Help! (I got a different answer using another way)

Admins please help!
When I did Your Turn I used a way where:
ab+bc+ca = 3+6+8
2(a+b+c) = 17
a+b+c = 17/2 = 8.5
And I got this different answer.
I totally understand Professor Loh’s way of doing the question, but I don’t get how we could get different answers.
I am currently trying to check for any silly mistakes. 
@RZ923 Hi there! I think I see it. Going from the line
$$ ab + bc + ca = 3 + 6 + 8$$
to the line
$$ 2 \left( a + b + c \right) = 17,$$
there is the assumption that two \(a's\) are added together, two \(b's\) are added together, and two \(c's\) are added together. In actuality, though, \( ab \neq a + b, bc \neq b + c,\) and \( ca \neq c + a.\)
Yours is the solution you would use if the question looked instead like this:
$$\begin{aligned} a + b &= 3 \\ a + c &= 6 \\ b + c &= 8 \\ \end{aligned} $$Instead we should actually multiply the \(ab, bc,\) and \(ac\) together, like this:
$$ (ab)(bc)(ac) = 3 \times 6 \times 8 $$
which gives us
$$ a^2 b^2 c^2 = 144 $$
and we can see rather easily that \(abc = 12.\) 
Thanks!