Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

1975 AHSME Problems/Problem 12

Revision as of 06:00, 17 May 2016 by Quantummech (talk | contribs) (→‎Solution)

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 \box{(B): a=3x or a=-2x}.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1975_AHSME_Problems/Problem_12&oldid=78552"