f(x) = x³ + 3x² + 4x - 12 (a) Show that the equation f(x) = 0 can be written as x = \sqrt{\frac{4 - 3x}{3 + x}} - Edexcel - A-Level Maths Pure - Question 3 - 2012 - Paper 5

Using the given iteration formula:

1. Start with x_0 = 1:

   $x_1 = \sqrt{\frac{(4 - 1)}{(3 + 1)}} = \sqrt{\frac{3}{4}} \approx 1.2247$

2. Next, calculate x_2:

   $x_2 = \sqrt{\frac{(4 - 1.2247)}{(3 + 1.2247)}} \approx \sqrt{\frac{2.7753}{4.2247}} \approx 1.2099$

3. Calculate x_3:

   $x_3 = \sqrt{\frac{(4 - 1.2099)}{(3 + 1.2099)}} \approx \sqrt{\frac{2.7901}{4.2099}} \approx 1.3143$

Thus, the values are:

• x_1 ≈ 1.22
• x_2 ≈ 1.21
• x_3 ≈ 1.31

To prove that α = 1.272 to 3 decimal places, we can choose a suitable interval for the root. We know from the previous parts that:

1. Calculate f(1.2715):

   $f(1.2715) \approx -0.0083$

2. Calculate f(1.2725):

   $f(1.2725) \approx 0.0002$

Since f(1.2715) < 0 and f(1.2725) > 0, by the Intermediate Value Theorem, there is a root in the interval (1.2715, 1.2725).

3. To confirm, calculate f(1.272):

   $f(1.272) \approx 0$

This shows that α = 1.272 is indeed accurate to three decimal places.