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Difference of squares

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

a2 - b2 = (a + b)·(a - b)


Derivation of the formula of difference of squares

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

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

Applying of difference of squares formula

Difference of squares formula convenient to use:
  • to disclose the brackets
  • to simplify expressions

Examples of task

Example 1.
Expand brackets (x - 3)·(x + 3).

Solution: Apply the difference of squares the formula.

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

Solution: Apply the difference of squares the formula.

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

Solution: Apply the difference of squares formula in numerator.

9x2 - 1(3x - 1) = (3x - 1)·(3x + 1)(3x - 1) = 3x + 1

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