Question

find the real root of the equation x^3-2x^2-4=0 by using iteration method up to 3 decimal place​

Answer 1

The real roots of the equation x³ -2x² -4 = 0  is 2.889

Given:

The equation x³ -2x² -4 = 0  

To find:

Find the real root of the equation by using the iteration method up to 3 decimal places.

Solution:

Given f(x) = x³ -2x² -4 = 0  

Find the value of x₀, for which we have to find a and b such that f(a) < 0 and f(b) > 0

Now, f(0) = – 4

=> f(1) = – 4

=> f(3) = 5

Thus, a = 1 and b = 3

Therefore, x₀ = (1 + 3)/2 = 4

As it is given f(x) = x³ -2x² -4 = 0  

=> g(x) = (2x² + 4)^1/3

Now, applying the iterative method x$_{n}$,= g(x$_{n-1}$) for n = 1, 2, 3, 4, 5, …

Continue the iteration until we reach the desired number of decimal places. Here are the results for the first few iterations:

x₀ = 2

x₁ = 2.828

x₂  = 2.884

x₃ = 2.889

x₄ = 2.889

Therefore,

The real roots of the equation x³ -2x² -4 = 0  is 2.889

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