The square of the difference of two expressions is equal to the square of the first minus twice the product of the first and the second plus the square of the second.

 

The square of the difference of two expressions

 

Important!

 

Квадрат разности

Derivation of the Formula

 

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

Derivation of the formula of the square of the

 

Step 1

 

The square of the expression is the expression mutiplied by itself:

Derivation of the formula of the square of the

Step 2

 

Expand the brackets:

Derivation of the formula of the square of the

Step 3

 

As the product remains the same by reordering its factors, then:

Derivation of the formula of the square of the

Step 4

 

Collect like terms and multiply the elements:

Derivation of the formula of the square of the

Step 5

 

As a result, we can derive:

Derivation of the formula of the square of the

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

 

The square of the difference of two expressions  

 

Step 1

 

Consider:

Derivation of the formula of the square of the

Step 2

 

Let us write in the form:

Derivation of the formula of the square of the

Step 3

 

Apply this to the expression under consideration:

Derivation of the formula of the square of the

Step 4

 

Simplify:

Derivation of the formula of the square of the

Step 5

 

Factorize the common factors; and put -b outside the second bracket:

Derivation of the formula of the square of the

Step 6

 

Factorize the expression (a-b):

Derivation of the formula of the square of the

Step 7

 

As a result of transformations we have a product of the expression by itself, and it is the square of this expression:

Derivation of the formula of the square of the

Hence, we have proved that:

Derivation of the formula of the square of the

The Formula has been derived.