Math Problem Statement

The iterative formula below will give an approximate solution to the equation x^3 + 2x^2 − 3x = 12. Use this formula to work out an approximate solution to the equation starting with x₁ = 0, and calculate the difference between the rounded solution and the actual value when substituted back into the equation.

Solution

Let's break this problem into two parts, as outlined in the question.

Part (a)

We are given the iterative formula: xn+1=3xn+12xn+2x_{n+1} = \sqrt{\frac{3x_n + 12}{x_n + 2}} and need to compute the approximate solution starting with x1=0x_1 = 0, giving the answer to 3 decimal places.

  1. Starting with x1=0x_1 = 0: x2=3(0)+120+2=122=62.449x_2 = \sqrt{\frac{3(0) + 12}{0 + 2}} = \sqrt{\frac{12}{2}} = \sqrt{6} \approx 2.449

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Iterative Methods
Algebraic Equations
Roots of Equations

Formulas

x_{n+1} = sqrt((3x_n + 12) / (x_n + 2))
f(x) = x^3 + 2x^2 − 3x

Theorems

Fixed Point Iteration

Suitable Grade Level

Grades 10-12