If f(x)=ax^2+bx+c, where a, b and c are integers, is b=0?

[Max@Math Revolution]

[GMAT math practice question]

If f(x)=ax^2+bx+c, where a, b and c are integers, is b=0?

1) f(49)=f(-49)=0
2) f(0)f(49)=0

[GMATGuruNY]

[GMAT math practice question]

If f(x)=ax^2+bx+c, where a, b and c are integers, is b=0?

1) f(49)=f(-49)=0
2) f(0)f(49)=0

Statement 1:
Since (49, 0) and (-49, 0) are both solutions, the quadratic must be as follows:
f(x) = (x-49)(x+49) = x² - 49²
In the resulting quadratic, b=0.
Thus, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
If x-49 is a factor of the equation, then f(49) = 0.
Case 1: f(x) = (x-49)(x+49) = x² - 49², with the result that f(0)f(49) = f(0) * 0 = 0
In this case, b=0, so the answer to question stem is YES.
Case 2: f(x) = (x-49)(x+1) = x² - 48x - 49, with the result that f(0)f(49) = f(0) * 0 = 0
In this case, b=-48, so the answer to the question stem is NO.
INSUFFICIENT.

The correct answer is A.

[Max@Math Revolution]

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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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question.

In order to have b =0, f(x) must be symmetric about the y-axis. Thus, condition 1) is sufficient.
By the factor theorem, condition 1) tells us that f(x) = a(x-49)(x+49) = a(x^2 - 49^2) = ax^2 - 49^2a and b = 0.
It is sufficient.

Condition 2)
If f(0)f(49) = 0 then either f(0)= 0 or f(49) = 0.
Note that f(0) = c.
If a = 1, b = 0 and c = 0, then f(0) = 0, so f(0)f(49) = 0, and the answer is 'yes'.
If a = a, b = 1 and c = 0, then f(0) = 0, so f(0)f(49) = 0, and the answer is 'no'.
Thus, condition 2) is not sufficient, since it does not yield a unique solution.

Therefore, A is the answer.
Answer: A