# Sydney's question at Yahoo! Answers regarding root approximation

#### MarkFL

Staff member
Here is the question:

Approximate the solution to the following equation that satisfies the given condition: x^(4) + x - 3 = 0 ; x is greater than or equal to 0

Here is a link to the question:

I have posted a link there to this topic so the OP can find my response.

#### MarkFL

Staff member
Hello Sydney,

First, let's take a look at a plot of the function:

We see the positive root is near $x=1$, so that will be a good initial guess.

Newton's method gives us the recursion:

$\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

In our case, this is:

$\displaystyle x_{n+1}=x_n-\frac{x_n^4+x_n-3}{4x_n^3+1}=\frac{3(x_n^4+1)}{4x_n^3+1}$

where $x_0=1$.

Now, I have a TI-89 Titanium graphing calculator, so what I do is enter the following:

1 [ENTER] Result: 1
(3(ans(1)^4+1))/(4ans(1)^3+1) [ENTER] Result: 1.2
[ENTER] Result: 1.16541961577
[ENTER] Result: 1.16403726916
[ENTER] Result: 1.16403514029
[ENTER] Result: 1.16403514029

Since the last two approximations are the same, we have exceeded the limit of accuracy of the calculator, and we know the required root, to 12 digits, is:

$x\approx1.16403514029$

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