Find a real solution for: x^3+10x=80

We can use Newton's method. Writing f(x)=x^3+10x-80, and f'(x)=3x^2+10 and writing as a sequence we can see:

For a reasonable first guess we can use the fact that when x=3, the LHS of the original equation is 57 and when x=4, LHS=104. So let's split the difference and choose x=3.5
This gives us...




so probably close enough... 

How do you solve square roots of irrational numbers w/o a calculator? for example sqroot of 68.

There are many ways. One way is to sort of work backwards:
64 < 68 < 81 (close to 64),
so sqrt(68) is close to 8.
Then by trial and error, determine that
8.2 < sqrt(68) < 8.3
Continue this process as long as you are able.

how to solve 10th root of 1.6

From numerical methods we can write

and use Newton's method to get:


which converges very fast:

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I have tried to solve the following equation but unfortunately couldn't managed to do it. This is the simplest form it gets: (a/x)+lnx=b How can I find x? Is there a similar function that I can use to simplify it?

this equation has no algebraic solution. Problems of this type are solved by graphing and/or using numerical techniques. So any numeric solution depends on the values of a and b - which must be supplied before anything else can be determined.

how to solve this equation? x + ln (x^(2+x)) = 0

First we rewrite 



 
Immediately we notice that no matter what we do we end up with exponential/logarithmic term as well as algebraic terms. This equation cannot be solved using algebra because it is neither strictly algebraic nor exponential and so no operations can be done to solve directly for x. In cases like this we can use numerical or graphic methods: 

 


After tinkering with the window settings and such we can approximate the intersection as much as we need. 
x is about 0.75941 



:)

Approximate sqrt(5) using Newton's Method

Newton's Method is an efficient way to approximate square roots. We choose our initial point as 2 because we know that root 5 is closer to 2 than to 3. The number crunching is done by calculator (HP50g) and the table summarizes the results.

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