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1975 AHSME Problems/Problem 12

Revision as of 15:50, 4 September 2020 by Brian22 (talk | contribs) (→‎Solution: Fixed formatting on the answer)

Problem 12

If $a \neq b, a^3 - b^3 = 19x^3$, and $a-b = x$, which of the following conclusions is correct?

$\textbf{(A)}\ a=3x \qquad \textbf{(B)}\ a=3x \text{ or } a = -2x \qquad \textbf{(C)}\ a=-3x \text{ or } a = 2x \qquad \\ \textbf{(D)}\ a=3x \text{ or } a=2x \qquad \textbf{(E)}\ a=2x$

Solution

We can factor $a^3-b^3=19x^3$ into: \[(a-b)(a^2+ab+b^2)=19x^3.\] Substituting yields: \[x(a^2+ab+b^2)=19x^3\] \[a^2+ab+b^2=19x^2.\] This is equal to: \[(a-b)^2+3ab=19x^2\] \[x^2+3ab=19x^2\] \[ab=6x^2.\] Checking with the possible answers, along with $a-b=x$ yields the only answer to be $\boxed{(\textbf{B}): a=3x\text{ or }a=-2x}.$

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