partial fraction - complex roots

Question

i really can't understand how to manage this P(s) in order to apply inverse laplace transform

$P(s) = \frac{2s-5}{1.5s^2-3s+4}$

i've tried this expansion:

$P(s) = \frac{A}{s-1-1.29i} +\frac{B}{s-1+1.29i} $

but i can't manage the complex number in the system, please show all passages

Answer 1

If the denominator has distinct roots $r_k$, and the numerator has lower degree than the denominator, then the partial fraction decomposition is of the form $ P(s) = \sum_k a_k/(s - r_k)$ where $a_k$ is the residue of $P(s)$ at $s=r_k$, which is $\lim_{s \to r_k} (s - r_k) P(s)$.

Answer 2

$$P(s) = \frac{2s-5}{1.5s^2-3s+4}$$ $$P(s) = \frac{4s-10}{3(s^2-2s+8/3)}$$ $$P(s) = \frac{4s-10}{3((s-1)^2+5/3)}$$ $$P(s) = \frac 4 3\frac{s-1}{(s-1)^2+5/3}-2\frac{1}{(s-1)^2+5/3}$$ $$P(s) = \frac 4 3\frac{s-1}{(s-1)^2+5/3}-2\sqrt {\frac 35}\frac{\sqrt {\frac 53}}{(s-1)^2+5/3}$$ It's easy now to take the inverse Laplace Transform.