The difference of the squares of two expressions is equal to the product of the difference of the expressions and their sum.

 

The difference of the squares of two expressions

 

Important! the difference of the square is not the square of the difference:

 

The difference of the squares of two expressions

Or:

The product of the sum of two expressions and their difference is equal to the difference of the squares of these expressions.

The difference of the squares of two expressions

Derivation of the formula of the difference of the squares

 

Let us prove the formula from left to right, i.e. prove that:

Derivation of the formula of the difference

 

Step 1

 

To prove this formula, we will use an artificial method: add and subtract the same expression ab. As a result, nothing will change as it yields zero:

Derivation of the formula of the difference

Step 2

 

Simplify:

Derivation of the formula of the difference

Step 3

 

Simplify:

Derivation of the formula of the difference

Step 4

 

Factorize:

Derivation of the formula of the difference

Step 5

 

Factorize the expression (a-b):

Derivation of the formula of the difference 

Hence, we have proved that:

Derivation of the formula of the difference

Step 6

 

ТNow we will prove the reverse, i.e. we will prove that:

 Derivation of the formula of the difference

Expand the brackets:

Derivation of the formula of the difference

The formula has been derived.