# Proof 15: ‘Square of binomial’ and ‘a sum times a difference’

A ‘binomial’ is a word for two added terms, such as: $a+b$. The square of a binomial is naturally $(a+b)^2$.

$$(a+b)^2=(a+b)(a+b)=aa+ab+ba+bb=a^2+b^2+2ab\quad_\blacksquare$$

A small mnemonic rhyme would be: ‘The binomial square is the first term squared plus the second term squared plus the product doubled’. Note that if one of the numbers is negative, the $2ab$ term will automatically get a negative sign: $(a-b)^2=a^2+b^2-2ab$. So, the signs will take care of themselves.

‘A sum times a difference’ means: $(a+b)(a-b)$.

$$(a+b)(a-b)=aa+a(-b)+ba+b(-b)=a^2-\cancel{ab}+\cancel{ab}-b^2=a^2-b^2 \quad_\blacksquare$$

A mnemonic rhyme would be: ‘A sum times a difference is the first term squared minus the second term squared’.

## Proof 14: Motion equations

In motion along a path, $a$ is acceleration, $v_0$ and $v$ are initial and final (current) speed, and $t_0$ and $t$ are initial and final…

## Proof 18: Root rules of algebra

Requirements: Remember the sign rule: both ‘positive times positive gives positive’ and ‘negative times negative gives positive’. Only ‘negative times positive (or opposite) gives negative’.…

## Proof 17: Power rules of algebra

Definitions: The pattern looks as if reducing the exponent means removing an $a$ term: \vdots\\a^4=a\cdot a \cdot a \cdot a\\a^3=a\cdot a \cdot a\\ a^2=a\cdot a…

## Proof 1: Rectangle, square, box and cube area and volume

‘A room is $4\,\mathrm m$ long and $2\,\mathrm m$ wide’; a rectangle-shape. For each of the 4 metres there are 2 square metres across. That…

## Proof 31: Elastic-collision speed “conservation”

We call a collision elastic if the total kinetic energy is conserved (same before as after the collision) and there are no other energies involved.…