This method is much like long division except that each step the divisor is changed.

Remember the steps for long division.

1.-       Divide

2.-       Multiply

3.-      Subtract

Example 1  

Step A

 Separate the number 4225 into two-digit “groups” beginning at decimal point.

Step B

For the first trial divisor, take the square root of the largest perfect square root of the largest perfect square which is less than the first group ( √ 42 ).

In this case is 6 because 6² or 36 is the largest perfect square less than 42.

Write the 6 on top of 42 then  multiply 6*6 and subtract the answer from 42.

 6 is the first digit of the square root of  4225

Step C

 Bring down the next  two digits.

Step D

In this step we find the next quotient or the last digit of the square root of 4225.

First take the quotient (6) and  add it to the old divisor (6) to get the "partial" new divisor.

It is called "partial" divisor  because we still have to find the last digit to complete this divisor. This last digit is also the last digit of the  square root of 4225.  The parenthesis ( ) indicates the location of this digit.

                                            Old quotient + old divisor = 6 + 6 = 12 .

                                            12 is  the "partial" new divisor.

                                            Set down the 12 so as to leave a space for ( ).

Step E

The last digit ( ) has to be 9 or less.  

If you multiply 12( ) * ( ) the answer should be equal or less than the remainder 625.

There are many tricks to find ( ).

Estimate using the following values:  120 for 12( ) and 600 for 625

Now think!   How many 120's go into 600 ?     five (5)

Then  multiply 125 * 5 = 625 ,  this is good because the product is less or equal than 625.

Note

Lets assume that the product of 125 * 5 is a number more than the remainder 625,  then you have to try the numbers 4, 3, 2,or 1 respectively until the product of 12( ) * ( ) is less or equal than 625.

In other words

124*4

123*3

122*2

121*1

EXAMPLE 2

Step A

Separate the numbers into groups of two.

Add four zeros to get two decimals.

Step B

For the first divisor take the square root of the largest perfect square root which is

less than 47. 

Step C

Multiply, subtract and bring down the next two digits.

Step D

To get the next "partial" trial divisor follow the next two steps.

---- Add the previous quotient (6) to the last divisor (6) to get  12(  ).

----The (  ) in 12(  ) indicates the missing digit to be found.

Step E

To find the missing digit approximate how many 120's go into1141.

A better way, approximate how many 12's go into 114.

Step F

-----Multiply and subtract as in long division.

-----Then bring down the next two zeros. 

Step G

To find the next "partial" trial divisor add the previous divisor ( 128 ) to the previous

quotient ( 8 ) to get 136.

 Step H

To find the missing digit in parenthesis, approximate how many 1360's go onto 11700.

A better way, approximate how many 13's go into 117.

Step I

-----Multiply and subtract as in long division.

------Then bring down the next two zeros.

Step J

To find the next "partial" trial divisor add the last quotient (1368) to the last divisor (8).

Step K

To find the missing digit inside the parenthesis approximate how many 13760's

go into 75600.

A better way, approximate how many 13's go into 75.

Step L

-----Multiply and subtract as in long division.

-----There are no more digits to bring down.