Originally Posted by

**loquaci** "Using the General Iterative Method estimate the greatest positive solution to the following equation:** x^4 - 2x^2 - x - 3 = 0"**

( ^ means to the power of , e.g x^2 is the same as x squared)

I need help simplifying the equation. I'm fine with the General Iterative Method, and I've attempted to simplify the equation so that I can then use it in the General Iterative Method, but I have doubts about whether my answer is right. Please help! I need to learn this before Monday 30th April (Australian EST).

Could you tell us what the "General Iterative Method" is?

Descartes rule of signs tells us that x^4-2x^2-x-3=0 has exactly one positive real solution.

For large x, f(x) = x^4-2x^2-x-3 is positive, so if we put x=10, we find that f(x)~=9000, so

the root is less that 10.

Rearrange the equation as:

x = FourthRoot(x^2+x+3)

and use the iterative scheme:

x_{n+1} = FourthRoot(x_n^2+x_n+3)

and we find:

Code:

n x_n
0 10.0000
1 3.82028
2 2.44965
3 2.04388
4 1.91323
5 1.87022
6 1.85598
7 1.85125
8 1.84969
9 1.84916
10 1.84899
11 1.84893
12 1.84891
13 1.84891
14 1.84891

So the (largest and only) positive root of x^4-2x^2-x-3=0 is ~1.84891

RonL