Question

If a+b+c = 9 and ab+bc+ca = 40, find a^2 + b^2 + c^2

Answer 1

`a^2+b^2+c^2=1`

Step-by-step explanation

By definition.

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

so

Let

`\begin{gathered} (a+b+c)=9 \\ ab+bc+ac=40 \\ \text{now, replace} \end{gathered}`

`\begin{gathered} (a+b+c)^2=a^2+b^2+c^2+2((ab)+(bc)+(ac)) \\ 9^2=a^2+b^2+c^2+2(40) \\ 81=a^2+b^2+c^2+80 \\ \text{subtract 80 in both sides} \\ 81-80=a^2+b^2+c^2+80-80 \\ 1=a^2+b^2+c^2 \end{gathered}`

hence

`a^2+b^2+c^2=1`

I hope this helps you