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 Post subject: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:38 am 
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If ax² + bx³ = 5, where a and b are non-zero numbers, what is the value of a + b?
(1) ax² = a
(2) bx³ = b

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(B) The important thing to remember here is that, when a number is squared, it will always result in a positive, regardless of whether the original number was negative or positive. This is not the case with a number that is cubed. A cube will only be positive if the original integer was positive; otherwise, it will be negative. That having been said, statement (1) is insufficient. While we know that x can be nothing other than 1 or -1 in order for ax² to equal a, we are not told specifically if x is 1 or -1.

Statement (2), however, is sufficient. Because x is cubed, we know that it had to be 1 from the beginning. Statement (2) tells us that a + b = 5. The answer is (B).
-------------

The right answer is D.
ax² = a, so x² = 1, so x = 1. And so it is similar to statement (2).


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:39 am 
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ax² = a
x² = 1
x = 1 or x = -1

If x = 1, then
ax² + bx³ = a × 1² + b × 1³ = a + b = 5

If x = -1, then
ax² + bx³ = a × (-1)² + b × (-1)³ = a – b = 5
But we don't know what a + b equals to in this case (if x = 1).

Therefore statement (1) by itself is NOT sufficient.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:40 am 
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By default, all the numbers in GMAT are real. Therefore, both x = 1 and x = Image are valid solutions for (2). While 1 yields the right answer, Image does not. In conclusion, we have to use Statement 1 to make sure, that x = 1, not Image.
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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:42 am 
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The values Image and 1 are equal.

The second statement yields x³ = 1. This equation has only one solution: x = ³√1 = 1.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:42 am 
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In case of 1, if x is -1, then we get ab = 5 and when x is 1, we get a + b =5. Since we get only 1 value of a + b from (1), I feel the answer to this should be D.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:43 am 
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Quote:
Since we get only 1 value of a + b from (1)
The situation when x is -1 does NOT give us any specific value of a + b. It gives only the value of ab. This is NOT sufficient.
For example, at least two of the possible variants are:
If a = 4, b = 5, then ab = -1, a + b = 9.
If a = 5, b = 6, then ab = -1, BUT a + b = 11.

Therefore statement (1) does NOT give us a definite value of a + b.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:43 am 
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a, b and x could be rational numbers, it is not mentioned that they are integers. Hence, the inference that x could only be 1 or -1 is wrong; e.g. x = 1/2, a = 4 and b = 32.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:43 am 
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questioner wrote:
a, b and x could be rational numbers, it is not mentioned that they are integers. Hence, the inference that x could only be 1 or -1 is wrong; e.g. x = 1/2, a = 4 and b = 32.

The proposed values do NOT fit in any statement:
Statement (1) will transform into 4 × (1/2)² = 4. This is NOT correct.
Statement (2) will transform into 32 × (1/2)³ = 32. This is NOT correct.

The basic question statement does NOT specify that x is an integer, but each additional statement gives us some specific possible values of x. Statement (1) yields x = -1 and x = 1. Statement (2) yields x = 1 only.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:44 am 
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I desagree with the given answer, we have:

If ax² + bx³ = 5, where a and b are non-zero numbers, what is the value of a + b?
(1) ax² = a
(2) bx³ = b

I agree that with (1) we cannot answer the question, but (2) isn't enough either.

number 2 says that x³ = 1 (with b non-zero), which has 3 solutions, not 1:

x³ - 1 = 0
(x – 1) × (x² + x + 1)=0
Either (x – 1) = 0 or (x² + x + 1) = 0. Only x – 1 = 0 gives x = 1.
But the 3 solutions are:

1, -1/2 + (√3 / 2) × i, -1/2 – (√3 / 2) × i

The second and third solution, when squared, don't equal 1, so we need the first statement to know that x can't take these values. Therefore, the answer is C and not B.


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 Post subject: Re: GMAT Algebra
PostPosted: Mon Apr 08, 2013 11:45 am 
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Quote:
But the 3 solutions are:

1, -1/2 + (√3 / 2) × i, -1/2 – (√3 / 2) × i
Your solution would be true, if x were a complex variable. But x is a real variable, not a complex one.

NEVER assume in GMAT that we deal with complex numbers (variables).
ALWAYS assume that a number (variable) is a real number, if nothing else is stated.


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