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 $\fbox{(\textbf{B}):$ (Error compiling LaTeX. Unknown error_msg)a=3x $or$ a=-2x$}.$ (Error compiling LaTeX. Unknown error_msg)

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1975 AHSME Problems/Problem 12

Problem 12

Solution