Question

If $a^2+b^2+c^2=16$ and $ab+bc+ca=10$, find the value of $a+b+c$.

Answer 1

Given, $a^2+b^2+c^2=16$ and $ab+bc+ca=10$

Using identity,
$(a+b+c{)}^2=a^2+b^2+c^2+2ab+2bc+2ca$
$\Rightarrow (a+b+c{)}^2=16+2(ab+bc+ca)$
$\Rightarrow (a+b+c{)}^2=16+2(10)$
$\Rightarrow (a+b+c{)}^2=16+20$
$\Rightarrow (a+b+c{)}^2=36$
$\Rightarrow a+b+c=\pm \sqrt{36}$
$\therefore a+b+c=\pm 6$

Hence, the value of $a+b+c$ is $\pm 6$.