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

1972 AHSME Problems/Problem 6

Revision as of 23:07, 21 December 2020 by Eschlaton (talk | contribs) (Created page with "== Problem 6 == If <math>3^{2x}+9=10\left(3^{x}\right)</math>, then the value of <math>(x^2+1)</math> is <math>\textbf{(A) }1\text{ only}\qquad \textbf{(B) }5\text{ only}\...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem 6

If $3^{2x}+9=10\left(3^{x}\right)$, then the value of $(x^2+1)$ is

$\textbf{(A) }1\text{ only}\qquad \textbf{(B) }5\text{ only}\qquad \textbf{(C) }1\text{ or }5\qquad \textbf{(D) }2\qquad \textbf{(E) }10$

Solution

We can rearrange to get $3^{2x}-10\left(3^{x}\right)+9=0$. Factor or use the quadratic formula to find that $3^{x}= 9\text{ or }1$, which makes $x = 2\text{ or }0$ and our answer \[\boxed{\textbf{(C) }1\text{ or }5}.\]

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1972_AHSME_Problems/Problem_6&oldid=140167"