# Square of the difference

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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 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) = 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:

- 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}= x^{2}- 2·3·x + 3^{2}= x^{2}- 6x + 9Example 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 ^{2}- 6x + 1

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

^{2}- 6x + 1

^{2}

*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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