Three Basic Algebrae Formulae That All Math Students Should Learn

Every Algebra has seen the following three basic formulae at some time or the other -

1. \((a+b)^2 = a^2 + 2ab + b^2\)

2. \((a-b)^2 = a^2 - 2ab + b^2\)

3. \(a^2 - b^2 = (a+b)(a-b)\)

Proving these formulae

Formulae 1 and 2 are really the same, if you think of (a-b) as (a+(-b)), then you can use the first formula to still get the same output as in 2. However, before we remember these formulae, let's think where are these formulae coming from.

Let's consider the first formula first:

\((a+b)^2 = (a+b) *(a+b) \)

Multiply the two brackets on the right hand side

\((a+b)^2 = a*a + a*b+ b*a + b*b \)

\((a+b)^2 = a^2 + ab + ba + b^2\)

The middle two terms are the same and equal 2ab. Thus the formula is 

\((a+b)^2 = a^2 + 2ab + b^2\)

The Second Formula

The second formula can be seen as (a+(-b))

\((a-b)^2 = (a+(-b))^2 = a^2+2*a*-b + (-b)^2\)

\((-b)^2\) is the same as \(b^2\), while the middle term is -2ab. Thus

\((a-b)^2 = a^2 - 2ab + b^2\)

The Third Formula

The third formula can also be found by multiplying the right hand side of the formula

\((a^2 - b^2) = (a+b)*(a-b)\)

\((a^2-b^2) = a*a - a*b + b*a - b*b\)

The middle two terms cancel out, leaving you with

\((a^2-b^2) = (a^2 - b^2)\)

Uses of these formulae 

There can be several uses of these formulae. Some examples:

1. \((101)^2 = (100+1)^2 = 100^2 + 2*100*1 + 1^2 = 10000+200+1 = 10201\)

2. \(99^2 = (100-1)^2 = 100^2 - 2*100*1 + 1^2 = 10000 - 200 +1 = 9801\)

3. \( 101^2 - 99^2 = (101+99)*(101-99) = 200*2 = 400\)

I have found these formulae handy over the years. Hope you will too!