Iteration

Show that the equation x³ + 2x − 6 = 0 has a solution between x = 1 and x = 2

Show that the equation x³ + 2x − 6 = 0 can be rearranged to give x =

Starting with x₀ = 1.45 use the iteration formula x = twice to find an estimate

for the solution of x³ + 2x − 6 = 0 Give your answer correct to 4 decimal places.

Starting with x₀ = 1.5

a solution of x⁴ – x² – 5 = 0

The values of x₁, x₂ and x₃ found in part (a) are estimates of the solution of an equation of the form x³ + ax + b = 0 where a and b are integers.

Find the value of a and the value of b.

Show that the equation x³ + x = 7 has a solution between 1 and 2

Starting with x₀ = 2,

Starting with x = 1, use the iteration formula x = three times to find

an estimate for the solution of x³ + 7x − 5 = 0

By substituting your answer to part (c) into x³ + 7x − 5, comment on the accuracy of your estimate for the solution to x³ + 7x − 5 = 0

with x₀ = 2.5

find the values of x₁, x₂ and x₃

Explain the relationship between the values of x₁, x₂ and x₃ and the equation x³ + 2x² + 4 = 0