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

Square of a difference

For the square of a difference, we have: \[(\blue a-\green b)^2=\blue a^2-2\blue a \green b+\green b^2\] |
\[\begin{array}{rcl} (\blue{x}-\green{3})^2 &=& \blue{x}^2 - 2 \blue{x}\cdot \green{3} + \green{3}^2 \\ &=& x^2 - 6 x + 9 \end{array}\] |

#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}#

#\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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