Question

The value of $x$ such that ${3^{2x}} - 2\left( {{3^{x + 2}}} \right) + 81 = 0$ is$a.1 \\ b.2 \\ c.3 \\ d.4 \\ e.5 \\$

Hint: Substitute ${3^x}$ to any other variable say (t) then factorize the equation to reach the answer.

The given equation is
${3^{2x}} - 2\left( {{3^{x + 2}}} \right) + 81 = 0$
Now simplify the above equation as ${3^{x + 2}}$ is written as ${3^x} \times {3^2}$ and ${3^{2x}}$ is written as ${\left( {{3^x}} \right)^2}$.
${3^{2x}} - 2\left( {{3^x} \times {3^2}} \right) + 81 = 0 \\ \Rightarrow {3^{2x}} - \left( {2 \times 9 \times {3^x}} \right) + 81 = 0 \\ \Rightarrow {\left( {{3^x}} \right)^2} - 18 \times {3^x} + 81 = 0 \\$
Now let ${3^x} = t...........\left( 1 \right)$, substitute this value in above equation
$\Rightarrow {t^2} - 18t + 81 = 0$
Now factorize this equation
$\Rightarrow {t^2} - 9t - 9t + 81 = 0 \\ \Rightarrow t\left( {t - 9} \right) - 9\left( {t - 9} \right) = 0 \\ \Rightarrow \left( {t - 9} \right)\left( {t - 9} \right) = 0 \\ \Rightarrow {\left( {t - 9} \right)^2} = 0 \\ \Rightarrow \left( {t - 9} \right) = 0 \\ \Rightarrow t = 9 \\$
Now from equation (1)
${3^x} = t = 9$
Now we know 9 is written as ${3^2}$
$\Rightarrow {3^x} = {3^2}$
So, on comparing
$x = 2$
Hence option (b) is correct.

Note: In such types of questions first simplify the given equation then substitute ${3^x} = t$ this makes the equation simple after this factorizes the equation and calculates the value of t, then re-substitute the value of t and simplifies, we will get the required answer.