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 \\ $
Answer
Verified
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.
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