Question

Solve the following a)give the solution in calculator ready form (the exact value of x) and b)approximate the solution to the nearest hundredth.8. 3^2x - 12 * 3^x -28 =0

Answer 1

Given the following exponential equation:

`3^(2x)^{}-12\cdot3^x-28=0`

Note: the given equation is a quadratic equation

Let u = 3ˣ

So, the equation will be:

`\begin{gathered} u^2-12u-28=0 \\ \text{factor}\rightarrow(u-14)(u+2)=0 \\ u-14=0\rightarrow u=14 \\ u+2=0\rightarrow u=-2 \end{gathered}`

so,

`\begin{gathered} 3^x=14 \\ or,3^x=-2 \end{gathered}`

The negative result will be rejected because the range of the exponential function is greater than zero

So, the exact value of x will be as follows:

`x=(\log 14)/(\log 3)`

The approximate solution using the calculator will be:

x = 2.40