Equations obtained in real life by people such as scientists and engineers often are not easy to solve. Algebraic techniques such as transposing terms, factorising and using the quadratic formula are not suitable. Another method is to use numerical methods which are available to approximate an answer or root of the equation to a certain level of accuracy. Two such methods, studied in this course, are the Bisection method and the NewtonRhapson method.
Roots of an Equation
The root(s) of an equation are the value(s) of the variable, which make the equation true.
The roots of an equation are the solutions of f(x) = 0. This is where the graph of the function f(x) crosses the xaxis.
e.g. The three roots of the equation 3x^{3} + 3x^{2} − 3x − 1 = 0 are shown on the diagram below.
When the graph of a function crosses the xaxis its sign changes from positive to negative.
e.g. To show that there is a root between x = 2 and x = 1, evaluate the function at both values.
f(2) = 7
f(1) = 2
One value is positive and one value is negative ⇒ there is a root between x = 2 and x = 1.
The Bisection Method
In the graph above for the equation 3x^{3} + 3x^{2} − 3x − 1 = 0 there is also a root between x = 0 and x = 1. We can use the bisection method to find the value of this root to a required number of decimal places.
The concept described above is used in this process. By bisecting the interval containing the root (x_{m}), the root can be found more accurately. x_{m} = (x_{1} + x_{2} )/2
To approximate the root to 1 decimal place the bisecting process is repeated until x_{2} − x_{1} < 0.1 (in the end column).
Each repetition is called an iteration.
x_{1}

f(x_{1})

x_{2}

f(x_{2)}

x_{m}

f(x_{m})

Notes

x_{2 }x_{1}

0

1

1

2

0.5

1.375

x_{m} replaces x_{1}

1

0.5

1.375

1

2

0.75

0.297

x_{m} replaces x_{1}

0.5

0.75

0.297

1

2

0.875

0.682

x_{m} replaces x2

0.25

0.75

0.297

0.875

0.682

0.8125

0.152 
x_{m} replaces x2

0.125

0.75

0.297

0.8125

0.152

0.78125

0.082

0.0625

The f(x) values do not have to be to many decimal places it is the sign that is important.
The process is stopped because x_{2} − x_{1} < 0.0625, thus the approximate value of the root is0.78125 which is 0.8 to 1 decimal place. This is done because both x_{1} and x_{2} now round to 0.8.
The Bisection method is accurate but lengthy. To estimate the root in the example above to 2 decimal places would require several more iterations.
Graphical Representation of the Bisection Method
The graph below shows the intervals used in the example above and illustrates how the root (where the curve crosses the xaxis) is approached. The more iterations taken, the more accurate becomes the approximation.Look at an interactive spreadsheet (Microsoft Excel) illustrating the bisection method.
(Windows users, right click and "Save target as..." to save the files on your computer.)