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How to Use Newton's Method to Find Roots of Equations

How to Use Newton's Method to Find Roots of Equations
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  • 0:04 Newton's Method
  • 2:06 Solving the Equation
  • 3:20 Graphing the Equation
  • 4:39 Complex Equations
  • 6:09 Lesson Review
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Lesson Transcript
Instructor: Erin Ryan
Finding the roots of equations usually requires the use of a calculator. However, in this lesson you'll use Newton's Method to find the root of any equation, even when you can't solve for it explicitly.

Newton's Method

The steps to using Newtons Method
Newtons Method Steps

Remember that Newton's Method is a way to find the roots of an equation. For example, if y = f(x), it helps you find a value of x that y = 0. Newton's Method, in particular, uses an iterative method. That is, you make some guess and you use it to find another guess - a better guess. The method is just a form of linearization (estimation). You estimate x sub (n + 1) - that's your next guess - is going to be equal to your current guess x sub n - y sub n / f '(x sub n). That is, your next guess is equal to your current guess minus the current value of y divided by the derivative.

In this example, we guess x sub 0, find the value of y at x sub 0 and the value of the derivative at x sub 0. We use that information to find a new estimate: x sub 1. Again, we find y at x sub 1, we find the derivative at x sub 1, and we use that information to find another guess, x sub 2. And, we'll continue this until our new value of x gives us y = 0.

In practice, you're going to start with some initial guess, x sub 0. You're going to find the derivative,f '(x) of your equation, and then you're going to use Newton's equation to estimate x sub 1. You're going to estimate x sub 2 (using x sub 1), and so on until your x values converge. They're going to converge somewhere where y = 0. This usually works (we're not going to get into the complex cases where Newton's Method doesn't work in this course).

So, how do I use Newton's Method? I generally make a table to keep track of all my variables. Here's my table:

x y y'

I've got x in one column, y in one column, and the derivative, y ', in another. I make my first guess, which would be the first row of this table. I'm going to make a guess for x, and I'm going to plug it in. Then, I'll find what y and y ' are for that value of x. I'm going to use all that information in the Newton equation to find the next row (particularly, the next x). I'm going to continue from there.

Using Newtons Method for the equation f(x) = x^2 + 3x - 4
Newtons Method Example 1

Solving the Equation

Let's do this. We have the equation f(x) = xˆ2 + 3x - 4. When I graph this, I know that it makes a parabola. I already think there might be two roots, and I can probably solve for them by hand; but for this case, I'm going to use Newton's Method.

I'm going to use x = 0 for my first guess. I'll use my Newton equation, x sub n+1 = x sub n - y sub n / f ' (x sub n). I can also write this as x sub n - f(x sub n) / f '(x sub n), and all I've done is say that f(x sub n) is the same as y sub n. Let's plug in our f(x) and our f '(x) into that equation.

First, what is f '(x)? Let's take the derivative of f(x) and I get 2x + 3. I used the power rule to differentiate xˆ2 as 2x. Then I've got the derivative of 3x, which is just 3.

Let's plug in f(x) and f '(x) into our Newton equation. I get x sub n+1 = x sub n - (x sub nˆ2 + 3x sub n - 4) / 2x sub n + 3. Now I've got an equation. I just need my table, and away we go.

Graphing the Equation

So, here's my table:

x y y'

I'm going to use an initial guess, x sub 0 = 0. That goes on the first line, 0 for x. When x = 0, f(x) (or y) is -4 and y ' = 3. Let's put those into my table as well:

x y y'
0 -4 3

Let's use Newton's equation to find x sub 1. If my initial guess x sub 0 = 0, I plug in 0 into the equation. I get x sub 1 (my next guess) is 4/3. Let's put that into the table. Let's say that's about 1.3. If x is 1.3, I can find y by plugging 1.3 into my f(x), and I get 1.8. y ' (by plugging 1.3 into this equation) is about 5.66.

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