Question

Solve the equation for x: ${3^{2{\text{x}}}} - {10.3^{\text{x}}} + 9 = 0$
A. 0, 2
B. 2, 2
C. -2, 2
D. 3, 2

Answer 1

- Hint: In this question, we need to convert the given equation in a suitable quadratic form by substituting another value for x and then use splitting the middle term or quadratic formula to solve the equation. The quadratic formula for a general quadratic equation is given by-
$x = \dfrac{{ - {\text{b}} \pm \sqrt {{{\text{b}}^2} - 4ac} }}{{2{\text{a}}}}$

Complete step-by-step solution -
We have been given the equation-
${3^{2{\text{x}}}} - {10.3^{\text{x}}} + 9 = 0$
When observed closely, we can see that the term $3^x$ is being used in two terms, once as its square and second as itself. So we will replace it by a suitable term as-
Let, $t = 3^x$, then
$t^2 = 3^{2x}$

On substituting these values of t we get-
$\begin{align}
  &{{\text{t}}^2} - 10{\text{t + }}9 = 0 \\
\end{align} $
By splitting the middle term,
$\begin{align}
  &{{\text{t}}^2} - {\text{t}} - 9{\text{t + }}9 = 0 \\
  &{\text{t}}\left( {{\text{t}} - 1} \right) - 9\left( {{\text{t}} - 1} \right) = 0 \\
  &\left( {{\text{t}} - 1} \right)\left( {{\text{t}} - 9} \right) = 0 \\
\end{align} $

So, t = 1 and t = 9
But we know that,
$\begin{align}
  &{3^{\text{x}}} = {\text{t}} \\
  &{3^{\text{x}}} = 1,\;{3^{\text{x}}} = 9 \\
  &{3^{\text{x}}} = {3^0},{3^x} = {3^2} \\
  &{\text{x}} = 0,2 \\
\end{align} $

So, the solution of the equation is x = 0 and x = 2. The correct option is A.

Note: In such types of questions, it is important to analyze which value we should substitute. This is because when we substitute the right value, the equation becomes quadratic just like in this question and can be solved easily, Also, students often forget to find the final solution and just write the values of the substituted variable. For example, in this question students may just find the value of t = 1, 9 and forget to find the value of x.