afa13

This is my take on your query, other experts can pitch in their views.

All Real numbers are either Rational or Irrational

Rational number can be further divided into - Natural Number, Whole Numbers, Integers

Now Integers can be fractions also, you simply need to add 1 to the denominator. For eg. 2=2/1 (fraction), or 4/2 or 6/3 and so on. this means that every rational number can be written as a fraction.

Rational numbers can fall in more than one category - a Whole number can be an integer, similarly integers can be fractions.

So in this question if x is not a positive integer then we can safely assume that it will not be a positive fraction.
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Joined: 12 Jul 2017
Posts: 33
Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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chetan2u wrote:
Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

TIP- whenever you see a SQUARE ROOT or a MODULUS on one side, and the choices/statements give you no specific values but type of integers they are, such as zero, negative etc, CHECK if the other side is NEGATIVE

So here whatever be in the equation, the LHS is a square root and a square root can never be NEGATIVE, so check RHS ..
RHS is a+x...
Combined we know x is non Positive, so max value will be 0 and a is NEGATIVE so a+x will be NEGATIVE..
Thus RHS is NOT equal to LHS..
Sufficient

C

Chetan2u,
when the statement says "not a positive integer", is it safe to assume that it is either zero or a negative integer. I mean why do we rule out the fact that x is either a positive fraction or a negative fraction?
Do u understand what my concern is?
thank u in advance.
SVP  V
Joined: 26 Mar 2013
Posts: 2322
Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

Dear Bunuel

This question triggers general question when I see something like Statement 1.

(1) x is not a positive integer.

Does it mean that x could be positive fraction or decimal such as 1/2 or 2.35 or any like numbers? or only mean that x< 0 or x =0?

IMO: nothing the statement limits the the option of positive fractions or positive decimals??
Math Expert V
Joined: 02 Sep 2009
Posts: 57995
Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Mo2men wrote:
Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

Dear Bunuel

This question triggers general question when I see something like Statement 1.

(1) x is not a positive integer.

Does it mean that x could be positive fraction or decimal such as 1/2 or 2.35 or any like numbers? or only mean that x< 0 or x =0?

IMO: nothing the statement limits the the option of positive fractions or positive decimals??

x is not a positive integer means that x is a positive number bur not an integer (1/2, $$\sqrt{2}$$, ...) or x is a negative number or x is 0. So basically x is any number but 1, 2, 3, 4, ...
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GMAT 1: 760 Q51 V42 GPA: 3.82
Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Condition 1) & 2):
Since $$\sqrt{6a+x^2} ≥ 0$$ and $$a < 0$$, $$x≤0$$, the left side is greater than or equal to 0 and the right side is less than 0.
Thus both sides never equal each other.
Both conditions together are sufficient.

Since this question is not one of key questions, we choose C.
Therefore, the answer is C.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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Bunuel wrote:
Is $$\sqrt{6a + x^2} = a + x$$ ?

(1) x is not a positive integer.
(2) a is a negative integer.

IMO C solve the question first 6a+x^2=a^2+2ax+x^2
6a=a^2+2ax
now come to statement
statement 1 x is not positive then x can be 0 or negative insufficient
statement 2 a is negative no information about x insufficient.
together we can say that not equal so sufficient.
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Re: Is (6a + x^2)^(1/2) = a + x ?  [#permalink]

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