Question

How do you solve \[{{3}^{2x+1}}-26\left( {{3}^{x}} \right)-9=0\]?

Answer 1

Hint: To solve the given equation first we have to assume the appropriate substitution to make the given complex equation simple. Here we can substitute \[{{3}^{x}}\] as y in the given equation, then we can simplify the obtained quadratic equation to get the values of x from the given equation.

Complete step by step answer:
The given equation to solve is as follows,
\[{{3}^{2x+1}}-26\left( {{3}^{x}} \right)-9=0\]
We can simplify this equation as below,
\[3{{\left( {{3}^{x}} \right)}^{2}}-26\left( {{3}^{x}} \right)-9=0\]
We can now substitute y = \[{{3}^{x}}\], so we can write the above equation as,
\[3{{\left( y \right)}^{2}}-26\left( y \right)-9=0\]
Now we can factorise the above equation as follows,
\[\Rightarrow 3{{\left( y \right)}^{2}}-27\left( y \right)+1\left( y \right)-9=0\]
Let us see the common terms and take them aside,
\[\begin{align}
  & \Rightarrow 3y\left( y-9 \right)+1\left( y-9 \right)=0 \\
 & \Rightarrow \left( 3y+1 \right)\left( y-9 \right)=0 \\
\end{align}\]
Now equate each to zero to get the values of y,
\[\]
\[\Rightarrow y=\dfrac{-1}{3}\]
Or \[\left( y-9 \right)=0\]
\[\Rightarrow y=9\]
Now let us re-substitute y as \[{{3}^{x}}\] to get the value of x,
\[y=\dfrac{-1}{3}\Rightarrow {{3}^{x}}=\dfrac{-1}{3}\]
\[\Rightarrow x={{\log }_{3}}\left( \dfrac{-1}{3} \right)\]
Or
\[y=9\]\[\Rightarrow {{3}^{x}}=9\]
\[\Rightarrow {{3}^{x}}={{3}^{2}}\]
On comparison we can get the value of x as,
\[\Rightarrow x=2\]
From this the value of x present in the equation can either be 2 or \[{{\log }_{3}}\left( \dfrac{-1}{3} \right)\]

Note: The equation given here seems to be difficult because the variable is in the power place. So the student should be able to substitute \[{{3}^{x}}\]as y otherwise the question gets clumsy to solve it. Students should solve the quadratic equation to get the value of y which can be further reduced to get the value of x. The simplification must be done carefully to make sure that the result obtained at the end is correct.