As the curve crosses the x axis, the value of the function f(x) changes sign, for example:
The values of f(x) in the table above confirms there are roots between [-2,-1.5] [-1,-0.5] and [0, 0.5].
I am going to concentrate on finding a more accurate value of the root between [-2,-1.5]. I will achieve this by reducing the size of the intervals to a width of 0.1. The graph and table below show that the root is between [-1.8,-1.9].
By reducing the width of the interval even further, we can continue to find a more accurate interval. The interval is reduced to 0.01 and the root is between [-1.85,-1.86].
The interval is reduced to 0.001, the root is between [-1.854,-1.855].
The interval is reduced to 0.0001, the root is between [-1.8546,-1.8547].
The interval is reduced to 0.00001, the root is between [-1.85463,-1.85464].
The interval is reduced to 0.000001, the root is between [-1.854633,-1.854638].
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].
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].