Question

What is the formula of $a^2+b^2+c^2$?

Answer 1

Find the formula for $a^2+b^2+c^2$.

Since, For any real number $x,y,$ and $z$,

$\begin{array}{rcl}{\left(x+y+z\right)}^{\mathbf{2}}& \mathbf{=}& {\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}{\mathit{z}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathit{y}\mathbf{+}\mathbf{2}\mathit{y}\mathit{z}\mathbf{+}\mathbf{2}\mathit{z}\mathit{x}\end{array}$

Substitute $x,y,$ and $z$, with $a,b,$ and $c$ respectively in the above algebraic identity:

$$\begin{gathered} {\left(a+b+c\right)}^2=a^2+b^2+c^2+2ab+2bc+2ca \\ \Rightarrow a^2+b^2+c^2={\left(a+b+c\right)}^2-2ab-2bc-2ca \\ \Rightarrow a^2+b^2+c^2={\left(a+b+c\right)}^2-2\left(ab+bc+ca\right) \end{gathered}$$

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