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The Change of Sign method locates the root of an equation by where it crosses the x-axis.

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Introduction

Method 1: Change of Sign

Change of Sign

The Change of Sign method locates the root of an equation by where it crosses the x-axis. The point at which the curve crosses x-axis is the root. When a function changes sign in a certain interval, we can see that the root, or the place where the curve crosses the axis is within that interval. Using a Decimal Search method, it is possible to find intervals to varying degrees of accuracy to help pinpoint the position of a root.

I am going to use the equation:

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Middle

1

-2

1.1

-0.205

1.2

2

1.3

4.645

The above table shows that there is a change of sign between 1.1 and 1.2. Therefore, we now know that the root lies in the interval [1.1,1.2]

image01.png

We can extend the search further to find the interval of the change of sign with greater accuracy.

x

y

1.1

-0.205

1.11

-0.00344

1.12

0.20224

1.13

0.412085

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Conclusion

2 (x+1.4) would overlook the second root. As you can see from the graph below, there are two roots; one in the interval [-2,-1] and one in the interval [-1,0]. However, the table only shows one change of sign in the interval [-2,-1].

image04.png

image05.png

2.   The Change of Sign Method would also fail with an equation where all of the roots fall within the same interval, such as in the equation y = x3-1.7x2+0.84x-0.108. The table below shows only one change of sign in the interval [0,1]. This would indicate that there is only one root, rather than three, in this interval.

image06.png

image07.png

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