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

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.

The values and 1 are equal.

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

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.

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.

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.