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).
By default, all the numbers in GMAT are real. Therefore, both x = 1 and x = are valid solutions for (2). While 1 yields the right answer, does not. In conclusion, we have to use Statement 1 to make sure, that x = 1, not .
The situation when x is -1 does NOT give us any specific value of a + b. It gives only the value of a – b. This is NOT sufficient. For example, at least two of the possible variants are: If a = 4, b = 5, then a – b = -1, a + b = 9. If a = 5, b = 6, then a – b = -1, BUT a + b = 11.
Therefore statement (1) does NOT give us a definite value of a + b.
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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