
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
512.1k+ 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
Glucose when reduced with HI and red Phosphorus gives class 11 chemistry CBSE

The highest possible oxidation states of Uranium and class 11 chemistry CBSE

Find the value of x if the mode of the following data class 11 maths CBSE

Which of the following can be used in the Friedel Crafts class 11 chemistry CBSE

A sphere of mass 40 kg is attracted by a second sphere class 11 physics CBSE

Statement I Reactivity of aluminium decreases when class 11 chemistry CBSE

Trending doubts
When people say No pun intended what does that mea class 8 english CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How many ounces are in 500 mL class 8 maths CBSE

Which king started the organization of the Kumbh fair class 8 social science CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

Advantages and disadvantages of science
