The square of a sum of two expressions is equal to the square of the first, plus twice the product of the first and second, plus the square of the second.

 

The square of a sum of two expressions

 

 Important!

The square of a sum of two expressions

The square of the sum formula shall be used for finding squares of large numbers.

Derivation of the formula of the square of the sum. Method 1

 

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

Derivation of the formula of the square

 

Step 1

 

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

Derivation of the formula of the square

Step 2

 

Expand the brackets:

Derivation of the formula of the square

Step 3

 

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

Derivation of the formula of the square

Step 4

 

Collect like terms and multiply the elements:

Derivation of the formula of the square

Step 5

 

As a result, we can derive:

Derivation of the formula of the square of the sum. Method 1

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

Derivation of the formula of the square

 

Step 1

 

Consider:

Derivation of the formula of the square

Step 2

 

Let us write in the form:

Derivation of the formula of the square

Step 3

 

Apply this to the expression under consideration:

Derivation of the formula of the square

Step 4

 

Simplify:

Derivation of the formula of the square

Step 5

 

Factorize:

Derivation of the formula of the square

Step 6

 

Factorize the expression (a+b):

Derivation of the formula of the square

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

Hence, we have proved that:

Derivation of the formula of the square of the sum. Method 1

The formula of the square of the sum has been derived.