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

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by Max@Math Revolution » Mon Apr 29, 2019 5:16 pm

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[GMAT math practice question]

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

1) f(10) = f(-10)=0
2) f(0)f(10)=0
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Source: — Data Sufficiency |
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by Max@Math Revolution » Tue Apr 30, 2019 11:39 pm
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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.
If b = 0, then the graph of f(x) is symmetric about the y-axis.
Condition 1) tells us that b = 0 because there are two possible ways that this graph can lie in the xy-plane.

Image

This implies that f(x) = a(x+10)(x-10)= a(x^2-100) = ax^2 - 100a and the middle term of f(x) is 0.
Thus, condition 1) is sufficient.

Condition 2)
Condition 2) is equivalent to the statement that f(0) = 0 or f(10) = 0.
If f(x) = x^2 - 100, then f(10) = 0 and b = 0.
If f(x) = x^2 - 10x, then f(0) = 0 and b = 10.
Condition 2) is not sufficient since it does not yield a unique answer.

Therefore, A is the answer.
Answer: A
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