8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d17a1b86-d758-4470-834a-b32a41f90c89-4_478_937_251_450}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
- Find an equation for the normal to the curve at \(P\).
The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
- Show that \(1 < q < 2\).
- Show that \(q\) is a solution of the equation
$$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
- Use the iterative formula
$$x _ { n + 1 } = \frac { 12 } { 7 } \ln \left( 2 x _ { n } + 5 \right) - 2 ,$$
with \(x _ { 0 } = 1.5\), to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.