The derivation of the sum formula follows directly from the banana method:\[\begin{array}{rclcl}\left(\blue a+\green b\right)^2&=&\left(\blue a+\green b\right)\left(\blue a+\green b\right) &\phantom{x}&\blue{\text{from square to product}}\\&=& \blue a^2+{\blue a}{\green b}+{\green b}{\blue a}+\green b^2&\phantom{x}&\blue{\text{brackets expanded}}\\&=& \blue a^2+2\blue a \green b+\green b^2&\phantom{x}&\blue{\text{terms added}}\end{array}\]

Also this difference formula can be derived from the banana method directly:

\[\begin{array}{rclcl}\left(\blue a-\green b\right)^2&=&\left(\blue a-\green b\right)\left(\blue a-\green b\right) &\phantom{x}&\blue{\text{from square to product }}\\&=& \blue a^2-{\blue a}{\green b}-{\green b}{\blue a}+(-\green b)^2&\phantom{x}&\blue{\text{brackets expanded }}\\&=& \blue a^2-2\blue a \green b+\green b^2&\phantom{x}&\blue{\text{terms added and }(-1)^2=1}\end{array}\]

We can also derive the difference formula from the sum formula by replacing \(\green b\) by \(-\green b\). In the worked out solutions of the exercies we will do this regularly.