Trial and Improvement

improvement method, We make an estimate of
what the unknown quantity might be and test the
estimate out.

Solve         x2 = 20       to 2 decimal places
42 = 20        and          52 = 25
So x is between 4 and 5,    try 4.5  
( put your calculations in a table)
x           x2  
4.5       20.25        Too big         try 4.4
4.4       19.36        Too small     try 4.45
4.45     19.80        Too small     try 4.46
4.46     19.89        Too small     try  4.47
4.47     19.98        Too small     try 4.48
4.48     20.07        Too big


 4.47 is the nearest so x = 4.47 to 2 decimal places


Solve x2 +1 = 11 to 2 decimal places
32 + 1 = 10 and 42 + 1 = 17
So x is between 3 and 4, try 3.5

x          x2 + 1  
3.5       13.35      Too big         try  3.4
3.4       12.56      Too big         try 3.3
3.3       11.89      Too big         try 3.2
3.2       11.24      Too big         try 3.1
3.1       10.61      Too small     try 3.15
3.15     10.92      Too small     try 3.16
3.16     10.99      Too small     try 3.17
3.17     11.05      Too big
  

3.16 is the nearest so x = 3.16 to 2 decimal; places