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Finding the root of an equation

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Introduction

Finding the root

Change of sign method This is the graph of the equation y=3x3+4x2-3x-1. The change of sign method will be used to find out the value of the root in this graph which lies between -1 and 0. The change of sign method requires finding out where a change of sign occurs in the y-co-ordinate and establish where, within the gap found out, the next change of sign will occur. This will result in a narrowing down of the group of values the root could lie between which will result in an answer of a satisfactory degree of accuracy.

The first part of the investigation involves finding out where, between -1 and 3, a change of sign occurs.

 X Y -1 3 -0.9 2.753 -0.8 2.424 -0.7 2.031 -0.6 1.592 -0.5 1.125 -0.4 0.648 -0.3 0.179 -0.2 -0.264 -0.1 -0.663 0 -1

There is a change of sign and therefore a root in the region between -0.3 and -0.2. The next part of the investigation is to find out where a change of sign occurs between -0.3 and -0.2.

 X Y -0.3 0.179 -0.29 0.133233 -0.28 0.087744 -0.27 0.042551 -0.26 -0.00233 -0.25 -0.04688 -0.24 -0.09107 -0.23 -0.1349 -0.22 -0.17834 -0.21 -0.22138 -0.2 -0.264

Middle

0.002145

-0.26

-0.00233

There is a change of sign and therefore a root in the region between -0.261 and -0.260. The next part is to establish where, between -0.261 and -0.26, a change of sign occurs.

 X Y -0.261 0.0021453 -0.2609 0.0016978 -0.2608 0.0012503 -0.2607 0.0008029 -0.2606 0.0003556 -0.2605 -0.0000918 -0.2604 -0.0005391 -0.2603 -0.0009864 -0.2602 -0.0014336 -0.2601 -0.0018808 -0.26 -0.0023280

There is a change of sign and therefore a root in the region between -0.2606 and -0.2605. The next part of the investigation is establishing where, between -0.2606 and -0.2605, a change of sign occurs.

 X Y 0.2606 0.000355557 -0.26059 0.000310821 -0.26058 0.000266086 -0.26057 0.000221351 -0.26056 0.000176616 -0.26055 0.000131882 -0.26054 0.000087148 -0.26053 0.000042414 -0.26052 -0.000002319558 -0.26051 -0.000047053 -0.2605 -0.000091785

There is a change of sign and therefore a root between -0.26053 and -0.26052. As shown visibly, and as can be proved mathematically, the root of the equation which lies between -1 and 0 is closer to -0.26052 than it is to -0.26053. Therefore, the root of the equation which lies between -1 and 0 is -0.26052 to 5 significant figures.

Because of the nature of the answer when given to five significant figures, the amount of error is +0.000005. This makes the percentage error of the answer is 0.

Conclusion  This shows the process of Newton-Rapshon iterations aiming to find out the value of the root between -1 and 0 which lies closer to zero. The starting value chosen in this case in 0. This has resulted in a value of -0.22708 to 5.s.f. being obtained for the root.

The other root lies between 1 and 2.  The starting value for this root was 2. This root has been found to be 1.7368 to 5.s.f.

However, there are occasions where finding the root using the Newton-Rapshon method doesn’t always work. This is the graph y=2x3+2x2-2x-1. If the root between -2 and -1 was under investigation and the starting value of -1 was used, the gradient of the tangent at this point is zero. Therefore, no tangent can be drawn from this point to intercept the x-axis because the tangent at this point is horizontal.

y=2x3+2x2-2x-1; When y=-1, x=1

dy/dx=6x2+4x-2; When x=-1, dy/dx=6-4-2=0

y=mx+c; y=-1, m=0, x=1, c=?

y-mx=c; -1-0=-1 Therefore, equation of tangent is y=-1

When y=0, y cannot =0. Therefore the Newton-Rapshon method cannot be applied here.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

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