# Sunday Maths: Simplifying Squares

This is a simple method to make it easier to calculate squares in your head (or on paper). The key is the following equation:

x² = (x+y)(x-y) + y²

If we multiply the terms in parenthesis you can see how these are equal:

x² = (x+y)(x-y) + y² x² = x² +yx -yx -y² +y² x² = x²

We can use this to simplify the calculation of any x² by choosing a value of y such that (x-y)(x+y) becomes easy to calculate. For example:

99² = ? Using y = 1 99² = (99-1)(99+1) + 1² 99² = (98)(100) + 1 99² = 9800 + 1 99² = 9801

This technique works best when x is close to a power of 10, such as the above example. In more difficult cases it may be necessary to apply this technique recursively to calculate y² as well:

72² = ? Using y = 22 72² = (72-22)(72+22) + 22² 72² = (50)(94) + (22+2)(22-2) + 2² 72² = 9400 ÷ 2 + (24)(20) + 2² 72² = 4700 + 480 + 4 72² = 5184

Note also the use of 50=100÷2 to simplify the calculation of 50×94.