### Instruction: If the sum of the squares of nonnegative real numbers $a,b,$ and $c$ is $13$, and $ab + bc + ca = 6$, then what is the sum of $a,b,$ and $c$?### Response: This problem involves the squares and products of three numbers, so I wonder if there is a way to relate them to some familiar identity or formula.
One thing that comes to mind is the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$.
If I apply this identity to the given numbers, I get $(a + b + c)^2 = 13 + 2 \cdot 6 = 25$.
Taking the square root of both sides, I get $a + b + c = \pm 5$.