Question

Find the root correct to three decimal places of x

3 −4x−9 = 0 lying between 2 and

3 by using Regula Falsi method.​

Answer 1

Answer:

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Answer 2

Given:

x³ - 4x - 9 = 0

Range: between 2 and 3

To find:

Find the root correct to three decimal places of x by using Regula Falsi method

Solution:

Here, x³ - 4x - 9 = 0

Let, f(x) = x³ - 4x - 9

1st Iteration:

Here f(2) = -9 < 0 and f(3) = 6 > 0

∴Now, roots lies between x₀ = 2 and x₁ = 3

$x_2 = x_0 - f(x_0)\frac{x_1 - x_0}{f(x_1)-f(x_0)} \\\\x_2 = 2 - (-9)\frac{3 -2}{6 - (-9)}$

x₂ = 2.6

f(x₂) = f(2.6) = -1.824 < 0

2nd Iteration:

Here f(x₂) = f(2.6) = -1.824 < 0 and f(3) = 6 > 0

∴Now, roots lies between x₀ = 2.6 and x₁ = 3

$x_3 = x_0 - f(x_0)\frac{x_1 - x_0}{f(x_1)-f(x_0)}$

x₃ = 2.6933

f(x₃) = f(2.6933) = -0.2372 < 0

3rd Iteration:

Here f(x₃) = f(2.6933) = -0.2372 < 0 and f(3) = 6 > 0

∴Now, roots lies between x₀ = 2.6933 and x₁ = 3

$x_4 = x_0 - f(x_0)\frac{x_1 - x_0}{f(x_1)-f(x_0)}$

x₄ = 2.7049

f(x₄) = f(2.7049) = -0.0289 < 0

4th Iteration:

Here f(x₄) = f(2.7049) = -0.0289 < 0 and f(3) = 6 > 0

∴Now, roots lies between x₀ = 2.7049 and x₁ = 3

$x_5 = x_0 - f(x_0)\frac{x_1 - x_0}{f(x_1)-f(x_0)}$

x₅ = 2.7063

f(x₅) = f(2.7063) = -0.0035 < 0

5th Iteration:

Here f(x₅) = f(2.7063) = -0.0035 < 0 and f(3) = 6 > 0

∴Now, roots lies between x₀ = 2.7063 and x₁ = 3

$x_6 = x_0 - f(x_0)\frac{x_1 - x_0}{f(x_1)-f(x_0)}$

x₆ = 2.7065

f(x₆) = f(2.7065) = -0.0004 < 0

Approximate root of the equation x³ - 4x - 9 = 0 using Regula Falsi method is 2.706