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

[RZ923]

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.

[debbie]

@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 $abc = 12.$

[RZ923]

Thanks!