Calculate x from x^3+3=x^2

We'll use numerical methods to find the solution. Graphing  and  together shows that there is only one point of intersection - slightly less than -1. So we know that there is only 1 real solution. 
We use Newton's method: 
 
 

then 
 
We wont bother simplifying because we'll do the calculations by calculator: 
guess x=-1 

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 to solve 10th root of 1.6

From numerical methods we can write

and use Newton's method to get:


which converges very fast:

---

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.

If you need help with math, I can help you. Visit my mobile website: www.mathhelp.mobi

or email me at services@mathhelp.mobi