Question

${\text{a}}^2{\text{+b}}^2{\text{+c}}^2$ is equal to ?

Answer 1

Let us multiply $\mathrm{a}+\mathrm{b}+\mathrm{c}$ by itself, we get;

$({\text{a+b+c)(a+b+c)=a}}^2{\text{+ab+ac+ba+b}}^2{\text{+bc+ca+cb+c}}^2$ (after multiplying the brackets)

We know that $\text{ab=ba,ac=ca,bc=cb}$ and put these values we get

$$\begin{gathered} {\text{(a+b+c)(a+b+c)=a}}^2{\text{+ab+ac+ab+b}}^2{\text{+bc+ac+bc+c}}^2 \\ {\text{=a}}^2{\text{+b}}^2{\text{+c}}^2\text{+2ab+2bc+2ac} \end{gathered}$$

Now we get $({\text{a+b+c)}}^2{\text{=a}}^2{\text{+b}}^2{\text{+c}}^2\text{+2ab+2bc+2ca}$

Hence ${\text{a}}^2{\text{+b}}^2{\text{+c}}^2$is equal to ${\text{(a+b+c)}}^2\text{-2ab-2bc-2ca}$