
Square of the differencePage navigation:
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} = a^{2}  2ab + b^{2} Derivation of the formula of square of the differenceThe proof of the formula is very simple. To prove the formula is sufficient to multiply the expression: (a  b)^{2} = (a  b)·(a  b) = a^{2}  ab  ba + b^{2} = a^{2}  2ab + b^{2}
Applying of square of the difference formula
Square of the difference formula convenient to use:
Examples of taskExample 1. Expand brackets (x  3)^{2}.
Solution: Apply the square of the difference formula. (x  3)^{2} = x^{2}  2·3·x + 3^{2} = x^{2}  6x + 9
Example 2. Expand brackets (2x  3y^{2})^{2}.
Solution: Apply the square of the difference formula. (2x  3y^{2})^{2} = (2x)^{2}  2·(2x)·(3y^{2}) + (3y^{2})^{2} = 4x^{2}  12xy^{2} + 9y^{4}
Example 3. Simplify the expression Solution: Apply the square of the difference formula in numerator. 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 69^{2}.
Solution: 69^{2} = (70  1)^{2} = 70^{2}  2·70·1 + 1^{2} = 4900  140 + 1 = 4761
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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