Find $a^2 + b^2+c^2$

Question

Given $a^2+2b = 7$

$b^2+4c = -7$

$c^2+6a = -14$

Find $a^2 + b^2 + c^2$

The answer was an Integer

I tried to solve it by making $a$ the subject of the equation and substituting in others but equations became too complex(got $a^8$!) and difficult to solve.

Answer 1

HINT:

Completing the square, $$(a+3)^2+(b+1)^2+(c+2)^2=7-7-14+9+1+4=0$$

Now if $a,b,c$ are real, what can we derive from here?