Question

The value of $x$ such that $3^{2x}-2\times 3^{x+2}+81=0$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$
• E. $5$

Answer 1

The correct option is B

$2$

Explanation for the correct option:

Given: $3^{2x}-2\times 3^{x+2}+81=0$

$\begin{array}{crc}\Rightarrow & 3^{2\mathrm{x}}-2\times 3^{x+2}+81& =0\\ \Rightarrow & 3^{2x}-2\times 3^x\times 3^2+81& =0\\ \Rightarrow & 3^{2x}-18\times 3^x+81& =0\end{array}$

Let $3^x=t$

$\begin{array}{crc}\Rightarrow & t^2-18t+81& =0\\ \Rightarrow & t^2-9t-9t+81& =0\\ \Rightarrow & t\left(t-9\right)-9\left(t-9\right)& =0\\ \Rightarrow & {\left(t-9\right)}^2& =0\\ \Rightarrow & t& =9\end{array}$

Back substituting,

$$\begin{gathered} \Rightarrow 3^x=9 \\ \Rightarrow 3^x=3^2 \\ \therefore x=2 \end{gathered}$$

Hence option B is the correct option.