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).
To simplify a complex fraction like this, I need to eliminate the imaginary part from the denominator. One way to do that is to multiply the fraction by the conjugate of the denominator, which is $2+3i$. This will give me a difference of squares in the denominator, and some complex terms in the numerator. So, I have $\dfrac{5+12i}{2-3i} \times \dfrac{2+3i}{2+3i} = \dfrac{(5+12i)(2+3i)}{(2-3i)(2+3i)}$. Expanding the numerator, I get $10+15i+24i+36i^2 = 10+39i-36 = -26+39i$. Expanding the denominator, I get $4-9i^2 = 4+9 = 13$. So, the fraction becomes $\dfrac{-26+39i}{13}$. To write this in the form $a+bi$, I need to divide both the real and the imaginary parts by 13. This gives me $\dfrac{-26}{13}+\dfrac{39}{13}i = -2+3i$. 