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

Given:

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

To do:

We have to find the value of $a + b + c$.

Solution:

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

Therefore,

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$

$= 16 + 2 (ab+bc+ca)$

$=16+2\times10$

$= 16 + 20$

$= 36$

$= 6^2$

$\Rightarrow (a+b+c)=\sqrt{6^2}$

$=\pm 6$

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