Question

What is the formula for ${\left(a-b-c\right)}^2$?

Answer 1

Determine the formula of ${\left(a-b-c\right)}^2$.

Rewrite the expression ${\left(a-b-c\right)}^2$:

${\left(a-b-c\right)}^2=\left(a-b-c\right)\left(a-b-c\right)$

Apply the distributive property, $a\left(b-c\right)=ab-ac$:

$\begin{array}{rcl}{\left(a-b-c\right)}^2& =& a\left(a-b-c\right)-b\left(a-b-c\right)-c\left(a-b-c\right)\\ & \Rightarrow & {\left(a-b-c\right)}^2=a^2-ab-ac-ab+b^2+bc-ac+bc+c^2\\ & \Rightarrow & {\left(a-b-c\right)}^2=a^2+b^2+c^2-2ab-2ac+2bc\end{array}$

Therefore, the formula is $\begin{array}{rcl}{\left(a-b-c\right)}^2& =& a^2+b^2+c^2-2ab-2ac+2bc\end{array}$.