How do you use Newton's method to find the approximate solution to the equation x+sqrtx=1?

1 Answer
Steve M
Feb 16, 2017

The solution is x=0.3820 to 4dp.

Explanation:

We want to solve:

x+sqrt(x) = 1 => x+sqrt(x) -1 =0

Let f(x) = x+sqrt(x) -1 Then our aim is to solve f(x)=0.

First let us look at the graphs:
graph{x+sqrt(x) -1 [-2, 3, -2, 2]}

We can see there is one solution in the interval 0 le x le 1, so will use x=1 as our initial approximation

To find the solution numerically, using Newton-Rhapson method we will need the derivative f'(x).

\ \ \ \ \ \ \f(x) = x+sqrt(x) -1
:. f'(x) = 1+1/2x^(-1/2)
\ \ \ \ \ \ \ \ \ \ \ \ \ \= 1+1/(2sqrt(x))

The Newton-Rhapson method uses the following iterative sequence

{ (x_1,=1), ( x_(n+1), = x_n - f(x_n)/(f'(x_n)) ) :}

Then using excel working to 8dp we can tabulate the iterations as follows:

enter image source here

We could equally use a modern scientific graphing calculator as most new calculators have an " Ans " button that allows the last calculated result to be used as the input of an iterated expression.

And we conclude that the solution is x=0.3820 to 4dp.

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