Newton's Method Formula
Newton's method formula is used to approximating solutions to equations. Newton's method formula is given by Newton to calculate the roots of a polynomial equation by the iterations from one root to another. Calculating the roots by this method is a lengthy process for the higher degree of a polynomial but for the smaller degree of polynomials, this method gives results very quickly and close to the actual roots of the equation. Newton's method formula is explained below along with the solved examples.
What is Newton's Method Formula?
Newton's method formula is used to find the roots of a polynomial equation. If we know any root of the equation we can find the successive roots of the equation using this method. Newton's method formula is given as:
x_{1}= x_{0 }− f(x_{0}) / f′(x_{0})
where,
 x_{0} is the initial value.
 f(x_{0}) is value of function at x_{0.}
 f'(x_{0}) is the first derivative of the function at x_{0}.
Let's take a quick look at a couple of examples to understand Newton's method formula, better.
Solved Examples Using Newton's Method Formula

Example 1: Find the next root of the given equation x^{3 }− 7x^{2 }+ 8x − 3 using Newton's method formula starting with x_{0} = 5.
Solution:
To find: Root of the given equation
Given, x_{0} = 5Using Newton's method formula,
Answer: The next root of the given equation is 6.
x_{1}= x_{0 }− f(x_{0})/f′(x_{0})
x_{1}= 5_{ }− f(5)/f′(5)
x_{1}= 5  (13)/(13) = 6 
Example 2:
Find the next root of the given equation x^{3}−x^{2 }15x+1 using Newton's method formula starting with x_{0} = 3.5.
Solution:
To find: Root
Given, x_{0} = 3.5Using Newton's method formula,
Answer: The next root of the given equation is 3.443.
x_{1}= x_{0 }− f(x_{0}) / f′(x_{0})
x_{1}= 3.5_{ }− f(3.5) / f′(3.5)
x_{1}= 3.5  (1.625)/(28.75) = 3.443
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