# Square Roots

## 1   Solving Square Roots, Three Methods

### 1.1   An Approximation by Formula

We shall try to find the square root of 121, which we presume we do not know the answer to.

We need to find a number close to, and below, 121 that we know the square root of. In this case it will be 100, the square root being 10.

Let x = 100

The difference between 121 and 100 = 21

Let delta x = 21

Formula:

Sqrt( x + delta x ) approx = sqrt(x) + (delta x / (2 * sqrt(x))

Example: sqrt(100 + 21) approx = sqrt(100) + (21/(2 * sqrt(100)))
approx = 10 + 21/20 approx = 10 + 1 approx = 11

Regards to 'Dan' who posted the method in .

### 1.2   Longhand

This method is similar to long division, some practice will make the process a little easier.

Let's try to find the square root of 545.6 correct to three decimal places...

Starting at the decimal point and working left and right we partition the number into periods of two digits each.

Each partition will give a one digit portion of the root.

545.6 will result in this format:

```               05 45.60 00 00
The first period is 5. The largest square less than or equal

2
05 45 .60 00 00
4
Subtract 4 from 5 and bring down the next period.

2
05 45.60 00 00
4
1 45
We now double the extraction, which at this point is 2 and place it to the left of the subtraction.

We must now find the largest digit n so that 4n times n is as close as possible to 145.

In this case it is 43 times 3 (123).

2
05 45.60 00 00
4

1 45
4n x n          1 23
Place the value of n above the second period then subtract 123 from 145 and bring down the third period.

2   3
05 45.60 00 00
4
1 45
4n x n          1 23
22 60
Now we repeat the process, double the extraction (2 times 23 = 46), find the largest digit so that
46n times n is as  close as possible to 2260.

In this case it is 464 times 4.

2   3. 4
05 45.60 00 00
4
1 45
4n x n          1 23
22 60
46n x n                   18 56
I'll allow you to complete the process.....
```

### 1.3   Guess, Divide, Average

We pick, let's say, 1515 to find the square root of.

Now take a rough guess at the root, say 50.

Square = 1515

Guess = 50

Square/Guess = 30.3

Average = ( 50 + 30.3 )/2 = 40.15

This result becomes the new guess.

Square = 1515

Guess = 40.15

Square/Guess = 37.733 (3 dp)

Average = ( 40.15 + 37.733 )/2 = 38.942

Square = 1515

Guess = 38.904

Square/Average = 38.942

Average = ( 38.942 + 38.904 )/2 = 38.923

Square = 1515

Guess = 38.923

Square/Guess = 38.923

Your calculator will confirm this is correct to three decimal places.