## I. Introduction

Welcome to our guide on how to find square roots! In this article, we’ll cover several methods for calculating square roots, from classic formulas to using online calculators. Whether you’re a student, a math enthusiast, or just curious about math, you’ll find helpful tips and examples for each method. Let’s get started!

## II. The Mathematical Formula Method

The simplest method for finding the square root of a number is to use the mathematical formula for it. The formula for taking the square root of “x” is:

### √x = y

Where “y” is the square root of “x”. Using this formula is straightforward:

- Identify the number you want to find the square root of (x)
- Write down the formula √x = y
- Calculate the value of “y”

For example, the square root of 16 is:

√16 = y

√16 = 4 (since 4 squared is 16)

Using this formula is easy, but there are tips you can use to be more efficient:

- Simplify the square root as much as possible
- Rename the radicand (x) as a perfect square if possible

## III. The Guess-and-Check Method

The guess-and-check method may seem primitive, but it’s actually a helpful technique for finding square roots. The idea is that you guess a number that’s close to the square root of the number you’re looking for, and then adjust your guess until you’ve found the exact value.

For instance, to find the square root of 30, you might guess that it’s between 5 and 6. Then, you test each number:

- Is 5 too low? Test 6.
- Is 6 too high? Test 5.5.
- Is 5.5 too low? Test 5.75.
- Is 5.75 too low? Test 5.875.
- Is 5.875 too high? Test 5.8125.
- Continue until you find the exact square root.

We can see that this process requires some patience and accuracy, but it can be improved with these tips:

- Use the previous result as a starting point to reduce the number of guesses
- Record each guess and the result to avoid repeating calculations

## IV. The Repeated Subtraction Method

The repeated subtraction method is similar to the guess-and-check approach, but instead of testing each number, you subtract the odd numbers starting from 1 until you reach 0 or a negative number. The number of times you’ve subtracted (counted) the odd numbers is the square root of the original number.

For instance, to find the square root of 50, you would subtract odd numbers starting with 1, until you reach the end:

- 50 – 1 = 49
- 49 – 3 = 46
- 46 – 5 = 41
- 41 – 7 = 34
- 34 – 9 = 25
- 25 – 11 = 14
- 14 – 13 = 1

We can see that we’ve counted 7 odd numbers, so the square root of 50 is 7. This method can be improved with these tips:

- Write down the odd numbers and their results to avoid repetition
- Round up the number to make it a perfect square to simplify the calculations

## V. The Long Division Method

The long division method is a classic approach for finding square roots, and it’s still used today in some situations. The idea is to divide the number you’re looking for the square root of by its perfect square divisors (starting with the largest) until you reach a quotient that’s less than the divisor. The divisor and the remaining quotient make up the square root.

For instance, to find the square root of 72, we can use the divisors of 36:

72/36 = 2, remainder 0

Therefore, the square root of 72 is 6 since the quotient is less than 36.

This method can be useful when you need to find the square root of large numbers or when other methods are not available. Here are some tips to make it easier:

- Start with the largest divisor and work your way down to save time
- Round off the number you’re finding the square root of to the nearest perfect square to make the calculation simpler

## VI. The Calculus Method

The calculus method is the most advanced technique for finding square roots, and it’s not practical for most everyday scenarios. It involves using the derivative of a function to determine the square root of a number. We won’t go into the details of the formula here, but we’ll show you some examples:

The square root of 9 can be found using the derivative of the function f(x)=x²-9:

f(x) = x²-9

f'(x) = 2x

f'(3) = 6

Therefore, the square root of 9 is 3.

This method is not recommended unless you’re familiar with calculus and are dealing with complex equations.

## VII. The Online Calculator Method

If you’re not a math whiz and don’t want to spend time calculating square roots, you can always use an online calculator. There are many calculators available online that can quickly calculate square roots, including Google Calculator, Wolfram Alpha, and Symbolab. Simply type in the number you want to find the square root of, and hit enter. You’ll get the result instantly.

This method is convenient but may not be reliable if the internet connection is down or the website is not trustworthy. Here are some tips to avoid issues:

- Use reputable websites and apps
- Verify the result using other methods if possible

## VIII. Conclusion

Now that you’ve seen several methods for finding square roots, you may be wondering which one to choose in different situations. Here’s our advice:

- For small numbers, use the mathematical formula or guess-and-check method
- For larger numbers, use the repeated subtraction or long division method
- For complex equations, use calculus if you’re familiar with it
- If you’re in a hurry or don’t want to do calculations, use an online calculator

We hope this guide has been helpful and that you’ll try different methods to find square roots.