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

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

(→‎Solution)
 
(4 intermediate revisions by 2 users not shown)
Line 14: Line 14:
 
<cmath>x^2+3ab=19x^2</cmath>
 
<cmath>x^2+3ab=19x^2</cmath>
 
<cmath>ab=6x^2.</cmath>
 
<cmath>ab=6x^2.</cmath>
−
Checking with the possible answers, along with <math>a-b=x</math> yields the only answer to be \fbox{(B): <math>a=3x</math> or<math> a=-2x</math>}.
+
Checking with the possible answers, along with <math>a-b=x</math> yields the only answer to be <math>\boxed{(\textbf{B}): a=3x\text{ or }a=-2x}.</math>
 +
 
 +
==See Also==
 +
{{AHSME box|year=1975|num-b=11|num-a=13}}
 +
{{MAA Notice}}

Latest revision as of 17:09, 19 January 2021

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 by
Problem 11 		Followed by
Problem 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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1975_AHSME_Problems/Problem_12&oldid=142819"