Answer:
a = 13 and b = 12-5i
Step-by-step explanation:
We need a common denominator
3+2i 5-i
---------- + ---------
3-2i 2+3i
(3+2i) (2+3i) (5-i)(3-2i)
------------------ + --------------------
(3-2i) (2+3i) (2+3i) (3-2i)
Foil the numerators
6 +4i+9i+6i^2 +15-3i-10i+2i^2
------------------ --------------------
(2+3i) (3-2i)
Combine like terms
21 +8i^2
------------------
(2+3i) (3-2i)
We know that i^2 = -1
21 +8(-1)
------------------
(2+3i) (3-2i)
21 -8
------------------
(2+3i) (3-2i)
13
------------------
(2+3i) (3-2i)
Foil the denominator
13
---------------
6 +9i -4i -6i^2
Combine like terms
13
----------------
6+ 5i -6(-1)
13
----------
12 +5i
We know have
13
-------------
12 + 5i
Multiply by the conjugate
13 ( 12-5i)
------------- * -------------
12 + 5i 12 -5i
13 (12-5i)
--------------
144 +25
13(12-5i)
-------------
169
12-5i
------------
13
The parentheses are (12-5i)/13
We need the reciprocal to make the equation become 1
which is 13/ 12-5i
a = 13 and b = 12-5i
