1982 AHSME Problems/Problem 17

Revision as of 22:02, 16 June 2021 by Aopspandy (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)


How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^{x}+3=0$?

$\text {(A)} 0 \qquad  \text {(B)} 1 \qquad  \text {(C)} 2 \qquad  \text {(D)} 3 \qquad  \text {(E)} 4$


Let $a = 3^x$. Then the preceding equation can be expressed as the quadratic, \[9a^2-28a+3 = 0\] Solving the quadratic yields the roots $3$ and $1/9$. Setting these equal to $3^x$, we can immediately see that there are $\boxed{2}$ real values of $x$ that satisfy the equation.