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### Non-Linear Equation

```
Date: 2/13/96 at 19:58:38
From: Budi Rachman
Subject: Non-Linear Equation

Can you give me a detailed solution for non-linear equations like
this:

5x^2 + log(x) = 0
x = ?

```

```
Date: 7/29/96 at 16:32:17
From: Doctor Jerry
Subject: Re: Non-Linear Equation

Equations similar to this equation can't be solved in what is called
closed form.  They must be solved numerically, with an algorithm.

There are several algorithms, some of which depend upon calculus.
I'll give you some results from applying Newton's method, which
and I'll talk about the bisection method.

Let f(x) = 5x^2+log(x).  I assume log means natural logarithm.  If you
graph f you'll see that it crosses the x-axis near 0.5.  The first
guess at the zero c of f(x) will be x1=0.5.

The next guess is x2 = x1-f(x1)/f'(x1) = 0.42045...

For f(x) = 5x^2+log(x), the derivative function f' is f'(x)=10x+1/x.

The next guess is x3 = x2-f(x2)/f'(x2) = 0.417797...

The calculations continue in this pattern and converge quite rapidly
to the zero c, which is approximately 0.417795.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Calculus

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