Monday, April 21, 2014

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

We'll use numerical methods to find the solution. Graphing x%5E3%2B3 and x%5E2 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: 
f%28x%29=x%5E3-x%5E2%2B3 
df%2Fdx=3x%5E2-2x 

then 
x%5Bn%2B1%5D=x%5Bn%5D-%28x%5Bn%5D%5E3-x%5Bn%5D%5E2%2B3%29%2F%283x%5Bn%5D%5E2-2x%5Bn%5D%29 
We wont bother simplifying because we'll do the calculations by calculator: 
guess x=-1 
x%5B2%5D=-6%2F5
x%5B3%5D=-47%2F40
x%5B4%5D=-1.1745595

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