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Square of the difference

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

(a - b)2 = a2 - 2ab + b2


Derivation of the formula of square of the difference

The proof of the formula is very simple. To prove the formula is sufficient to multiply the expression:

(a - b)2 = (a - b)·(a - b) = a2 - ab - ba + b2 = a2 - 2ab + b2

Applying of square of the difference formula

Square of the difference formula convenient to use:
  • to disclose the brackets
  • to simplify expressions
  • to calculate the squares of large numbers without using a calculator or multiplication in column

Examples of task

Example 1.
Expand brackets (x - 3)2.

Solution: Apply the square of the difference formula.

(x - 3)2 = x2 - 2·3·x + 32 = x2 - 6x + 9
Example 2.
Expand brackets (2x - 3y2)2.

Solution: Apply the square of the difference formula.

(2x - 3y2)2 = (2x)2 - 2·(2x)·(3y2) + (3y2)2 = 4x2 - 12xy2 + 9y4
Example 3.
Simplify the expression 9x2 - 6x + 1(3x - 1).

Solution: Apply the square of the difference formula in numerator.

9x2 - 6x + 1(3x - 1) = (3x - 1)2(3x - 1) = 3x - 1
Note that using square of the difference formula is easily find the squares of large numbers without using a calculator or multiplication in a column.
Example 4.
Apply the square of the difference formula to calculate 692.

Solution:

692 = (70 - 1)2 = 702 - 2·70·1 + 12 = 4900 - 140 + 1 = 4761

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