x2=14. Take the square root of both sides: x=±√14 (there are two square roots).
So x=3.7417 or -3.7417 approx. You can also find the square root using a type of long division method:
First find the nearest integer perfect square to 14, which is 9, the square root of which is 3. Note how the number 14 is presented with the decimal point and zeroes in pairs.
Subtract 9 from 14 to get 5 and bring down a pair of zeroes. Write 3 in the answer box at the top, corresponding to its square which we just subtracted. The decimal point comes next to align with the decimal point after 14. Now we can ignore the decimal point in further calculations.
So we now have 500. We double the answer so far, that is, we double 3=6 and multiply by 10 to give us 60. We need to guess what number between 0 and 9 we have to add to 60 so that the product of the number with the sum of 60 and the number are as close as possible to 500 (but not greater than 500). It turns out that the number is 7 so 67×7=469 which is as close as we can get to 500. Subtract from 500 to get 31 and. bring down the next pair of zeroes: 3100. Double the answer so far to get 2×37=74. Multiply by 10: 740 and guess at the number to be added to it (as before). This time 744×4=2976 to be subtracted from 3100=124. Tack on the zeroes: 12400 and continue doubling and multiplying by 10 as before. The process goes on indefinitely until you have the required decimal accuracy.
3 . 7 4 1
3 ) 14 . 00 | 00 | 00 |...
-9
67 ) 5 00
-4 69
744 ) 31 00
-29 76
7481 ) 1 24 00
- 74 81
49 19 00 ...
There are other ways. These days it's best to use a calculator!
The method can be used to find the square root of any positive number. If the number happens to be a perfect square, the "remainder" will eventually become zero, so completing the operation.
