Question

What is the formula of $x^2+y^2+z^2$?

Answer 1

Formula for $x^2+y^2+z^2$.

Use the formula ${\left(a+b\right)}^{\mathbf{2}}\mathbf{=}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}$ to find${\left(x+y+z\right)}^2$:

Put $a=x+y,b=z$

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

Hence, the required formula is $\begin{array}{rcl}{x}^2+y^2+z^2& =& {\left(x+y+z\right)}^2-2xy-2yz-2zx\end{array}$