The square of a sum or a difference

Algebra: Notable Products

The square of a sum or a difference

Notable products are particular cases of the banana method, which are used so regularly that they take a special place.

Square of a sum

For the square of a sum we have: \[(\blue a+\green b)^2=\blue a^2+2\blue a \green b+\green b^2\]

Example

\[\begin{array}{rcl} (\blue{x}+\green{3})^2 &=& \blue{x}^2 + 2 \blue{x}\cdot \green{3} + \green{3}^2 \\ &=& x^2 + 6 x + 9 \end{array}\]

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

Square of a difference

For the square of a difference, we have: \[(\blue a-\green b)^2=\blue a^2-2\blue a \green b+\green b^2\]

Example

\[\begin{array}{rcl} (\blue{x}-\green{3})^2 &=& \blue{x}^2 - 2 \blue{x}\cdot \green{3} + \green{3}^2 \\ &=& x^2 - 6 x + 9 \end{array}\]

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.

Expand the brackets in \((-x-5)^2\) and simplify as much as possible.

#x^2+10x+25#

#\begin{array}{rclcl}(-x-5)^2&=&(-x)^2+2\cdot (-x)\cdot -5+(-5)^2\\&&\phantom{xxx}\blue{\text{sum formula for squares}}\\&=&x^2+10x+25\\&&\phantom{xxx}\blue{\text{reduced}}\end {array}#

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