$8 \cdot 2$ If $3^{2 x}+9=10 \cdot 3^{x}$, then the value of $\left(x^{2}+1\right)$ is
(A) 1 only.
(B) 5 only.
(C) 1 or 5.
(D) 2.
(E) 10.
(23rd American High School Mathematics Examination, 1972)
[Solution] Let $y=3^{x}$, then the given equation is equivalent to the following equation.
$y^{2}-10 y+9=0$, which is $(y-9)(y-1)=0$.

Therefore, $y=3^{x}=9$ or $1, x=2$ or $0$. So $x^{2}+1=5$ or $1$. Hence, the answer is $(C)$.