# Method for [Mentally] Calculating Squares

## Proposal

Let *a* and *b* be real numbers for which *a*^{2} is readily known and *b*^{2} has yet to be calculated. Also let *h* be "step size", and be equal to *b - a*. Then:

b^{2} = a^{2} + 2(h)(a) + h^{2}

## Example

Find: 17.1^{2}

Nearby 17.1 is 20, and we know that 20^{2} is 400. 'h' then is 17.1-20, which is -2.9, and (-2.9)^{2} is 8.41, so now we can mentally solve for 17.1^{2}.

17.1^{2} = 400 + 2(-2.9)(20) + 8.41 = 400 - 116 + 8.41 = 292.41

## Proof of Method

b^{2} = a^{2} + 2(h)(a) + h^{2}

b^{2} = a^{2} + (2)(b - a)(a) + (b - a)^{2}

b^{2} = a^{2} + (2ab - 2a^{2}) + (b^{2} - 2ab + a^{2})

b^{2} = a^{2} + 2ab - 2a^{2} + b^{2} - 2ab + a^{2}

b^{2} = ~~(a~~^{2} - 2a^{2} + a^{2}) + ~~(2ab - 2ab)~~ + b^{2}

**b**^{2} = b^{2}

## About

I noticed this pattern in an effort to gain an edge in a *"Can you guess the square root of... "* game that I like to play with people in math classes. I'm sure that hundreds of people have noticed this relation before me, however I do not know who found it first. Please contact me if you know!