Question

What is the formula for ${\left(x-y-z\right)}^2$?

Answer 1

Determine the formula for ${\left(x-y-z\right)}^2$.

It is known that the formula for ${\left(a+b\right)}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}$.

Rewrite the expression for ${\left(x-y-z\right)}^{2}as{\left(x+\left(-y-z\right)\right)}^2$ and use the mentioned formula to simplify:

$\begin{array}{rcl}{\left(x+\left(-y-z\right)\right)}^2& =& x^2+{\left(-y-z\right)}^2+2x\left(-y-z\right)\\ & \Rightarrow & {\left(x+\left(-y-z\right)\right)}^2=x^2+{\left(-y\right)}^2+{\left(-z\right)}^2+2\left(-y\right)\left(-z\right)-2xy-2xz\\ & \Rightarrow & {\left(x+\left(-y-z\right)\right)}^2=x^2+y^2+z^2+2yz-2xy-2xz\end{array}$

Therefore the formula for${\left(x-y-z\right)}^2=\begin{array}{rcl}& & x\end{array}\begin{array}{rcl}& & \end{array}$.