Difference of squares
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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 Solution: Apply the difference of squares formula in numerator.
Factoring: Some special cases
Square of the sum
Square of the difference
Difference of squares
Cube of sum
Cube of difference
Sum of cubes
Difference of cubes
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