Newtons Method sqrt(7)

Enter Number

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Given S = 7:

Calculate:√7 using the Newtons Method

Build Newtons Method

The square root of a number can be represented
ƒ(x) = x² - S

Take the Derivative of this

ƒ'(x) = 2x

Since the square root > 0, start with x₀ = 1

Calculate x₁

x₁ = x₀ + (ƒ(x₀) - S)/ƒ'(x₀)

x₁ = 1 + (1² - 7)/2(1)

x₁ = 1 + (1 - 7)/2

x₁ =1 + -6/2

x₁ = 1 + -3

x₁ = 4

Calculate x₂

x₂ = x₁ + (ƒ(x₁) - S)/ƒ'(x₁)

x₂ = 4 + (4² - 7)/2(4)

x₂ = 4 + (16 - 7)/8

x₂ =4 + 9/8

x₂ = 4 + 1.125

x₂ = 2.875

Calculate x₃

x₃ = x₂ + (ƒ(x₂) - S)/ƒ'(x₂)

x₃ = 2.875 + (2.875² - 7)/2(2.875)

x₃ = 2.875 + (8.265625 - 7)/5.75

x₃ =2.875 + 1.265625/5.75

x₃ = 2.875 + 0.22010869565217

x₃ = 2.6548913043478

Calculate x₄

x₄ = x₃ + (ƒ(x₃) - S)/ƒ'(x₃)

x₄ = 2.6548913043478 + (2.6548913043478² - 7)/2(2.6548913043478)

x₄ = 2.6548913043478 + (7.0484478379017 - 7)/5.3097826086957

x₄ =2.6548913043478 + 0.048447837901702/5.3097826086957

x₄ = 2.6548913043478 + 0.0091242601575365

x₄ = 2.6457670441903

Final Answer

x₄ = 2.6457670441903

You have 1 free calculations remaining

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What is the Answer?

x₄ = 2.6457670441903

How does the Newton Method Calculator work?

Free Newton Method Calculator - Calculates the square root of a positive integer using the Newton Method
This calculator has 1 input.

What 3 formulas are used for the Newton Method Calculator?

ƒ(x) = x² - S
ƒ'(x) = 2x
x_n = x_{n - 1} + (ƒ(x_{n - 1}) - S)/ƒ'(x_{n - 1})

For more math formulas, check out our Formula Dossier

What 3 concepts are covered in the Newton Method Calculator?

algorithm

A process to solve a problem in a set amount of time

newtons method

another numerical method for solving an equation f...

square root

a factor of a number that, when multiplied by itself, gives the original number
√x

Example calculations for the Newton Method Calculator

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