Square Root by Factoring Method

A square root is a result that multiplies itself to get the same number. The factors of the number under the root are taken and then arranged in a pair of two in the prime factorisation method. The square root of 16 is, for example, = =4. We obtain the square root of a perfect square at this point. However, when calculating the square root of an imperfect square, we usually use the long division method.

Quadratic equations of the type “x2 = b” can be solved using the square root method. Since the square root of an integer can be either negative or positive, this procedure can produce two results. If an equation can be written in this way, the square roots of x can be used to solve it.

Put the Equation into the Proper Form

To isolate x2, the second factor on the left side (-49) must be eliminated from the equation x² – 49 = 0. By adding 49 to both sides of the equation, this is easily accomplished. It’s critical to remember to make these modifications on both sides of the equal sign; otherwise, you’ll obtain an inaccurate result. For the square root approach, x² – 49 (+ 49) = 0 (+ 49) provides an equation in appropriate form: x² = 49.

Find the Roots

x² is composed of an element (x) that has been squared (x x) or multiplied by itself (x x). To put it another way, finding the square root entails determining the number (x or -x) that is the squared number’s root. In the equation x² = 49, √49 = +/- 7, giving the result x = +/- 7.

Isolate the Square

You might be given an equation in the form ax² = b to solve using this method. By multiplying both sides of the equation by the reciprocal of “a,” you may isolate x². 1/a is the reciprocal of “a,” and the sum of these terms equals 1. To get the reciprocal of a fraction, such as 3/4, simply turn the fraction upside down: 4/3

Long-form Factoring

You may not always be able to tell which elements of a square root are squares. You can consider finding out the square root by the factoring method by breaking it down into its separate factors. Here are the steps for long-form factor √225, for example:

1. Find the factors – Because the most evident factor of 225 is five, you’d begin with √225 = √(5 x 45). You may make things even easier by discovering the 45 factors: √(5 x 5 x 9). You can simplify, and your final long-form factor will be √(5 x 5 x 3 x 3).
2. Remove any duplicate factors – When the same integer appears twice as a factor, list it only once outside the square root symbol. Because we have two 5s and two 3s in this situation, the equation will be 5 x 3.
3. Complete the remainder of the equation – The final step is to complete the equation: 5 x 3 = 15 in this situation.

Prime Factorisation Method

To find the square root of natural numbers:

Step 1: Convert the provided number into a product of prime factors.

Step 2: Make a pair of prime numbers that are the same.

Step 3: Take one of each pair’s prime numbers and multiply them together.

Step 4: The square root of the supplied number is the product.

Let’s have a look at some examples.

Example 1. Prime factorisation is used to find the square root of 256.

Solution: Consider finding out the square Root by the factoring method. Transform 256 into prime factors.

√ 256  = √ 2×2×2×2×2×2×2×2

        = √ 2×2×2×2×2×2×2×2

     = 2×2×2×2 (Taking one prime number from each pair)

So, √ 256 is equal to 16.

Understanding the Square Root by Factoring Method

Using the prime factorisation approach, below are some examples of finding the square root.

1. Find the square root of 484 by the prime factorisation method.

Solution:

When we solve 484 as a prime product, we get

√484 is equal to 2 × 2 × 11 × 11 

√484 is equal to √(2 × 2 × 11 × 11) 

is equal to 2 × 11

Therefore, √484 = 22

2. Find the square root of 324.

Solution:

By prime factorisation, we get the square root of 324.

324 is equal to 2 × 2 × 3 × 3 × 3 × 3

√324 is equal to √(2 × 2 × 3 × 3 × 3 × 3)

is equal to 2 × 3 × 3

Therefore, √324 = 18

3. Find out the square root of 1764.

Solution:

By prime factorisation, we get the square root of 1764.

1764 is equal to 2 x 2 x 3 x 3 x 7 x 7.

√1764 is equal to √(2 x 2 x 3 x 3 x 7 x 7)

is equal to 2 x 3 x 7

Therefore, √1764 is equal to 42.

Conclusion

A number’s square root is a value that, when multiplied by itself, returns the original number. The square root is a way of squaring a number that is the inverse of squaring it. As a result, squares and square roots are two ideas that are connected. The square root is known as the radical, and the number beneath it is known as the radicand. The square root symbol is used to denote the square root of any natural integer.