9.
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The diagram 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 an iterative process based on the equation above with a starting value of 1.5 to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.