A root of equation $$ax^2 + bx + c = 0$$ (where $$a, b$$ and $$c$$ are rational numbers) is $$5 + 3\surd3$$. What is the value of $$\frac{(a^2 + b^2 + c^2)}{(a + b + c)}$$?

$$ax^2 + bx + c = 0$$ has $$5 + 3\surd3$$ and so the other root is $$5 - 3\surd3$$ since these roots occur in pairs
sum of the roots= $$5 + 3\surd3$$+$$5 - 3\surd3$$ 
=10
Product of the roots=($$5 + 3\surd3$$)($$5 - 3\surd3$$)
=-2
Sum of the roots=-b/a
Product of the roots=c/a
-b/a=10
b=-10a  c/a=-2
c=-2a
$$\frac{(a^2 + b^2 + c^2)}{(a + b + c)}$$

=$$\frac{(a^2 + (-10a)^2 + (-2a)^2)}{(a -10a -2a)}$$

=$$\frac{(105(a)^2)}{( -11a)}$$

=$$\frac{(-105(a))}{( 11)}$$