# How do you use Newton's method to find the approximate solution to the equation #x^4=x+1,x<0#?

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Steve M
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Dec 29, 2016

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We have:

# x^4=x+1 => x^4-x-1 = 0 #

Let

First let us look at the graph:

graph{x^4-x-1 [-3, 3, -5, 8]}

We can see there is one solution in the interval

In order to find the solution numerically, using Newton-Rhapson method, we need the derivative

# f(x) = x^4-x-1 => f'(x) = 4x^3-1 # ,

and the Newton-Rhapson method uses the following iterative sequence

# { (x_0,=0), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :} #

Then using excel working to 5dp we can tabulate the iterations as follows:

And we conclude that a solution is

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