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A function f(x) satisfies 2f(x)+3f(1-x)=x^2.

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A function f(x) satisfies 2f(x)+3f(1-x)=x^2.

by Max@Math Revolution » Mon Oct 07, 2019 11:28 pm
[GMAT math practice question]

A function f(x) satisfies 2f(x)+3f(1-x)=x^2. What is the expression of f(x)?

A. (x^2 +6x +3)/7
B. (x^2 -6x +3)/7
C. (x^2 -6x +1)/3
D. (x^2 -6x +3)/5
E. (x^2 +6x +3)/5
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by Max@Math Revolution » Thu Oct 10, 2019 12:12 am
=>

We have 4f(x)+6f(1-x)=2x^2 when we multiply the equation in the original condition by 2.
If we substitute x by 1-x, then we have 2f(1-x)+3f(x) = (1-x)^2.
If we multiply the equation by 3, we have 6f(1-x)+9f(x) = 3(1-x)^2.
If we subtract equations [4f(x)+6f(1-x)=2x^2] - [6f(1-x)+9f(x) = 3(1-x)^2], then we have -5f(x) = 2x^2 - 3(1-x)^2, -5f(x) = 2x^2 - 3(1-2x+x^2), -5f(x) = 2x^2 - 3 + 6x - 3x^2, -5f(x) = -x^2 + 6x - 3 or f(x) = (1/5)(x^2-6x+3).

Therefore, the answer is D.
Answer: D
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