Rekenen met variabelen *: Notable Products
The square of a sum or a difference
Notable products are particular cases of the multiplication formula, which are used so regularly that they take a special place.
Sum formula for squares For the square of a sum we have the following sum formula \[(a+b)^2=a^2+2ab+b^2\]
The derivation of the sum formula is as follows: \[ \begin{array}{rclcl}\left(a+b\right)^2&=&\left(a+b\right)\left(a+b\right) &\phantom{x}&\color{blue}{\text{from square to poduct}}\\&=& a^2+{a}{b}+{b}{a}+b^2&\phantom{x}&\color{blue}{\text{removing brackets}}\\&=& a^2+2a b+b^2&\phantom{x}&\color{blue}{\text{take terms together}}\end {array}\]
Difference formula for squares For the square of a difference applies the difference formula \[(a-b)^2=a^2-2ab+b^2\]
The difference formula \[(a-b)^2=a^2-2ab+b^2\] follows from the sum formula by replacing \(b\) by \(-b\) (and we will do the same in the solutions of the excercises), but it is good to know both formulas by heart.
The formulas are quite useful, even if the two-terms are more complicated.
For, \[ \begin{array}{rclcl}(-2u+9v)^2&=&(-2u)^2+2\cdot (-2u)\cdot (9v)+(9v)^2&&\color{blue}{\text{sum formula for squares}}\\&=&4u^2-36u\cdot v+81v^2&&\color{blue}{\text{simplified}}\end {array}\]
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