1. How to solve high-degree equations
Natural Science Department – Duy Tan University
Newton’s Method
In this section, we will learn:
How to solve high-degree equations
using Newton’s method.
Lecturer: Ho Xuan Binh
Da Nang-11/2014

2. Natural Science Department – Duy Tan University
INTRODUCTION 1
Suppose that a car dealer offers to sell you a car for $18,000 or for payments of $375 per month for five years.
You would like to know what monthly interest rate the dealer is, in effect, charging you.
To find the answer, you have to solve the equation
48x(1 + x)60 - (1 + x) = 0
How would you solve such an equation?
Newton’s method  

3. The geometry behind Newton’s method is shown here.
Natural Science Department – Duy Tan University
NEWTON’S METHOD 2
The geometry behind Newton’s method is shown here.
We start with a first approximation x1, which is obtained by one of the following methods:
Newton’s method

4. Natural Science Department – Duy Tan University
NEWTON’S METHOD 2
Consider the tangent line L to the curve y = f(x) at the point (x1, f(x1)) and look at the x-intercept of L, labeled x2.
Newton’s method  

5. If f’(x1) ≠ 0, we can solve this equation for x2:
Natural Science Department – Duy Tan University
SECOND APPROXIMATION 3
As the x-intercept of L is x2, we set y = 0 and obtain: 0 - f(x1) = f’(x1)(x2 - x1)
If f’(x1) ≠ 0, we can solve this equation for x2:
Newton’s method

6. SUCCESSIVE APPROXIMATIONS 4
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SUCCESSIVE APPROXIMATIONS 4
If we keep repeating this process, we obtain a sequence of approximations x1, x2, x3, x4, . . .
Newton’s method

7. SUBSEQUENT APPROXIMATION 5
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SUBSEQUENT APPROXIMATION 5
In general, if the nth approximation is xn and f’(xn) ≠ 0, then the next approximation is given by:
Newton’s method

8. Natural Science Department – Duy Tan University
NOTE 6
Consider the situation shown here. You can see that x2 is a worse approximation than x1. It might even happen that an approximation falls outside the domain of f, such as x3.
Newton’s method

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Example 7
Starting with x1 = 2, find the third approximation x3 to the root of the equation x3 – 2x – 5 = 0.
Newton’s method

10. Thank you for your attention