The solutions of

are $\alpha$ and $\beta$, where $\beta >\alpha .$

A researcher is comparing the efficacy of root-finding algorithms, and wants to compare two different methods for approximating $\beta$ using a starting value of $x=1.6$

1. Use the quadratic formula to find the exact value of $\beta$

2. Using the iterative formula

   $$x_{n+1}=\sqrt{6-2x_n},x_0=1.6$$

   find two further approximations for $\beta .$

3. Use the Newton-Raphson method to find two further approximations, again starting from $x=1.6$

4. By considering appropriate percentage errors, compare the efficacy of the two methods.

Let $P$ be the percentage error when using the iterative method in (b), and $Q$ be the percentage error when using the Newton-Raphson method in (c).

Give the value of $\frac{P}{Q}$, correct to $1$ significant figure.