- Using an Iterative Formula

The equation LaTeX: x^3-5x=24 has a root between 3 and 4. Use an iterative process to approximate this root correct to 2 decimal places.


STEP 1: Make an LaTeX: x the subject of the equation.

LaTeX: x=\sqrt[3]{5x+24} or LaTeX: x=\frac{x^3-24}{5}

STEP 2: Choose the approximate equation. (The one which converges).

Therefore, we choose LaTeX: x=\sqrt[3]{5x+24} since LaTeX: x=\frac{x^3-24}{5} diverges.

STEP 3: Rewrite equation as an iterative formula

LaTeX: x_{n+1}=\sqrt[3]{5x_n+24}

STEP 4: Choose an initial approximation, LaTeX: x_0. (We will choose the midpoint of the interval LaTeX: [3, 4])

LaTeX: x_0=3.5

STEP 5: Apply the iterative formula until the desired number of decimal places is attained.

LaTeX: x_0=3.5
LaTeX: x_1=\sqrt[3]{5x_0+24}=\sqrt[3]{5\left(3.5\right)+24}=3.4621

Though we only need 2 decimal places, we use more decimal places than required until we get to the final answer.

Since LaTeX: x_1, to 2 decimal places is NOT the same as LaTeX: x_0 we repeat the iterative process.

LaTeX: x_2=\sqrt[3]{5x_1+24}=\sqrt[3]{5\left(3.4621\right)+24}=3.4569

Since LaTeX: x_1 and LaTeX: x_2 are BOTH approximately 3.46 to 2 decimal places we can state that the root of the equation LaTeX: x^3-5x=24 between 3 and 4 is approximately 3.46.