Difference between revisions of "1975 AHSME Problems/Problem 12"

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 $\boxed{(\textbf{B}): a=3x\text{ or }a=-2x}.$

See Also

 1975 AHSME (Problems • Answer Key • Resources) Preceded byProblem 11 Followed byProblem 13 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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