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
383.4k+ views
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.
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.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is pollution? How many types of pollution? Define it

Change the following sentences into negative and interrogative class 10 english CBSE

Why do noble gases have positive electron gain enthalpy class 11 chemistry CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE

Write an application to the principal requesting five class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers
