1982 AHSME Problems/Problem 17

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Problem

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

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

Solution

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.

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