STEP BY STEP GUIDE ON HOW TO FACTORIZE USING THE DIFFERENCES OF TWO SQUARES METHOD + DIAGRAMS

Objectives: 

1. Be able to tell if you can use the difference of two square methods to solve the expression.

2. Be able to solve expressions using the method.

3. Be able to tell what is the difference of two squares.

What is the difference of two squares?

The difference of two squares is an expression that two square is being subtracted from each other and when factored it can be seen in the foil method format.

for example : a^2 - b^2 = (a + b) (a - b)

This method is based on the pattern and can be verified when the parentheses are expanded. 

Historically it is said that the Babylonians were the ones who used the difference of two squares to calculate multiplications.

This method is used in today's world to solve mathematical problems and in other jobs that require a degree in mathematics.

To identify if an expression can be calculated with the difference of two squares method  you would have to : 

1. Identify if the numbers are square numbers.

2. If you have identified that the numbers/co-efficient aren't square numbers then the difference of the two squares method cannot be used.

3. Also identify if the exponents are odd, if the exponents are odd you cannot use this method.

4. It must be a binomial expression.

Written example: 

    6x^3 - 25

In the above expression, there isn't a perfect square for 6.

The exponent is "3" and is odd.

These are some common mistakes that you can make:

1. Even though one number/co-efficient is not a perfect square and the other is perfectly square students often still use the method.

2. Often forgot to identify if the numbers and exponents are perfectly squared and odd.

3. Often use the method to solve the expression when it has an addition sign adding the two squares.

Ways to fix these common mistakes: 

1. If one number/co-efficient is not perfectly squared and the other is don't use the method.

2. Always make sure you identify if the numbers are perfectly squared and the exponents are odd.

3. Always pay close attention to the sign used between the two squares.

Written example: 

4^2 - 25 

In the above-written example, the sign used is negative, the numbers are perfectly squared and the exponent isn't odd.

Here are a few steps by step examples along with diagrams: 

Example 1:

100 - 9x^2 

Step 1: Identify if the numbers/coefficient are perfectly squared.

Step 2: Identify if the exponents are odd.

Step 3: Identify the function sign that is between the square numbers.

Step 4:  Begin the calculation.

Step 5:  Find the square root of 100.

Step 6: Find the square root of -9x.

Step 7: Find the square root of x^2 

Step 8: Re-write your answers in parentheses with a different function sign between each of them.

Diagram Example : 

Example 2: 

2x^4 r - 72 y^4 r

Step 1: Identify if the numbers/coefficient are perfectly squared.

Step 2: Identify if the exponents are odd.

Step 3: Identify the function sign that is between the square numbers.

Step 4:  Begin the calculation.

Step 5: Identify what is the Highest Common Factor in both terms.

Step 6: Identify the common exponent(s) in both terms.

Step 7:  Write the Highest Common Factor and the most common exponent outside the bracket. 

Step 8: Factor your answer with the two terms that were given in the question.

Step 9: Re-write your answers with the HCF and the most common exponent on the outside and the factored answer inside the brackets.

Written Example :

 2x^4 r - 72y^4 r 

2r (x^4 - 36y^4) 

2r ( x^2 - 6y^2 ) ( x^2 + 6y^2 ) 

As seen in the written example above expression method involves the method of HCF.

IF you are interested in learning how to use the HCF method you can always feel free to check out our page titled: 

"STEP BY STEP GUIDE ON HOW TO FACTORIZE USING THE HCF METHOD"