How do you use Newton's method to find the approximate solution to the equation #x^3+5x-10=0#?

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Answer 1 · 2016-12-22T20:23:30

#x=1.423318 # to 6dp

Let #f(x) = x^3+5x-10# Then our aim is to solve #f(x)=0#

First let us look at the graphs:
graph{x^3+5x-10 [-5, 5, -20, 15]}

We can see there is one solution in the interval # 1 < x < 2 #.

We can find the solution numerically, using Newton-Rhapson method

# f(x) = x^3+5x-10 => f'(x) = 3x^2+5 #, and using the Newton-Rhapson method we use the following iterative sequence

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

Then using excel working to 6dp we can tabulate the iterations as follows:

enter image source here

And we conclude that the remaining solution is #x=1.423318 # to 6dp