# The decimal search

Extracts from this document...

Introduction

The Decimal Search method

I would like to solve the equation x³ + 2x² − 0.5 = 0. Let f(x) = x³ + 2x² − 0.5

The graph below shows the equation y = x³ + 2x² − 0.5. As you can see, there are three roots/solutions when f(x) = 0.

The equation x³ + 2x² − 0.5 = 0 cannot be solved algebraically so I will use the decimal search method.

This method looks for a change in sign of the value of f(x). The following table gives values of x and f(x).

x | -2.0 | -1.5 | -1.0 | -0.5 | 0 | 0.5 | 1.0 | 1.5 | 2.0 |

f(x) | -0.5 | 0.625 | 0.5 | -0.125 | -0.5 | 0.125 | 2.5 | 7.375 | 15.5 |

As the curve crosses the x axis, the value of the function f(x) changes sign, for example:

Middle

0.1225

0.1480

The interval is reduced to 0.001, the root is between [-1.854,-1.855].

x | -1.860 | -1.859 | -1.858 | -1.857 | -1.856 | -1.855 | -1.854 | -1.853 | -1.852 | -1.851 | -1.850 |

f(x) | -0.0157 | -0.0127 | -0.0098 | -0.0069 | -0.0040 | -0.0011 | 0.0018 | 0.0047 | 0.0076 | 0.0105 | 0.0134 |

The interval is reduced to 0.0001, the root is between [-1.8546,-1.8547].

x | -1.8550 | -1.8549 | -1.8548 | -1.8547 | -1.8546 | -1.8545 | -1.8544 | -1.8543 | -1.8542 | -1.8541 | -1.8540 |

f(x) | -1.1E-03 | -7.6E-04 | -4.7E-04 | -1.8E-04 | 1.1E-04 | 4.0E-04 | 6.9E-04 | 9.8E-04 | 1.3E-03 | 1.6E-03 | 1.8E-03 |

Conclusion

- 1.854640

-1.854639

-1.854638

-1.854637

-1.854636

-1.854635

-1.854634

-1.854633

-1.854632

-1.854631

-1.854630

f(x)

-6.7E-06

-3.8E-06

-9.3E-07

2.0E-06

4.9E-06

7.8E-06

1.1E-05

1.4E-05

1.6E-05

1.9E-05

2.2E-05

In error bounds x= 1.8546375 ± 0.0000005.

Where decimal search may fail

Let us solve the equation y = 4x³ + 5.84x² − 4.7104x − 7.20896.

f (x) = 4x³ + 5.84x² − 4.7104x − 7.20896

The following table shows the values of f(x) where is x between [-3, 3].

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

f(x) | -48.51776 | -6.42816 | -0.65856 | -7.20896 | -2.07936 | 38.73024 | 139.2198 |

The table confirms there is a root between [1, 2], as you can see from the graph of y = 4x³ + 5.84x² − 4.7104x − 7.20896 below, there is a root between [1, 2], but also a repeated root between [-1, -2]. The decimal search method fails to identify the repeated root between [-1, -2].

This student written piece of work is one of many that can be found in our GCSE Miscellaneous section.

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