Iteration

Show that the equation x⁴ + 3x² - 2x - 5 = 0 has a solution between x = 1 and x = 2.

Find this solution correct to 1 decimal place. You must show calculations to support your answer.

Show that the equation x³ - 3x - 4 = 0 has a solution between x = 2 and x = 3.

Use x = 2.5 to find a smaller interval for the solution to x³ - 3x - 4 = 0. You must show your working.

Find this solution correct to 1 decimal place. You must show your working.

The graph of y = x⁵ - 70x - 150 is sketched below. The root of the equation x⁵ - 70x - 150 = 0 is p.

Show that 3 < p < 4.

Find a smaller interval that contains the value of p. You must show calculations to support your answer.

Show that the equation x³ + x² - 5 = 0 has a solution between x = 1 and x = 2.

Find this solution correct to 1 decimal place. You must show calculations to support your answer.

The graph of y = x² + 6x - 2 is shown below. The roots of the equation x² + 6x - 2 = 0 are at p and q.

Calculate y when x = 1.

Without solving the equation, explain why q must lie between 0 and 1.

Explain why using a method of iteration is not the most appropriate way of finding a solution to this equation.

Show that the equation x³ - 5x - 1 = 0 has a solution between x = 2 and x = 3.

Find this solution correct to 1 decimal place. You must show your working.

Show that the equation x⁴ - x² - 9 = 0 has a solution between x = 1 and x = 2.

Find this solution correct to 1 decimal place. Show your working.

The graph of y = x³ - 7x - 12 is shown below. The root of the equation x³ - 7x - 12 = 0 is p.

Calculate y when x = 3.

Show that 3 < p < 4.

Find a smaller interval that contains the value of p. You must show calculations to support your answer.