Simplify $\dfrac{5+12i}{2-3i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary).
Simplify the following:
(12 i + 5)/(-3 i + 2)
Multiply numerator and denominator of (12 i + 5)/(-3 i + 2) by 2 + 3 i:
((12 i + 5) (3 i + 2))/((-3 i + 2) (3 i + 2))
(2 - 3 i) (2 + 3 i) = 2×2 + 2×3 i - 3 i×2 - 3 i×3 i = 4 + 6 i - 6 i + 9 = 13:
((12 i + 5) (3 i + 2))/13
(5 + 12 i) (2 + 3 i) = 5×2 + 5×3 i + 12 i×2 + 12 i×3 i = 10 + 15 i + 24 i - 36 = -26 + 39 i:
(39 i - 26)/13
Factor 13 out of 39 i - 26 giving 13 (3 i - 2):
(13 (3 i - 2))/13
(13 (3 i - 2))/13 = 13/13×(3 i - 2) = 3 i - 2:
=3i - 2 OR -2 + 3i
