Find $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{5}{2 - \sqrt{3}}}$, and write your answer in the form $\displaystyle \frac{A + B\sqrt{3}}{C}$, with the fraction in lowest terms and $A > 0$. What is $A+B+C$?
+128475 \( $\displaystyle{ \frac{2}{1 + 2\sqrt{3}} + \frac{5}{2 - \sqrt{3}}}$\)
We have
2 [ 2 - sqrt 3 ] + 5 [ 1 + 2sqrt 3 ]
_________________________ =
(1 + 2sqrt 3) ( 2 - sqrt 3)
4 - 2sqrt 3 + 5 + 10sqrt 3
______________________ =
2 + 4sqrt 3 - sqrt 3 - 2*3
9 + 8sqrt 3
_________ =
-4 + 3sqrt 3
(9 + 8sqrt 3) ( -4 -3sqrt 3)
______________________ =
(-4 + 3sqrt 3) ( -4 - 3sqrt 3)
-36 - 32sqrt 3 - 27sqrt 3 - 24*3
__________________________ =
16 - 9*3
-108 - 59sqrt 3
___________ =
-11
108 + 59sqrt 3
___________
11
A + B + C =
108 + 59 + 11 =
178
